ÍœbF « qOK×² «Ë W¹œbF « ‚dDmathcsbooks.com/assets/files/الطرق العددية والتحليل العددي... · ÍœbF « qOK×² «Ë W¹œbF « ‚dD « bO «

ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�«


bO��« bLŠ√ dJÐ uÐ√ —u²�b�«»uÝU(« ÂuKŽ r��

X¹uJ�« WF�Uł

5

”d‡‡‡NH�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W�b?I*«

‰Ë_« qBH�«W¹œbF�« �UÐU�(«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ øÍœbF�« qOK×?²�« u¼ U�

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ∫¡U?D?š_« Ÿ«u?½√

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ V¹d?I??²�« ¡UDš√

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ŸUD²??�ô« ¡UDš√

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W?Oz«b??²Ðô« ¡UDš_«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ¡UDš_« b�«uð Ë√ —UA²½«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ÍœbF�« ¡UG�ù«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ —«dI?²Ýô«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ‰Ë_« qB?H�« �UM¹d9

w½U¦�« qBH�«�ôœUF*« qŠ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ �ôœUF?*« —Ëcł

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ Wł—b�« WOŽUÐd�«Ë WO³OFJ²�« �ôœUF*«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ∫d½—u¼ WI¹dÞ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ œËbŠ …dO¦� rOOIð

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WO³O�d²�« WL�I�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ œËbŠ …dO¦� WI?²A� rOOIð

11

21

23

23

28

29

30

33

36

39

51

52

54

55

56

58

6

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W¹d¹dJ?²�« ‚dD�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ N-R Êu?��«— ≠ sðu?O½ W?I¹dÞ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ N-R WI¹d?Þ »—UIð

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ N-R WI¹dÞ w� QD)« d¹bIð

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ nOBM?²�« WI¹dÞ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ nOBM²�« W?I¹dÞ »—UIð

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ©WFÐU²²*« �U³¹dI²�«® W²ÐU¦�« WDIM�« WI¹dÞ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W²ÐU¦�« WDI?M�« WI¹dÞ »—UIð

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W²ÐU¦�« WDIM�« W?I¹dÞ w� QD)« d¹bIð

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ sJð≈ W?OKLŽ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ s�MHO²Ý WI¹dÞ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ∫WOD)« d?Ož W?O½ü« �ôœUF*«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ sðuO?½ WI¹dÞ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W?²ÐU¦�« WDIM�« W?I¹dÞ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ w½U¦�« qB?H�« �UM¹d9

Y�U¦�« qBH�«

WOD)« �ôœUF*« rE½

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W¹d¹dJ²�« ‚dD�«Ë …dýU³*« ‚dD�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ∫…dýU³*« ‚dD�« ∫ôË√

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ”ËU' ·c?(« WI¹dÞ ©√®

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ wze'« “UJð—ôUÐ ·c?×K� ”ËUł WI¹dÞ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ·c×K� Ê«œ—uł ≠ ”ËUł WI¹dÞ ©»®

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ∫·c?(« WI?¹dÞ vKŽ �UI?O?³Dð

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ �«œb?;« »U??�?Š ©√®

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W?�u?H?B?� ”uJF?� œU?−¹≈ ©»®

60

61

68

73

76

80

82

84

88

90

95

100

100

103

106

123

126

126

133

134

136

136

137

7

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W�u?HB� ‰U?LŽ≈ Ë√ qOK% ©Ã®

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ”ËU' ·c(« W?O�“—«uš Â«b�²ÝUÐ W?�uHB* LU qOK×²�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ wJ��u?AðË �Ëd�Ë q²?O�Ëœ ‚dÞ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ©WO¦K¦*« UNK�«uŽ v�≈® �U�uHBLK� dýU³*« qOK×²�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ b¹b×²�UÐ W³łu*« �U�uHB*« qOK×²� wJ��uAð WO�“—«uš

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ wJ��uA?ð WO�“—«u?š Â«b�²?ÝUÐ wD)« ÂUEM�« qŠ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ©�Ëd� WI¹dÞ® W¹dDI�« WOŁö¦�« �U�uHB*« qOK%

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W¹dDI�« wŁö?¦�« wD)« ÂUEM�« qŠ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ◊d?A�« WKOKF�« rEM�«Ë ◊d?A�« WM�?(« rEM�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ◊dA�« œb?Ž n¹dFð

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ Íd¹dJ²�« 5�×²�« Ë√ W¹d¹dJ²�« WOHB²�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ∫W¹d¹dJ²�« ‚dD�« ∫UO½UŁ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ©WFÐU?²²*« �U?³¹dI?²�«® wÐu�U?ł WI¹dÞ ©√®

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ‰b¹“ ≠ ”ËU?ł W?I¹dÞ ©»®

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ∫W¹d¹dJ²?�« ‚dD�« »—U?Ið

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W�uHB?* wHOD�« dDI�« nB½ n¹dFð

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ »—UI²K� W?OÝUÝ_« W¹dEM�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ Y�U¦�« qBH�« �UM¹d9

lÐ«d�« qBH�«

‰ULJ²Ýô« »U�Š

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WO�ULJ²?Ýô« !«dłô W¹œËbŠ

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ w�JF�« ‰U??LJ²?Ýô«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WO�U?LJ²Ýô« !«dłô WG?O� W�œ

W¹ËU??�??²*« ◊U??IM�« bMŽ ‰U??LJ?²?Ýô«® ‚Ëd??H�« Â«b??�??²??ÝUÐ ‰U??LJ²??Ýô« mO??�

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ©œU?????FÐ_«

141

145

147

149

150

152

154

156

159

166

169

172

173

175

178

180

186

192

215

219

221

224

8

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W¹e�d*«Ë WOHK)«Ë WO�U�_« ‚ËdH�« n¹dFðÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ‚ËdH�« �ö�UF� UNII% w²�« �U�öF�« iFÐÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ‚ËdH�« Â«b�²ÝUÐ ‰ULJ²Ýô«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WO�U�_« ‚ËdHK?� sðuO½ WGO�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WOHK)« ‚ËdHK� sðuO½ WGO�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ �dH¹≈ WGO�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ v�Ë_« ”ËU?ł W?GO?�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WO½U¦�« ”ËUł WGO�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ QD)« —U?A²½«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ w�JF�« ‰U??LJ²?Ýô«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ �U¹œËb(UÐ w¾¹e−²�« V¹dI²�«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WO³OFJ²�« W×¹dA�UÐ ‰ULJ²Ýô«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W²]³¦*« W×¹dA�«Ë WOFO³D�« W×¹dA�«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WOFO³D�« WO³OFJ²�« W×¹dA�« WO�“—«ušÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W²]³¦*« WO³OFJ²�« W×¹dA�« WO�“—«ušÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ lÐ«d�« qB?H�« �UM¹d9

f�U)« qBH�«

ÍœbF�« q{UH²�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ �U½UO³�« ‰Ëbł ◊UI½ bMŽ �UI²ALK� W¹œbŽ mO�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ �U½UO³�« ‰Ëb?−Ð X�O� ◊UI½ bMŽ �UI²A?LK� W¹œbŽ mO�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ �U½UO³�« ‰Ëbł ◊UI½ bMŽ �UI?²ALK� Èdš√ W¹œbŽ mO�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ÍœbF�« q{UH²�« mO� W�œ vKŽ ¡UDš_« dOŁQðÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ f�U)« qB?H�« �UM¹d9

225

228

234

234

235

235

235

236

240

241

243

245

246

252

255

257

267

274

277

279

282

9

”œU��« qBH�«

ÍœbF�« q�UJ²�«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ÍœbF�« q�UJ²�« v�≈ W?łU(«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ Íœb?F�« q�UJ²�« …dJ�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WDO�?³�« ·d×M*« t³ý …b?ŽU�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WDO�³�« Êu�³LÝ …bŽU�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ q�UJ²?K� Èd?š√ mO??�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ QD)« �«d?¹b??IðÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WOK�UJ²�« WDÝu²*« W?LOI�« W¹dE½ ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ WOK{U?H²�« WDÝu²*« WL?OI�« W¹dE½ ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ∫W³?�d*« q�UJ²�« mO�ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W³�d*« ·d×M*« t?³ý WGO� ∫ôË√ ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W³�d*« Êu?�³LÝ WG?O� ∫UO½UŁ ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ©W¹UNM�« s� q]łR*« »«d²�ô«® ÊuÝœ—UA²¹— WI¹dDÐ ¡UHO²Ýô«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ …œdHM*« �ö�UJ²�«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ”œU?��« qB?H�« �UM¹d9ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ o‡‡‡‡Šö*«

ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W�UŽ �«—U³²š« Ã–U/ ∫(1) r�— o×K*«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ ‰uBH�« �UM¹d9 WÐuł√ ∫(2) r�— o×K*«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W�UF�« �«—U³?²šô« Ã–U/ WÐuł√ ∫(3) r�— o×K*«ÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆÆ W¹eOK$ù«Ë WOÐd?F�« �U×KDB*« qO�œ ∫(4) r�— o×K*«

291

291

293

293

296

300

300

303

305

305

308

310

312

315

323

325

339

423

433

11

W‡‡�bI*«

vKŽË bL×?� UMO³½ 5KÝd*« -U?š vKŽ Âö��«Ë …öB�«Ë ¨5*UF?�« »— tK� bL(«Æ5FLł√ t³×�Ë t�¬

tK¼√Ë ÂöÝù« —UFý ÆÆwÐdF�« ÊU�K�«

t?ÐU²?� U?N?Ð ‰e½√Ë ¨d?O?³F?²?�«Ë W?½UÐù« W?�U?�?Ð WO?Ðd?F�« W?G?K?�« v�U?F?ð tK?�« e?]O?Ó� bI?�v?K?Ž Æ5�_« ÕËd?�« t?Ð ‰e?½ Æ5*U?F?�« »— q?¹e?M²?� t?½≈Ë˚ ¨5?³� w?Ðd?Ž ÊU?�K?Ð -U?)«Âö?Ýù« —UF?ý wÐd?F?�« ÊU�?K�U?� (*)˝5?³� w?Ðd?Ž ÊU�?K?Ð Æs¹—c?M*« s?� Êu?J²?� p?³K?�UN?²GK?Ð e²F?ð …e¹eF?�« W�_«Ë ÆÊËeO?L²¹ UN?Ð w²�« 3_« d?zUFý r?EŽ√ s� W?GK�«Ë ¨t?K¼√ËÍœUB?²�ô«Ë ÍdJ�?F�« UN�ö?I²Ý« vK?Ž ’d% UL� Íu?GK�« UN?�öI²Ý« v?KŽ ’d%ËÆUN?O�≈ WÐu�?M� w¼Ë UN?MŽ WO?³Mł√ `³?Bð v²Š U?N²G?� w� ◊dHð W?KO�c�« W?�_«Ë Æ¡«uÝU?M?¼ s?�Ë Æ—U?Ðœ≈Ë »U?¼– w?� Ád?�√ ÊU?� ô≈ X?D?×?½« U?�Ë ¨‰– ô≈ V?F?ý W?G?� X?�– U?�Ër¼dF?A¹Ë ÜNÐ r?N³�d¹Ë ¨…d?ÓLF²�?*« W�_« vKŽ ÎU?{d� t²G?� dLF²?�*« w³M?ł_« ÷dHÓ¹Î«u×� q²I�UÐ rNO{U� vKŽ rJ×¹Ë Î«bÐR� ÎUM−Ý t²G� w� rN²G� f³×¹Ë ÜNO� t²LEŽ

Æl³Óð Ád�_ U¼bFÐ s� r¼d�Q� ÜNFMB¹ w²�« ‰öž_« w� rNK³I²�� bOI¹Ë ÎU½UO�½Ë

WO�öÝù« W�_« WOB�ýË ÆÆWOÐdF�«

ë¹eF?�« UNÐU²?� WGKÐ Ócš_« U?¼œUO²Ž« b?MŽ ÎW]OK?ł Ëb³ð WO�ö?Ýù« W�_« WOB?�A�w?� W�U?N�« �ôU?:«Ë W?�UI?¦�«Ë Âu?KF?�« W?G� U?NK?F?łË ¨s¹œU?O*« v?²ý w?� U?N�«b?�²?Ý«Ës� U¼d?Ož UNÐ X?�b³²Ý«Ë ¨WG?K�« pKð W?�_« XKL¼√ ÊS?� ÆU¼UO½œË UN?M¹œ —u�√Ë UN?ðUOŠ

Æ195-192 ∫ ¡«dFA�« …—uÝ (*)

12

w²�« 3_« ÊUO� w� WOB�A�« pK²� ÊUÐË–Ë ÜN²OB�A� WFOC� p�c� ¨WO³Mł√ �UG�Ævðu*« w� U¼]bŽË ÜN� WOB�ý ô W�√ X×³�√ b� UN½√ c¾MOŠ rKŽ«Ë ÜNðUG� cšQð

‰Ë√ VFA�« ‰u?×²¹ sK� ¨s¹d?LF²?�LK� ‰Ë_« ·bN�« w?¼ W�_« WG� X½U?� Âdł ô¢«–≈ u¼Ë ¨t�U?�¬Ë t?HÞ«u?ŽË Á—UJ�√ s� ‰u?×?²�« QAM?� ÊuJ¹ –≈ ¨t?²G� s?� ô≈ ‰u×?²¹ U?�¨dJH�«Ë W?HÞU?FK� ÏV�Ó½ W?GK�U?� fOK� ¨t?O?{U� V?�½ s� lDI½« t?²G?� V�½ s� lDI½«Q??A½Ë ¨W??G� vKŽ TýU½ r?NM� Q??AM� rN??²M��√ X?HK²??š« u� b??Š«u�« »_« ¡UMÐ√ Ê≈ v²??Š Æ(*) ¢¡UÐ¬ WŁöŁ ¡UMÐQ� WHÞUF�« w� «u½UJ� ¨W¦�UŁ WG� vKŽ Y�U¦�«Ë ¨Èdš√ vKŽ w½U¦�«

w� W?GK�« dŁQÐ q�U?� wŽË vKŽ ‰Ë_« U?¼—b� w� t?K�« rN?LŠ— W?�_« ¡U?LKŽ ÊU?�ËvKŽ «b¹b?ý ÎU�dŠ «u?�dŠ p�c�Ë ¨rK�*« W?OB?�ý ¡UMÐ w� U¼dDšË ¨W?�_« s¹uJðu¼ Íc�« ‰b³²Ý«Ë Ü¼dOž v�≈ UNMŽ œUŠ s� vKŽ dOJM�« «ËœbýË ¨WM��«Ë Ê¬dI�« WG�w� v?²??Š »d???F�« dzU???F??ý d???O???O??G?ð Êu¼dJ¹ n?K��« ‰«“ U???�Ë Æd??O???š u¼ Íc�U?Ð v½œ√oDM�« “u?−¹ s��_« dzU?Ý Ê√ l� ÆÆW?łU?( ô≈ ¢WO?ÐdF�« d?O?GÐ rKJ²�«¢ u¼Ë ¨�ö?�U?F*«

ÆÂöÝù« kH(Ë ¨WłU(« dOG� U¼u¼d�Ë ¨WłU×K� U¼už]uÝ sJ�Ë Ü‡NÐU×�_ UNÐdO?š WOÐdF�« W?�_« qFłË ¨wÐd?F�« tO³½ tÐ YFÐË ¨wÐd?F�« ÊU�K�UÐ tÐU?²� ‰e½√ tK�«ÂöJ�« vKŽ Âb?Ið s?0 nOJ� ¨Âö?Ýù« kH?Š ÂU9 s� r¼—U?F?ý kH?Š —U?B?� ¨3_«øtMŽ ‰UI²½ô« nKJ¹Ë ¨t½u½U� sŽ tłd�¹Ë ¨t�b³¹Ë ÁdOG¹ ≠ t�uEM�Ë ÁœdH� ≠ wÐdF�«

w�öÝù« `²H�« œöÐ w� WOÐdF�« —UA²½«

rN½¬d?� ÊU?��Ë rN½U?�� Êu�d?²¹ Ê«bK³?�« s� «bKÐ Êu×?²?H¹ U?�bMŽ »d?F�« sJ¹ r�ËqÐ Ü?N½u?H�Q¹ s¹c�« Õu?²H*« d?B*« q¼√ vKŽ W?OÐd?F�« VKGð U/≈Ë ¨b?Š√ ÊU?�� qł√ s�vAH²ð Ê√ ÁdJ�« b?ý√ ÊuLK�*« ÁdJ¹Ë ÜNÐ Àb×²�«Ë U?N²Ý«—œË UNLKFð w� Êu?IÐU�²¹Ë

ÆtK¼√Ë »U²J�« WG� sŽ ÊUðbOF³�« W½UÞd�«Ë WL−F�« rNO�

ö?� ¨t?³?ŠU� l?� qłdK� …œU?Ž p�– d?OB?¹ v²?Š WOÐd?F�« d?O?GÐ »UD)« œU?O?²?Ž«ËU* ¨Êu?�b??I?²*« Êu?LK�*« ÊU??� «c?N�Ë ÆrłU?Ž_UÐ t??³?A?²�« s� t½S??� ÁËdJ� «c¼ Ê√ V¹—ULNK¼√ WG�Ë ¨ÊUÝ«dšË ‚«dF�« ÷—√Ë ¨W]O�Ë— ULNK¼√ WG�Ë ¨dB�Ë ÂUA�« ÷—√ «uMJÝ

32/3 wF�«d�« ‚œU� vHDB* rKI�« wŠË (*)


13

¨W?OÐd?F�« W?GK�« œö?³�« Ác¼ q?¼√ «Ëœu?Ž ∫ W¹dÐdÐ U?NK¼√ W?G�Ë ¨»d?G*« ÷—√Ë ¨W?O?Ý—U?�Ær¼d�U�Ë rNLK�� ∫ —UB�_« Ác¼ q¼√ vKŽ X³Kž v²Š

f�b½_« œöÐ w� W?�U?I?¦?�«Ë rKF�« W?G� w¼ X½U?� W?OÐd?F�« W?G?K�« Ê√ p�c?� ÂuKF?�ËÆ…d¼«e�« WO*UF�« WO�öÝù« …—UC(« ÂU¹√

WOÐdF�« WGK�« …œUOÝ

Ü?N?Ð ‰e½ -U?)« »U?²?J�« Ê√ v�≈ tK� r�U??F�« �U?GK� W??OÐd?F�« W??GK�« …œU?O??Ý lłdðË˝ÊuE�U( t� U½≈Ë d?�c�« UM�e½ s×½ U½≈˚ ∫ v�UFð ‰U?I� ¨tEH×Ð q?łË eŽ tK�« qHJðËU0 v×?BH�« W?OÐd?F�« fOI½ ô UMKF?−¹ Íc�« d?��« u¼ «c¼Ë ¨Wþu?H×?� WG� wN?� ¨(*)

d{U?(« UNKJý w� �UGK?�« Ác¼ dLŽ vB?�√ ÊS� ¨…d�U?F*« WO(« �U?GK�« w� Àb×¹�UGK�« l?� qŽUH²K� W?{dŽË ¨d?OG²�«Ë —uD²�« W?Lz«œ wN� ¨ÊU?�e�« s� 5½d� Èb?F²¹ ô…d?²?� w� j³ðdð r� U?N½_ Ü?łd?Š p�– q� w� b?& ôË ¨wDFðË U?NM� c?šQð ¨…—ËU?:«

ÆWOÐdF�« w� ‰U(« w¼ UL� ¨.d� »U²JÐ UNðUOŠ �«d²� s�

ÂöÝù« vKŽ …d�«R*« s� ¡eł Ê¬dI�« WG� WÐ—U×�

¨tK¼√Ë Âö?Ýù« v?KŽ …d?�«R*« s� ¡e?ł w¼ W?Ýd??ý UÐd?Š W?OÐd?F�« t??ł«uð Âu?O�«ËV½U?ł s� …d?�UJ�« qK*«Ë …b?ÝUH?�« q×M�«Ë ¨V½U?ł s� ÂöÝù« 5?Ð Ÿ«dB�« ÂU?� b?IK�mOK³ð w� tÐ— d?�QÐ rKÝË t?OKŽ tK�« vK� w³M�« t?O?� Ÿb� Íc?�« ‰Ë_« ÂuO�« cM� d?š¬s¹œU??O??� r¼√ s� ≠ ‰«e¹ ôË ≠ ÊU??�Ë Æ«c¼ ”UM?�« Âu¹ v�≈ b??²??A¹ ‰e¹ r� rŁ ¨t??²�U??Ý—w³M�« WG?�Ë ¨rOEF�« Ê¬dI�« WG?� ÜN�?H½ WOÐd?F�« WGK�« t�u?BšË Âö?Ýù« 5Ð Ÿ«dB�«W?O?Ðd?F�« WÐU??²?� v�≈ W??HK²?�??� �«u?Žœ �d?N?þ t½≈ v²?Š ¨rKÝË t??OKŽ tK�« v?K� .dJ�«ôbÐ W?O�U?F�« �U?−?NK�« Â«b�?²?Ý« v�≈Ë ¨° »d?G�UÐ ‚U×K?�« W�ËU; W?OMOðö�« ·Ëd?(UÐsŽ U?�U9 W?OÐdF�« ¡U?B?�≈ v�≈Ë ¨° ”UM�« vKŽ d?O?�O?²�« W?−?×Ð v×?BH�« W?OÐd?F�« s�bNA¹ tK�«Ë ¨° ÀU×Ð_« …d¹U��Ë Y¹b(« wLK?F�« —uD²�« W³�«u� ÈuŽbÐ ÂuKF�« Ê«bO�ÊS� ¨Ê¬d?I�« WG� vK?Ž ¡UCIK� —U?N½ qO� ÊuKL?F¹Ë ÊuDD�¹ rN� ¨ÊuÐ–UJ� 5I?�UM*« Ê≈Âö??Ýù« W?�√ vKŽ ¡U??C??I�« p�– b?F?Ð qN?�¹Ë ¨d?ŁbMðË �u9 Âb??�?²??�ð ô w²�« W??GK�«

° UN²�UIŁË U¼dJ�Ë UN½U�� V¹dGð bFÐ UN³¹dGðË9 ∫ d−(« (*)

W‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡�bI*«

14

tK�« s¹b� …dB½ ÆÆV¹dF²�«

Ê≈Ë Æs¹b�« «c¼ …d?BM� �«uD)« rEŽ√ s� V¹d?F²?�« Ê≈Ë ¨s¹b�« s� WOÐd?F�« Ê≈¨q‡łË eŽ tK�« v�≈ �U?ÐdI�« =q‡Ół√ s� WÐd� WM��«Ë »U?²J�« b�UI?� rNH� W?OÐdF�« rKFðÈdš√ —u?�√ t½—UIð ÊU?�K�«Ë ÆtOKŽ Vł«Ë ÷d?� t{d?� rK�*« tÐ rOI¹ U?� W�d?F� Ê≈Ë¨t¼dJ¹ U??L?O?�Ë tK�« t?³?×¹ U??L?O?� rOEŽ d?O?ŁQð U?N� �«œU?F�« ÊS?� ¨‚ö??š_«Ë ÂuKF�« s�ÃËd)« W¼«d?�Ë ¨rN�ULŽ√Ë rN�«u?�√ w� 5IÐU��« �«œU?Ž ÂËeKÐ WF¹dA�« �¡U?ł «cNK�ÊS?� w½U?�?½ù« dJH�« W½«e?š w¼ W??GK�« X½U?� «–≈Ë ÆW?łU?Š d??O?ž s� U¼d?O?ž v?�≈ U?NMŽsŽË w½U??�½ù« dJ?H�« sŽ Ò??×??B�« ÊU??O?³�« f?O?H?½ s� �d?šœ« b??� W??OÐd??F�« sz«e??š

Æ�UGK�« dzUÝ e−F¹ U� WO½U�½ù« ”uHM�«

”UM�« vKŽ ¡«bNý

U?NM¹œ d?A½ vKŽ qL?Fð Ê√ UN?OKŽ V−¹ .dJ?�« »U²J�« U?N½U?�KÐ ‰e½ w²�« W?�_« Ê≈U� v�≈ U¼uŽb?ð w¼Ë ¨Èdš_« 3_« 5Ð UNÐ«œ¬Ë UN�u?KŽË UNð«œUŽ dA½Ë UN?½U�� dA½Ë¨b?Š«Ë U?NM¹œ ¨…b?Š«Ë W?�√ 3_« Ác¼ s� qF?−?²� o?(« s¹œË Èb?N�« s� U?N?Ô?³½ tÐ ¡U?łÜDÝË W?�√ ÊuJ²?�Ë ¨…b?Š«Ë U?N?²?O?B?�?ý �U?�u?I?�Ë ¨…b?Š«Ë U?N?²?G?�Ë ¨…b?Š«Ë U?N?²K³?�ËÊ√ tOKF?� WO�öÝù« W³?BF�« Ác¼ w� qšb¹ Ê√ œ«—√ sL� ¨”UM�« vKŽ ¡«b?Ný «u½uJ¹ËUF?³ð tK� p�– w� ÊuJ¹Ë ÜN?²G� rKF²¹Ë Ü?N¹bNÐ Íb?²N¹Ë ÜN?²F¹dý l³?²¹Ë Ü‡NM¹œ bI?²F¹

ÆUŽu³²� ô

V¹dF²�« œUNł

U?N�?�* UM²�√ U?N� ÷dF?²ð w²�« W�—U?F�« WO?³¹d?G²�« WKL?(« …—uDš „—b½ UM¼ s��«œU?F�« v�≈ ¨Âö?Žù«Ë rOKF?²�« W?LE½√Ë Êu½U?I�«Ë oK)«Ë …b?OI?F�« s� ∫ ‰U?−?� q� w�¨¢W�_« V¹dFð¢ qł√ s� œuN'« ‰cÐ s� bÐ ôË Æ»«dA�«Ë ÂUFD�«Ë ”U³K�«Ë bO�UI²�«Ë…b?O?ŠË …«œ√ v×?B?H?�« W?OÐd?F�« W?GK�« ≠ «œ«d?�√Ë �U?�?ÝR?� ≠ U??NK� W?�_« œU?L?²?Ž« Í√Èu²�*« «c¼ W?OL¼√ dNEðË ¨rOKF²�«Ë r?KF�« d�√ p�– s�Ë ¨‰UÐ Í– d�√ q� w� dO?³F²K�«c¼Ë ¨jI� »d?F�« fO�Ë UNK� W?O�ö?Ýù« W�_« ∫ UM¼ W�_UÐ U½b?B� «–≈ V¹d?F²�« s�w� Êu(U?B�« UM�öÝ√ `$ Íc�«Ë ¨ÂuO�« 5L?K�*« tł«u¹ Íc�« r�C�« Íb?×²�« u¼


15

W??OÐd?F?�« W?GK�« t??F??� «uKL??ŠË ¨»d?F�« …d¹e??ł Ã—U??š v�≈ Âö?Ýù« «u?KL?Š 5Š Á“U??$≈ÆXÐdFðË »uFA�« XLKÝQ�

Ÿ—U?B?²ð U??L?� 3_U?� ¨Í—U?C?(« œU?N??'« s¹œU?O?� s� lÝ«Ë Ê«b?O?� V?¹d?F?²�« Ê≈Ê«b?O*« w� UC¹√ Ÿ—U?B?²ð ¨W¹dJ�F?�«Ë W¹œUB?²�ô«Ë W?O?ÝUO?��« s¹œU?O*« w� U¹—UC?ŠW?O�UF?H�« rOEŽ Õö?�� t½≈ qÐ ¨Èdš_« s¹œU?O*« sŽ «—UŁ¬Ë W?OL¼√ qI¹ ô Íc�« Íu?GK�«ÂU?L??²¼ô« w� d??��« rN??H½ UM¼ s�Ë ÆÈd??š_« Ÿ«d?B�« s¹œU??O?� w?� d?BM�« “«d??Š≈ w�»uFý 5Ð U?NðUG� dAM� ¨WO�d?ý Ë√ WOÐdž ¨W¹—ULF²?Ýô« ‰Ëb�« q� t²�Ë√ Íc�« rOEF�«

Æ÷—_«

UNMÞu?� w� WÝ«d?AÐ WOÐdF�« XÐ—u?Š ¨W¹uGK�« »d?(« Ë√ Ÿ«dB�« p�– —UÞ≈ w�ËwK�_« U?NKI?F� w?� ÂuO�« v²?Š W?OÐd?F�« ‰«eð ôË ¨WO?�ö?Ýù« œö?³�« q� w�Ë wK�_«Íb¹QÐ ≠ nÝú� ≠ ÂuO�« c?HMð w²�« »d?(« pKð ≠ Èu²?��Ë bO?F� q� vK?Ž ≠ tł«uð…—U??G�« Ác¼ b??{ V¹d??F??²�« œU??N?ł ÊS??� «c�Ë ÆrN?¹b¹QÐ rNðu??OÐ ÊuÐd??�¹ s?¹c�« U?N?zUMÐ√Æ—«b�« dI?Ž v�≈ q�Ë b� ËeG?�U� ¨wÐdŽË rK�?� q� vKŽ 5Ž ÷d� �UÐ b?� WK�UA�«v�≈ bK?�¹Ë œU??N?'« «c?¼ sŽ b??F??I¹ s� q�Ë Æt??²??×KÝ√Ë tD?DšË t�U??ł— œU??N??ł qJ�Ëq�«u?ŽË ¨pN²Mð U?NðU�u?I�Ë ¨‰U?²Gð W?�_« WO?B�?ý Èd¹ u¼Ë ô nO�Ë ¨r?Ł¬ u� ÷—_«∫ bŠ«Ë ‚bMš w� Ëb?F�« l� ÊuHI¹ s¹c�« p?¾�Ë√ U�√ ÆÆø fŽUI²?¹ rŁ ¨rD% UNJÝU9»U?�?( qL??F¹ sL?Ž W½U?O?š Êu?KI¹ ô rN?� Ü¼c?O??HMð Êu?Ýd?×¹Ë ¨tðUD?D�?� ÊË—“«R¹

∞ p¾�Ë√ tI×²�¹ ¡ôR¼ tI×²�¹ U�Ë ¨W¹dJ�F�« »d‡(« ÂU¹√ ËbF�«

¨�U?L?ł«d�UÐ nB?I�« s� vJ½√Ë b?Óý_ U?³¹ËcðË U?�?��Ë ô– Í—U?C?(« q²?I�« Ê≈W¹u¼ q²?IÐ v{d�« d¹d?³²� sJð U?L?N� �ö?FÓð Í√ q³?Ið ôË ¨”Ëƒd�« vKŽ �uO?³�« „œË

ÆÊUJ� q� w� WBšUý Á—UŁ¬ Èd½ Íc�«Ë W�_«

tM� W¹UG�«Ë ÆÆ»U²J�« «c¼

w� tK?� b??L??(«Ë �d??Nþ W??¦¹b??(« ÂuK?F�« Ê«b??O??� w� V¹d??F??²?�« œu??N??ł sL??{ËÂuKŽ ‰U?−?� w� ©W?L?łd?²?� Ë√ W?H?�R?�® W?OÐd?F�« W?GK�UÐ V²J�« i?FÐ …d?O?š_« �«uM��«V¹dFð WOKLŽ w� »uKD� u¼ U* W³�M�UÐ «bł WKOK� X½U� Ê≈Ë V²J�« Ác¼Ë ¨»uÝU(«

W‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡�bI*«

16

tK�« uŽb½ WOÐdŽ WOLKŽ W³²J* W³OÞ …«u½Ë —uJA� bNł UN½√ ô≈ ÜM²�QÐ ÷uNMK� ÂuKF�«Æ5¦ŠU³�«Ë 5Ý—«b�«Ë »öD�« QLþ ÍËdðË Èd¦²� UNO� „—U³¹ Ê√ qłË eŽ

v�≈ W??¦¹b??(« ÂuKF�« qIM?� q¹uD�« o¹dD?�« vKŽ W??F?{«u??²??� …uDš »U??²J�« «c¼Ë»U²J�« «c¼ ‰ËUM²?¹ YOŠ ¨»uÝU(« r?KŽË �UO{U¹d�« ‰U?−� w� p�–Ë ¨WOÐd?F�« UM²G�

ÆWOIO³D²�«Ë W¹dEM�« 5²OŠUM�« s� ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�« Ÿu{u�

‰uKŠ vKŽ ‰u?B(« ‚dÞ qOK%Ë n�ËË ‚U?I²ýUÐ oKF?²¹ rKŽ ÍœbF�« qO?K×²�«ËwH?� Æ…œU?²?F*« W¹d?³'« W?OKOK×?²�« ‚dD?�UÐ UN?KŠ …œUŽ V?FB¹ W?O?{U¹— qzU?�* W¹œb?Ž�U¹d?EM�«Ë W???OK?OK×???²?�« ‚dD�UÐ W?Þu???³???C*« ‰uK(« œU???−?¹≈ UMM?J1 ô qzU???�*« i?FÐ¨…œËb‡;« �ö?�UJ²�« Ë√ ¨W¹d³?'« �ôœUF*« iF?Ð qŠ bMŽ ö¦?� tł«u½ U?L� ¨…œU?²‡F*«sŽ Y×?³½ qzU?�*« Ác¼ q¦?� w� Æa�« ÆÆÆW?Oze?'« Ë√ …œU?²?F*« W?OK{U?H?²�« �ôœU?F*« Ë√Ë√ ZzU?²?½ v�≈ …UDF*« �U½U??O?³�« s� ‰u??�uK� �«uD)« …œb?×??� ‚dÞ Í√ �U?O??�“—«u?š¨W?O?�“—«u?š s� d??¦?�√ œu?łË W�U?Š w�Ë ÆWÞu?³?C*« ‰u?K×K� U?³¹d?Ið b?F?Ôð W¹œb?Ž ‰uKŠW½—U?IL?K� Âb�?²?�ð w²�« q�«u?F�« r¼√ s�Ë ÆU?NKC�√ v?�≈ ‰u�uK� U?NMOÐ q{U?H½ UM½S?�v�≈ »—U?I²?�« s�eÐ h²�?²?� W?Žd��« U?�√ ÆW?�b�«Ë W?Žd��« ö?�U?Ž �U?O�“—«u?)« 5ÐoKF?²?²� W?�b�« U?�√Ë ÆqC?�√ W?I¹dD�« X½U?� q�√ s�e�« «c¼ ÊU?� U?LK�Ë ¨»uKD*« q(«tOKŽ oKD?¹ ‚—UH�« «c¼Ë ◊u³?C*« q(«Ë tO�≈ UMK�Ë Íc�« ÍœbF?�« q(« 5Ð ‚—UH�UÐ

ÆqC�√Ë ‚œ√ WI¹dD�« X½U� dG�√ ‚—UH�« «c¼ ÊU� ULK�Ë ¨©QD)«®

»U²J�« ‰uB�

U¼d?OŁQð qOKI?ð WO?H?O?�Ë U¼—U?A?²½«Ë ¡UDš_« Ác?¼ q¦* W?HK²?<« Ÿ«u½ú� W?³?�M�UÐËÆW¹œbF�« �UÐU�(« sŽ ‰Ë_« qBH�« w� UNA�UMM�

‚dD�« i?FÐ g�UM¹ Íc?�« w½U??¦�« qB???H�« w� t�ËUM?²M� »—U??I??²?�« Âu??N??H??� U??�√ËÊu?�?�«— ≠ sðuO½ W?I¹dD� b?Š«Ë ‰u?N?−?� w� ©WO?D)« dO?ž® �ôœU?F*« q( W¹œb?F�«WI¹dÞË sJð≈ WI¹dÞË ©WFÐU²²*« �U³¹d?I²�«® W²ÐU¦�« WDIM�« WI¹dÞË nOBM²�« WI¹dÞËw� ©WOD?)« dOž® W?O½ü« �ôœUF*« qŠ ‚dÞ iFÐ qB?H�« «c¼ g�UM¹ rŁ ¨s�MH?²Ý

ÆbŠ«Ë ‰uN−� s� d¦�√


17

‚dD�« iF?Ð g�UMO?� W?O?D)« �ôœU?F*« rE½ qŠ ‚d?Þ Y�U?¦�« qB?H?�« ‰ËUM²¹Ë‚dD�« iFÐË ¨·c×K� Ê«œ—u?ł ≠ ”ËUł WI¹dÞË ¨”ËU?' ·c(« WI¹dD� …d?ýU³*«UL� Æ‰b¹“ ≠ ”ËU?ł WI¹dÞË ©WFÐU²²*« �U?³¹dI²�« WI¹dÞ® wÐu�U?ł WI¹dD� W¹d¹dJ²�«Ác¼ s�Ë ¨…d?ýU³*« ‚dD�« U?NO?� Âb�?²�ð w²�« �U?IO?³D²�« iFÐ qBH�« «c¼ ÷d?F¹W�uHB?� ‰ULŽ≈ Ë√ qOK%Ë ¨W�uHB?� ”uJF� œU−¹≈Ë ¨�«œb;« »U�Š∫ �U?IO³D²�«W�«œ V¹dIð W�Q�� ‰ËUM²¹ ‰ULJ²Ýô« »U?�Š sŽ lÐ«d�« qBH�«Ë ÆWHK²�� q�«uŽ v�≈ÃU²×½ w²�« �U?OKLF�« iFÐ ¡«dł≈ q=N�¹ p�– Ê√ YOŠ ¨j�Ð√ ‰«Ëœ s� W?HzUDÐ W�UŽ¨‰Ë«b'« s� ‰U?LJ²Ýô« vK?Ž 5F¹ jO�?³²�« p�– Ê√ UL?� ÆWK�UJ*«Ë WK{U?H*U� U?NO�≈ô WDI½ bMŽ U?N²?LO?� 5OFð UM� VKD¹ r?Ł ◊UIM�« iFÐ bMŽ U?� W�«œ rO� vÓDFÔ½ Y?OŠ¨W?O�U?LJ²?Ýô« !«d?łô W¹œËb?Š qB?H�« «c?¼ ÷dF?¹Ë Æ◊U?IM�« Ác¼ W?Žu?L?: wL?²Mð‚Ëd?H�«Ë W??O?�U?�_« ‚Ëd?HK� s?ðu?O½ w²?G?O??B?�® ‚Ëd?H�« Â«b?�??²?ÝUÐ ‰U?LJ²??Ýô« mO?�Ë�U¹œËb?(U?Ð w¾¹e?−??²�« V¹d?I?²?�« ‚dÞË ¨©”ËU?ł mO??�Ë ¨�d?H¹≈ W?G??O?�Ë ¨W?O??HK)«qBH�«Ë Æ©WO³OFJ²�« W×¹dA�UÐ ‰U?LJ²Ýô«Ë ¨wFOÐd²�«Ë wD)« w¾¹e−²�« ‰ULJ²Ýô«®W�«œ �UI²A� »U�( W¹œbŽ UGO� wDF¹ YOŠ ÍœbF�« q{UH²�« tŽu{u� f�U)«bMŽ U?�≈ �U?I?²?A*« Ác¼ »U?�?Š sJ1Ë ¨‰Ëb?ł w� ◊U?IM�« iFÐ bMŽ U?N?L?O?� ÂuKF?�‚dÞ iFÐ ÷d?F¹ d?Oš_«Ë ”œU?��« qB?H�«Ë Ætł—U?š WDI½ bMŽ Ë√ ‰Ëb?'UÐ WDI½…b?ŽU�Ë ¨Êu?�?³L?Ý mO?�Ë ·d?×M*« t³?ý mO?� Âb?�²?�ð w²�« pK²?� Íœb?F�« q�UJ²�«

Æ…œdHM*« �ö�UJ²�« rO� œU−¹≈ ‚dÞ qBH�« «c¼ g�UM¹ UL� ÆÊuÝœ—UA²¹—

V�UD�« 5Fð w²�« qzU?�*«Ë �UM¹d?L²�« s� d?O³?� œbŽ b?łu¹ qB� q� W¹U?N½ w�ËÆqBH�« p?�– w� �œ—Ë w²�« W¹œbF�« ‚dD�«Ë rO¼U?H*« »UF?O²Ý«Ë rN?� vKŽ tK�« Ê–SÐW¹U?N½ w� �UM¹dL?²�« WÐuł√ v�≈ Ÿu?łd�UÐ rN?H�« «c¼ W×?� s� b�Q?²�« V�UDK� sJ1Ë

Æ»U‡²J�«

�U??O?K� »öD� b???Š«Ë wÝ«—œ qB???� w� »U??²?J�« «c¼ …œU???� f¹—bð sJ?L*« s�ËÆWÝbMN�«Ë ÂuKF�«

t?Nłu?� UB�U?š ö?LŽ t?K³I?²¹ Ê√Ë ¨»U?²J�« «c?NÐ lHM¹ Ê√ d¹b?I�« wKF�« tK�« ‰Q?�½ÆWO�öÝù« UM²�√ ¡UMÐ_ UF�U½ ULKŽË ¨.dJ�«

W‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡�bI*«

18

t?³KÞË ¨W?OA?š tK� t?LKFð ÊS?� ¨rKF�« «u?LKFð® ∫ tMŽ tK�« w?{— q³ł sÐ –U?F?� sŽt�cÐË ¨W?�b??� t?LKF¹ ô s* t?L?O?KFðË ¨œU?N?ł tMŽ Y×??³�«Ë ¨Ò?³?�ð tðd??�«c?�Ë ¨…œU?³?Žw� fO?½_« u¼Ë ¨WM'« q¼√ q?³??Ý —UM�Ë ¨Â«d??(«Ë ‰ö??(« r�U??F??� t½_ ¨W?Ðd??� tK¼_¨¡«dC�«Ë ¡«d‡?��« vKŽ qO�b�«Ë ¨…uK)« w� Àb;«Ë ¨WÐd?G�« w� VŠUB�«Ë ¨W?AŠu�«dO?)« w� rNKF−O� Ü?�«u�√ tÐ tK�« l�d¹ ¨¡öš_« bMŽ s¹e�«Ë ¨¡«b?Ž_« vKŽ Õö��«Ëw� WJzö*« V?ždð ÆrN¹√— v�≈ wN?²M?¹Ë ¨rN�U?F?HÐ Èb??²?I¹Ë ¨r¼—UŁ¬ h²?I?ð W?Lz√Ë …œU?�¨t�«u¼Ë d×³�« ÊU²OŠË ¨fÐU¹Ë VÞ— q� rN� dHG²�¹Ë ¨rN×�9 UN²×MłQÐË ¨rNðöšÆrKE�« w� —UBÐ_« ÒÐUB?�Ë ¨qN'« s� »uKI�« …UOŠ rKF�« Ê_ ¨t�UF?½√Ë d³�« ŸU³ÝËt?O?� d?OJH?²�« Æ…d?šü«Ë U?O½b�« w� vKF�« �U?ł—b�«Ë ¨—U?O?š_« ‰“UM� r?KF�UÐ b?³?F�« mK³¹s� ‰ö??(« ·d?F¹ tÐË ¨ÂU??Š—_« q�uð tÐ ¨ÂU?O??I�« ‰b?Fð t??²?Ý—«b??�Ë ¨ÂU?O?B�« ‰b??F¹

[(*) d³�«b³Ž sÐ«] ©¡UOIý_« t�d×¹Ë ¡«b‡F��« tLNK¹ ¨tFÐUð qLF�« ÂU�≈ u¼Ë ¨Â«d(«

Æ5*UF�« »— tK� bL(« Ê√ U½«uŽœ dš¬Ë

ÆÍbM�dL�K� 5K�UG�« tO³MðË ¨w�«eG�« bL; rK�*« oKš (*)


‰Ë_« qBH�«

W¹œbF�« �UÐU�(«

Numerical Computations

¡UDš_« Ÿ«u½√ º

V¹dI²�« ¡UDš√

ŸUD²�ô« ¡UDš√

WOz«b²Ðô« ¡UDš_«

¡UDš_« b�«uð Ë√ —UA²½« º

ÍœbF�« ¡UG�ù« º

—«dI²Ýô« º

‰Ë_« qBH�« �UM¹d9 º

21

‰Ë_« qBH�«

W¹œbF�« �UÐU�(«Numerical Computations

ÍœbF�« qOK×²�« u¼ U�

!" #$% &'$( )**"+, -!./ +, 01213 #.(derivation) 1 4&51(description)

1 -&&&!./ (analysis) &&&6 #7(methods) ,&&&!**8 -1&&&./ 9&&&.8 -1&&&5/+ :(numerical

solutions) + :!&2$!7 -;$<3(mathematical problems) ! #76+$&( $&=./ >*$&8 ?"&5>*$&& "3+, :&&!7(@+, :&&!.!./ +, . :61(&&23+, -1&&./+, *$&&@!A $&&BBC3! D -;$&&<3+, E&&"( F&&GH

(exact solutions) (&3 4*$5B $3C I>*$ "3+, J$!7KB+,1 :!.!./ +, #76+$L -&/ *&B8 MN:+*$"3+$C I:!7(@+, JD*$"3+, E"(:

O:!3$< 3+, JD*$"3+, 1 (transcendental equations) :+*$"3+$C I:

10tanlnsin2 !

O$C +, E"( -/ *B8 13-3$C +, -L3 I >*1*/3+, JN: !

sin5

1

" 8 ! "# $% &

'&()*+,- ./01(2),- 34&(*+,- 5*6)'(7*+,- ./8-&)64- 7 9:,- ! 3(#(/6,- ;+ ( !- 34&(*+,- 5*6 $% &#<, ./8=>,- ./01(2)... ?@,.

A $B+ C D%6# $8(E+,- FGH< 3(/+=9- I J(Algorithms) K97 CH L(6/9M) &*) ./&&< $ 0% ! N8()# O,? '(7*+,- 3(#(/6,- J+ $ P 0, 3- 7I,- '&&%+

.7 61+,- $ 0%0, . Q&&*,- $%,- )0%,- ! ./&&*,- $ ( L-&%- L-&&< J R/ &S .,TE+0,), ./6/9M) .+/MR)./+(E)+ .,&(*+ $% ! "& &%+ $+(R ( 9BR! !) $B+) &-&<! '&*R

.,-& U/S $B+) &-&<! '&< ! "./96> .,&(*+, ./6/9M) L-9 G>y U/S 5*6 &#<x D/%y .V/P O0< $P%# 4 '9/IW- .,(%,- FGH CA (##! Q! "./01(2) .,&(*+ $% CH

# X- ./01(2),- .,&(*+,- $%, ./0/0%)+7MA $P%# ( .,-&,- U/S 5*6 O0<y Y&+,- CA QG,- Z/A U)[#6./01(2),- .,&(*+,- $%.(

0525 249 !"!"

22

A .6 07+,- .>9I+,- 3(+ 0*+,- &/, ), ./+=9- I J+ 9BR! & > .,(% C)N8()#,- (([01A! O,? $ P 0, ([#/6 $1(2# (##\A " . U&I)E) C),- $+- *,- UH! J+

.S&,- .<9E,- 3(/+=9- I,- J/6 .#9(M+0, . P ,- O,? Q&]) C),- .M/97,(A$ ? O,)#,- 96R! .<9E6 N8()9(M)) Q!converges ?$S! J+= CA 07+,- $%,- O, ( $12)

(H9/_ O0< . K&! N8()# O,? Q&]) C),- .M/97,- `,GR )$S! ([/A T7I,- J R/ Q! (

(H9/_ O0< $12) ..>/)#,- CA T7I O,? Q&]) a(7I! '&< &> ) L(+ +< .

AJ R) (+ L-9&(# LbB+ I&+,- 3(+ 0*+,-.0 '=[>! J+ C)T) '&(< ([#W .7 61++ c(/S9Id ! e # J . a(7I! $I&) ^(E%0, U&I)E) C),- ./+=9- I,- `,GR Y9I! . (+R J/9&P+,- J/GH J+ a(7I! O0< $+):) .>9I+,- 3(+ 0*+,(A `,G,

C,(),- C7/7I),- UE9,- `,G J/6/ . $0M) C),- ./+=9- I,- (minimizes) +# J+ a(7IW-(error growth) ,(6 (H9/_ J+ 9&>! J R) (H&,- )K/67).

A C),- .0B+W- 5*6 ;+ a(7If, .20)I+,- e- #W- g- JG\6 c9&# C0/ (+/([/0< ^0V),- ./2/R " ([%1 ).

A U//M) K(M):- CA hI0)/ Q&&*,- $/0%),((evaluation) N8()# ^(E%, K97'(7*+ ./&&< 3(#(/6 J+ .6 07+ ./&&< => Q&&*,- $/0%),- $*>/ -GH " U0*,- J+ L-a

O+E/ QG,- D/&%,-"3(+ 0*+,- $/V:) "(information processing) 3(#(/6,- D/% ",- '(7*+(given data) .0I&+,- 3(+ 0*+,- $B+)(input information) N8()#,- "

.>9I+,- 3(+ 0*+,- $B+) .6 07+,-(output information) UE9,- `,G $B+/ (+R "C,(),- C7/7I),-:

WKš«b�« �U�uKF*«

unput information

WO�“—«u)«

Algorithm

Włd<« �U�uKF*«

output information

WKšb*« ¡UDš_«

unput errors

WO�“—«u)« ¡UDš√

Algorithm errors

Włd<« ¡UDš_«

output errors

1ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�«

23

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«

!"# (i) 1234567 $%&'()* +, #- %).#.

(ii) 854211 /%&'()* +, #- %010.

(iii) 3.1415926 / +, #- %2*3%&#45.

(iv) 4.265 /6&&#45 6&* #.

Types of Errors ¡UDš_« Ÿ«u½√

Round-off Errors V¹dI²�« ¡UDš√ ∫ÎôË√

!"#$:

7(6- 8# 0x: 95:'.;* 9 (exact number)

x: %&.&#<= %*& (approximation) 99)>/ 0x

? 6@%&'()*% +(,* (absolute error) A(B. CD#E)F&:

!xx "#$% 0

-*% +(,*%./01 (relative error) % 1)/,. CD#E)F&:

00

1 xx

xr"$%$%

%2#$3!* (Rounding)

4#556 7*8 9) 556 2:$;3 (n) <#-1 )*% =9>$?% @) (significant digits /

figures / positions) . + # G;'*/H 6&*& +, #$H G&*I J)In K? + #/H ,*- L MH#7N G;'*/H + #n O9PH' O9P'. 9HQF& 6- '- #&&R= 6'9 'S ,*T U#=F& 6- ,*V A(@?(one

unit) G:=<*/H WQI/H 6,T HXV ,* !2P(truncated part) CN( 6* #.T- '- J -9P'O G;'*/H K?n . 6@? O9P' CN(/ :.;/,. M,&',2* G:=<*/H WQI/H HXS 6,T HX YH'

G;'*/H K? + #/Hn 9PH' O9P'. 9HQ& 6- 6T*& '- O- #&&R= 6'9 U#=& 6)9HQ& O9,5'.(

' %.2(/,.* 55 * 2#$3!'n <#$A *% =9>$?% @) (decimals) : 9). +, #$H 6@? + # G;'*/Hn – MH#,7N- ,Z>T J)I= K=/H'– %:,2.. CXP=.

) B9C(1-1):

1

24

%BD*:

)i( 1235000

(ii) 854000 (iii) 3.14159

(iv) 4.26 !4.27

!"#$:

"#! $%& 0x: '( )*+ ,,

x: -./)/%01 .+/ 23

: 456(+34 7(83

/ #9 :;0% &&$'x (' )"*+n #,#-. /0 ) 2/< !n =/>? @%A' B-%( 4C9#;D:

n!"# 105.0

<#;D 4C9 EFG+:

4631.2

4671.20

$

$

x

x

<#H:

2104.0004.0 !"$$

!#! @: 2105.0 !"%

#H< I3;13;) x #//%A' #/+-%3 =/>?.

!J";D 4C9 ;+ x = 2.4620

<#H:

12 10051.01051.00051.0 !! "$"$$

! #! @x -%3 =/>?,>4 @%A' B.


25

##! x.

#+ EFD #! $%&"a, b #+ # D+ ,,'n B-%.

> K%*34 :?;ab L6' E;+ +' @ 1>/2n B-%.

34 M%;8 .+N0a/b , ,>+ %/O ,,' L6' E;+ +' @ 1>/.

< ;P"Q) R%P- 4CH EFD L39 .+N034 K%*34 I1S/1" #+n I6/ ;+D ;+T3 ;"U+% V B-%:

4K%*34 ./6+W3 .)R%P0+34 .S/1"3: b ×a

4.+N034 ./6+W3 .)R%P0+34 .S/1"3: b ÷a

!"#$:

" #! $%&x X3 =/>? ,,'n @%A' B-%.

! ,,W34 B;-%x #+ %)D! Y,> 34 Z/> [*4 +34 :1>1 I134n!10 12320#4 536 7"89$3)\4 #! .]>F+ [+,W1 _ ./`4,1)_4 %;&?.(

<,,W34 EFG+:

! "#

$%&

! "#

& 400000.0001234.0

5

x

<FG V .>/>? ./%A' B;-%! .N+8 2/./ "W+ B;-%! .G.

),,W34 ;+"/: 000006.0001234.0 &

<#;/ "W+ #;+-% .>/>? ./%A' B;-%! .W)%! 2/.

37:*;:

'D< I(W/ .>/>?34 ./%AW34 B;-%\4 ,, %;+"/) 56(+34 7(834 BS> ! .+/- #' Y

I)N"34 7(834 .+/- #' Y%D< I(W/ ./ "W+34 B;-%\4 ,,'.

*<' ="#> '% ?2@A4 &8&7@"6B'% C2"<3$

Round-off error bounds for elementary operations

" X3 ,4,'\4 Ka%P0" ;""! $%&n "! V B-%2 ,4,'\;) (0< =N+/x 50>1 I134(%A34: 11

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

26

<#H: nabba !"#!" 105.0

nbaba !"#!' 105.0/

< 4C9 EFG+! ;"*%<#: n = 2, a = 0.56, b = 0.65

<#H: 36.0,364.0 $"$ baab 2105.0004.0 !"%$!" abba

4="#> '% 7D" 9 7"B26* ?2@A:

+ L6' ;"+N- BG ;+ b,,' I< ."/W+ .+/- ;")%* 4C9 ;""! E;/*;/% B 6W+34 # ,,W34 2N&" EFG+< V ./6?\4 .+/034 JS1" V:

( ) 76.006.006.067.0 $'"

D3./6+' :D I< K/%0134 c4%S9 [+ # ,- [+ EFG+< V ./6?\4 .+/034 ;","' d1"1 _A' #/+-%3 K/%0134 %I6/ ;+ .0);N34 .-FW34 ;"/(W1 V #//:

(0.76 × 0.06) ÷ 0.06 = 0.05 ÷ 0.06 = 0.83

* 0.76

/ e K/%0134 #' d1;"34 7(834 # D: |Total round-off error| = 0.83 – 0.76 = 0.07

'%7"B26*'% C2"<3$'% =" # 8 ="#> :

,- ./);N>34 J;/6+W34 K/1%1 %Gf/ (order of arith. operations) - + [ K/%0134– 9./`;T"34 .S/1"34 I< %//g1 L3.

<#! E;/*;/% B 6W+34 #+ EFG+:

(a × b) × c = a × (b × c)

3 4 I< ;+D V#;<%(34 h ;N1/ _ ,- K/%0134 [+ #D :;G+334 #/+-%3 Ka%0" Z/> I3;1#//%A':

(0.56 × 0.65) × 0.54 = 0.36 × 0.54 = 0.19

0.56 × (0.65 × 0.54) = 0.56 × 0.35 = 0.20

* 0.19


27

E4C%#"F 3 G&. HI 7'%& HI HB69 J@A #B

Max. relative error in a function of several variables

"&#! $% f I< .34,n %/g1+: nxxx ,,, 21 '

!#! @: ( )nxxxff ,,, 21 '+

c;(8\4 #! $%&" Y,%&+34 (individual errors) L6' Ie J4%/g1+34 iCe I<

K/1%134:

nxxx ,,, ,...,, 21

< .34,34 I< I6D34 7(834 # D/f - ;T- < ;+ ./";G34 .)1%34 , ,> :;+e9 [+– e:

nx

f

x

f

x

fxxxf

n,--,-,$,

.

.

.

.

.

....... 21

21

nx

f

x

f

x

fxxxf

n,--,-,$,/

.

.

.

.

.

....... 21max 21

I)N" 7(8 L?-! # D/ I3;13;)(m.r.e. + max. relative error) e :

0 1nx

f

x

f

x

f

ff

fxxx

n,--,-,$

.

.

.

.

.

.,...... 21

1

21

max

3 K2L(2-1):

!J4%/g1+34 .3_,) #/1/3;134 #/134,34 #+ :D I< I)N" 7(8 %)D! ,S :

nxxx ,...,, 21 ;T`;(8! :nxxx ,,, ,...,, 21.

nm

n

mm

n

xxxg

xxxf

...... 21

21

21

$

---$ '

%K*':

nxxxf ,--,-,$, .1....1.1 21max

n

n

xxx

xxx

f

f

---

,--,-,$

,

...

...

21

21

max

n

n

x

x

nx

x

x

x

g

gmmm

,,,,---$ ......

2

2

1

1

21max

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

28

37:*;:

/ .+/- I< I6W&34 7(834 #! E;+ +' ]>Ff L?-\4 ,>34 #+ %/GD) :-! # D/ 7(863 c;(8\4 #\ j3C – ./6+W34 ./>;"34 #+– E;*W) ;T*W) Ig6/ #\ 2S11

)W)34 KS + ;T*W) Z/>$ K3;N %8k4 .( l/*! 4C9 EFG+<20.000 Ka%- V,,' L39 ,,' :D4 @ ;N/ 7(863 L?-\4 ,>34 #H< V./%A' B;-%!000,20105.0 4 "" !

@ ;N/ @!1 I6D34 7(834 ;+"/) VY%/)D .+/- iCe (total error) Y,;' # D/ [- 1+34, ,> I< 0.005.

3 K2L(3-1):

!7(8 %)D! .+/- ,S .34,34 I< I)N"f Z/>:

( )z

xyzyxff

2

,, $+

9#! B6' 4C:

004.01

0005.03

003.02

&$

&$

&$

z

y

x

'%K*:

zz

yxy

z

xyx

z

y

zz

fy

y

fx

x

ff

f

,!,-,$

,.

.-,

.

.-,

.

.$,

$"$

..2

.

...

181

92

2

22

105.0004.0180005.012003.09max

$"-"-"$,f

00583333.018105.0

max

$$,

f

f

(8./`;T" _ ./6+W) ./T1"+ ./6+' :4,)1N4 #' mA;"34 7(834 e n;(1-_4 7

(infinite process 2 finite process)V .3,;W+ B4,81N;) ./6*;&1 .3,;W+ .6+;D+D

Truncation Errors ŸUD²�ô« ¡UDš√ ∫ÎUO½UŁ


29

n +S+) 2)/%01) , ,>+ :+;D1 K;N> V5 %<) J;<%>"+ i;)A! J;>;N+ n +S+D, ,>+34 :+;D134 ;T6G+/ I134 .>;N+34 E;)/%01 @ ;N/.(

3 K2L(4-1):

>4 %&34 .3,;W+ :4,)1N4 #' mA;"34 n;(1-_4 7(8 KN5 ( )1.01 $,$

,

!3$

,

,- xxx

yyx

x

yi

ii

)./6*;&134 .3,;W+3;:

xdx

dy$

3 .+/- K;N>y ,"'x = 0.2 #! B6' 4C9y=1 ;+,"'x=0.

%K*':

) ( !"#$%& '(&:

!

! 02.112

2.02.0

12

110

2

2

2

2

"#"

#"$"$"

#"$"

%&'

()*

y

xycy

cxyxdxdy

) ( )**+& '(&:

!

01.11.01.011.0

11.0

2.0

1.0.

112

0001

2

1

"+#"#"

""#"

,

#"-#"#

xyy

yxyy

yy

xyxxyyiiiii

",-!./0 1!2 3"45 6&-.&-#: Truncation error = 1.02 – 1.01 = 0.01

" 758 *.#0 9-:-5#& 6; <-!2= 6> ?@>(initial data) 6.& 9-:-5#&-4 A/ 3% BCD% -E5FG 'H(:<75F%+% 7#AI. 6; JKEI= L+# 9 .

Initial Errors WOz«b²Ðô« ¡UDš_« ∫ÎU¦�UŁ

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

30

G9 "!2 J*G 6; 75**+& 75#-M(& 75F%+& N.. J*- .% J"!2 3O LAP: ?@> 35F%G 6> 9 "!2& 7 %MQ& 7 z = x/y.

"3O LAP:yx .., 35!"#$%& 35**+F& 3-#5AQ. -%> x, y R5.A.& SFG T5(# U3O:

/0

#".

#".

yy

xx

#; 7!"#$%& 7%5Q& 3% B0*z :75#5AQ. 7%5/ RM( z. 5(T:

!

!

! !

! ! 1#%

&

'()

* 2#2#"

1#2

%&'

()* ##"

1##

#"

21,1###"

..".

...11

111

1

1

/

2

2

yyxyx

yxyx

yy

xx

erroroffroundyx

roundedyxz

//0

/0

/0

/0

.

Propagation of Errors ¡UDš_« b�«uð Ë√ —UA²½«

1#%&

'()

* 2##"

1#%&

'()

* 2#%&'

()*#3

2

2

.1.

.1.

yx

yz

yx

yyx

/0

/0

O6; 1!2& 3O )z 3% J*& ".% <-!2O 3% 3"4%x , y *5*I 1!2 3%"R5AQ.& 7I5.:.

!"#$:

@V 758 *.#0 9-:-5#& 6; 7!5M# 9 A5W. 9*O (initial data) JA5#4 9 A5W. S&V 3V '-Q5; U758-E:& X8-.:& 6;% &'#()% *+"+, *)-./)(ill conditioned problem).


31

: 7& *& 7%5/ R-M( *5A: -::O LAPf(x) T5(:

:x 6Q5Q( **G.

:f Q5Q( 7& *57.

)i( :3O LAPx. : 6#M: **G(rational number) **+F& R5AQ. ">x Y:O T5(3% 68-E: 0 **+# * *GO 35K2. Z5!.M5 RM-( *I"5 0 N-/A= 75A[+& .

3O )Oxx 2. : 7%5/ 6; 68 *.#0 1!2& ">x.

3"45 6&-.&-#"01%2 34% -&5)% 7& *& 7%5/ 6; '#-Q%& f ) *& ".%& 1!2& ">" 7%5/ 6; 68 *.#0 1!2& 3G X.-:& x:(

! !xfxf 2."11

)ii( : 3O LAP1f :& *& 3% !M#O 7& * 6> 7f T5(#O R\AQ. -E:f 6!+. )O7%5Q& 75#5AQ. 7%5/ f ] 3"4. J*-G1f 7+!.Q% ]"^/ 7F^MFM.% (truncated

power series) 7& *& _"4P% 3% (expansion of f) f [.

" 3"45 6&-.&-#5 67& 84% -& '#-Q%&:

! !xfxf .2."1 12

)iii( : 3O LAP !xf .2 :> RM-(&-# 7#"M(%& 7%5Q& 6) 9-#5AQ. < AI`# )O

roundings ;75#-M(& 9-5F%+& 6.(

#" 3"45 6&-.&-&59"#: )% - ) "!2& ">1 9-#5AQ.& 3% *& ".%&:(

! !xfxf .2."1 123

"6F5 -%5; a#M -% b52F. -::4%5:

! ! ! ! ! ! ! ! ! !xfxfxfxf

xfxfxfxf

xfxfxx

.2."1$.$.

.2."1$.$.

.2."1$.$

12321

121

1

"6F4& 1!2& 3"45 6&-.&-#:

! ! 3212 1#1#"12.1" xfxf

/;7&5<% =/ *>+ 5/)% 6%'?<% @"A' ) B7C

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

/ B7C(5-1):

; RM(:M -::O LA3/1e _"4P% 3% !Q; S&"= *"*( 7M%2& N *2.M-#7& *& xe .-M(& '4 3O LA; "#!Q; 75A[G N-/AO 7+#A1# ]AI.M 9- . N5/ *I"O

.2%& <-!2= FR-M( 6; 7P3/1e6F4& 1!2& *I"O ND 3%" U.

%D)B:

!31, "" xexf x

)i( ( '2*. 3O 3% B0*#; U 75A[G N-/AO 7+#A1# ]AI.M 9-#-M(& '4 3O T5x

&-# 9-#-M(& 6;Q7!"#$%& 7%531 75#5AQ.& 7%5Q&-# '2*..

3333.0".x

" S&V )*c5 @>5 01%2 3% -&; 7& *& 6f.

!6519600004.0

1

3333.03

1

...0000333.03333.0

...3333333.03333.0

3/13333.0

1

2"

2"

2"

2"$1$4

ee

ee

ee

) JA[+& _&@"R5AQ. N*G 6:+. 75A[G N-/AO JA[G 3O LAP# 75A[G N-/AO.(

)ii( " R-M( 3% B0*# -::O T5(xe *"*(& N5/ ,"%I% R-M(# 6P.4: U ]"/ 7FMFM.% 3% S&"= 7M%2& x 7& *& _"4P%&xe R-M(# )O U

7+!.Q%& 7FMFM.%& ,"%I%(truncated series)

!!4!3!2

11432 xxxxxf ####"

; S&V )*c5 @> 3`567& 84% -&

! ! xx exfxfe.. 2."1$.$ 121

xxxx ex.... 2###.#"41

!4!3!22

432

1

6275000003.0

...

...

!63333.0

!53333.0

!6!5

65

65

2"

567

89: ##2"

567

89: ##2" .. xx

32


33

)iii( " 3"* B-!"#$% B-#-M( 7+!.Q%& 7FMFM.%& ,"%I% R-M( 3% B0*# Y:O -%#R5AQ. )75A[G N-/AO JA[G '-%+.M-# )O(Q&-# -EG"%I% RM(: U5Q%& N7#dA:

!

3955.1

0005.00062.00555.03333.01

3333.01!4

3333.0!3

3333.0!2

3333.02

432

"

####"

####".xf

" R5AQ. 3"* 7+!.Q%& 7FMFM.%& ?@> ,"%I% 3O -%#)JA[+# )O N-/AO75A[G (">:

! 9630439552.11 ".xf

; R-M( -::4%559"#: )% -&:

$567

89: ##.# .

!4

4

...1 xx $ 567

89: ##.# .

!4

4

...1 xx

R5AQ.& Z% *R5AQ. 3"

9630400002.0

9630439552.13955.13

2"

2"1

" 7%5/ e#H.%0+E)% -&5):

2425000011.0

321

2"

1#1#"11

% 7/* S&V B-#5AQ. 355"-M.% 35**G 35# aA-P& R-M( )*c5 3O 34%%& 3 B *I 7$P2:% 75#M:(very poor relative accuracy) .5F%G 3O )O7 6.& fA!&

5 g)O 3% A5D4# '/O 35.5%4& 35# aAP& -E5; 3"4 7$P2:% 7/* S&V )*c. */ -%E:%7I5.:& 6; .)**+& <-W&h S%M5 @>".

;3O -:$A; @`: 21xxy 2"

;3`:

21

2

21

1

21

xx

x

xx

x

y

y

xxy

2

-#

2

-;

-4

-#-;-

Numerical Cancellation ÍœbF�« ¡UG�ù«

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

34

!"#$%&" '() "*+ 21 xx !"# $%&'" ( )*# +,"# -./ 0#+1 0#234 56 578 9.8,y / 0#+1 0#237.

6-./ #:; 0<=>:

4

2

4

1

105.05763.0

105.05764.0

!"#

!"#

x

x

6-?:

0001.00001.021

"# xx

@A13B!'" C2+D>"# A>3D"# EF.83 $%&'" ( )*# +,"# -@ E.

C2F 7 A>+&B8>"# A43 "# A7.B/ C+.G; HF.,! -@ 913 IJ.,"# K:L H=> 56"# -32#+D>"# M2% E@ N.4"O# .P36 9!1B! A78.!> A'=>*# -> -37B3 .>/ Q-372.DB>

A3".B"#.

!"#(6-1):

) ( !3D3D,"# -32:1"# +.13; +32! .!!@ R2S3A3!.="# A12+"# A"+.T>" -:

022 $#%% axx

,U3:

a$&&

/#E++T"# N.4"O# 0.7!B1> -32:1"# -3:L +.13O -3B78.!> -3B43 9B.

)! ( 7%A"+.T>"# E2:1 +.13O -3B43 "# -3B.L V:

02121.111.1112 #%% xx

$!%&:

)@( $ # 2

1aax

FA78.!> A43 K:L.

$ % # 2

2aax


35

)E++T"# N.4"O# ("; E+WB +) K:L(

$ $ $ % #

2

22 .

aa

aaaa

12 xaa

$#$

$#

a2 $# (without loss of significance)

JA43 "# X#+&B8# -@ Y, a2$ F@

1x$ 0.732DB E+W3("; X.)2*##A3F!T>" .P8S!.

)9( 02121.111.1112 #%% xx

10.111

545.55

2.3085

4.3086

555.55

2121.1

2

2

2

#

#$

#$

#

#

#$

x

a

a

a

a

F78!".7A " 2&Z# 2:1'2x A3' *# A43 "# .!>+&B8# #:; Q:

$ % # 2

2aax

! ,A>3D"# ('G H: 01000.2

#x

37FL X.)2*# -> -37>"# ++T'" %F7[>"# 2:1"# .>!: 010910.2

#x

)@32\G -3>)2" A,3, .P3'G .!' , 5B"# A13B!"# -@ E-3 %D6(.

@>A"]+T_>"# 2&*# A43 "# .!>+&B8# #:; .:

1

2 xx $#

?6 ('G H ,! .!!A>3D"# :010910.02

#x

LFX.)2@ A8>&" A,3, A>3) K:.

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

36

!"#(7-1):

a[ E++T"# N.4"O# 23=$B XF.DB A78.!> C2F 56 A3".B"# b3 "# -> </:

(i) cos (x + ') - cos x

,3 U' G 0#+1 234 ++.

(ii) sinh x – cosh x

, U3x G 0#+1 237/ ++.

(iii) f(x + ') – f(x)

, U3' G 0#+1 234 ++.

&$!%:

(i) ( ) ( )2

sin2

sin2coscos ''' % # % xxx

(ii) xxe

exxx

x

sinhcosh11sinhcosh%

###

(iii) 72F'3.B A'8'8B> cF/S> X#+&B8.

( ) ( ) ( ) ( ) ....21. 2 %**%*# % ''' xfxfxfxf

@#F Q 2#2DB8J# 2.3T> I.3>d2#F&"# -37 A!2.D>'" X+&B8B 5B"# 233.T>"# +e," E: 0.%387 0#234B -@ 5!T3 fg.B!"# 56 %387 234B ("; E+W3 A3g#+B7J# I.!.37"# 56

A3g.P!"# . A3>d2#F& .P3'G V'%3 A3 .&"# K:L VD,B 5B"# A3>d2#F&"#F '()*+

(stable) 5P6 J h#F Q, '()*+ (-(unstable) . C2DB8> -F/B I.3>d2#F&"# RT7F.& A3g#+B7# I.!.37 AGF>1>" A78!".7 AGF>1> E* i3"F A.

FA8#2+" 2.\B!# -37 A)<T"# QI.3>d2#F&"# A32#2DB8# -37F 932DB"# N.%&@ F>! F@ K2+) $%& -@ R2BS!+ )-@F Q I.7.8,"# N.!=@ .> A',2> 56 2PY + ++G +T7 $%&"#n

>"# ->TK2#+D> A',2>"# K:L +T7 A3".BB>"# A37.8,"# I.3' nE . -37 A)<T"# -GF-3$%&"#nE,+ A3'>T"# A3,.!"# -> -/"F Q A"$8>"# ('G +>BTB 2F C+G c.!P6

5'3 .>36 .>P6j2T! Q0#23=/ -#2PYB -.B"., c.!L:

Stability —«dI²Ýô«


37

*.-(/:

;-./ #:: +CnEn,

, U3C ('G +>BT3 J I7.=n -?6 Q0123&$ 4 k!; H.D33 52(linear).

"#$% &'( $)*+: +nnkE ,

, U3k > 1 5j8_@ $%&"# F>! -; H.D36(exponential).

FGF QN.%&l" 5%&"# F>!"# 9!1B -/>3 J C+.Ge -> H/ A>3) -F/B .>+!+,C

A"F7D> 0.>F>G -F/B fg.B!"# -?6 C234 . _*# F>!"# .>@F Qk7!1B 9136 N.%&l" 58"# -* c":F, +nk X3D" (B, C237/ kB>3) m7 Bn C234 "# k!?6 5".B".7F Q.378!

$%&"# X1, -G 2Y!"# R47 A"F7D> 23n fg.B! ("; E+W3+ . F -?6 #:L ('G 0N.!7B"# A3>d2#F&"#5 3 $%&"# F>! -F/6-F/B 0.3%& .P3 F>! -F/3 5B"# c'B .>!37 QC2DB8>

C2DB8> 23n -F/B 0.3j8_@ .P36 $%&"#)#H/\"# 2Y!.(

!"#(8-1):

! A3"#FB>"# 9.8, 9F'%>"# -@ R2S- .ny VF2S"# A"+.T> X#+&B8.7:

1101.100 % #

ny

nnyy

,U3: 01.0,110## yy

#FL C.%T>"# VF2S"# A"+.T>" X.T"# H,":

wÝ_« uLM�«

Exponential Growth

En = k

nc

wÒD)« uLM�«

Linear Growth

En = Cnc

En

n

1 2 3 4 5 6 7 8

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

38

( )nn

nccy

1001.100.

21%#

, -> H/ U321, cc I7.=E2.3B&# .7J# -3%2\"# VD,3 E:"# H,"#FB -33g#+

01.0,1 10 ## yy FL: ( )nny

1001#

) E@1,0 21 ## cc .(

FBG# -/>3 2334B" 0.g6./> N.%&*# UF+, 2.7"# H,"#&k78,! E:"# o. .

HT1 FL #:L 23=$BF01 /c E@ Q@ - c.!L 0#+,)FLFnc 100.1 ( ("; kB6.[; 913

ny . +,"# -@ m[#F"# ->Fnc 100.1 C+.3d a> AG287 kB>3) +#+dBn a'B73 -@ (";

(swamp) 2&Z# +,"#( )nc100

1.2

.

6[26 #:; <=>!A>3) 56 U+, 0.3g#+B7# 0$%& -@ .0y I,7 @ U3,7:

000001.110 6

0

*

0#%# yy

7 I'Y .>!31y -@ E@ Q 5L .>/: 01.0

1

*

1## yy

6 A>3) -?2y m7 B: 000099.0*

1#y

+7-> 0J 42

10100

12

#0123

45#y.

F -> H/" A78!".7 $%&"# +#+d343 , yy 'G H ,! -@ (";A>3) (:

0001.1*

5 6y

7-> 0J+ ( ) 105

10100

15

##y.

F+,"# +F1F" ( )nc 100.1 j8_@ 0#F>! $%&"# F>! (>83 0.3.


39

@ 237.TB"# X3) 56 578! $%& 27/@ A>3)F $%& 27/@ A>3) +1F(f) A3".B"#U3,: 00.4,00.3,00.2 ### zyx

I#234B>"# K:L a3>1F -@ E@ Q -332\G -3>)2 ("; A72D>:

maxmax

2

max105.0 zyx 7#7#!#7

)i( zyxf %# 3

)ii( z

yxf .#

)iii( 40

sin.y

xf #

"237TB"# A>3) 9.8,( )612 #f -G R3FTB".72 A3732DB"# A>3D".71.4 RFT! -@ .!!/>3 Q 56 F@ V7.8"# 237TB"# 56 .>; A3732DB"# A>3D"# K:P7

A3".B"# Ag6./>"# 237.TB"# +,@:

( )612

11

%#f ,

( ) ( )33

223

13,2232

%# # ff ,

27099

15,270994

%# # ff

6p A13B! H[6@ 5%T3 237TB E$

)67"8(: a[x -> 0J+72 $%&"# +1F@F Q237TB H/ 56f7 $%&"# A"J+7x7

$%&'" A>3) H)@ 5%T3 E:"# 237TB".6 QA"., H/ 56f7 237TB H[6@ FL.(

98,_B .>+!G 578! $%& 27/@ A>3) 98,#1p A)<T"# ->:

)4.1(21 21## kvpvp kk

(1-1) s¹d9

(2-1) s¹d9

(3-1) s¹d9

‰Ë_« qBH�« �UM¹d9

‰Ë_« qBH�« �UM¹d9 ∫‰Ë_« qBH�«1

40

!432/1 "#f !"#$%&'()3 *+",#+ *!'!-.$ *+!.' . *%&/%0 123 45 678%0 (!' (-&9) -:&'+%0 ;&</%0

-!'#$%0 023 45 !"#$%&' ($<0 *%&/ 45" =5&>+%0 -!'#$%0 ?0@835697 $#g.

0 45 678 -'>A ;</y B'<C/ 02Dy *9E#%0 (+422 tzxy # (A ?,) 02D3,2,1 ### tzx 45 F+G#%0 H&78I0 (A"x, z, t ;!$-$%0 F,) 430.1, 0.1, 0.2.

) ( A 45 678 -'>A *+!9 @J"x (A ?,) 02Dx ;</C$<*9E#%0 (+

2.. vuxy #

!"# 1.00.1 %#y

1.00.2 %#u

2.00.3 %#v

)$ ( A>6' ?3&<! B0-!K$+%0 L' 45 678%0 (+ 7<9 -xM 02&+%" N

O(!$!OP (!$!78 (!$%@&#+ (+ *O">+%0 *!%&$%0 *)"+J+%0 &O!@% (A -Q:

12

2

#"

#"

byx

ayx

/R!:

3

1

10500.3

105000.2

$

$

&%#

&%#

b

a

5 ;&</ &S' T!7$<O 4$%0 *9@%0 U,'$ ?>x + y 678 -'>A V"&<! ?> LA M*%0@%0 45 4'<O 678 -'>A": f(x , y) ' x + y M

D45 F+G#%0 *!'<O%0 H&78I0 (A ?,) 02 221 ,, pvv ;!$-$%0 F,) 43: 0.7 %, 0.75 %, 1 %

!I0 @/A @!--!'#$%0 *+!9 ;&</ W&8:

(5-1) s¹d9

(6-1) s¹d9

(7-1) s¹d9

(4-1) s¹d9


41

O (A -Qf(x) = cos x .-",!&$ *!-GO ?@8$<0:

! ! !xRxPxf nn "#

! ! ! ! !

!

! ! !

! ! !

! ! ! ! !xxxxfn

xxxR

xfn

xx

xfxx

xfxxxfxP

n

n

n

n

n

n

,;.!1

,!

...

!2

0

1

1

0

0

0

0

2

0

000

("

$#

$"

"))$

")$"#

"

"

**

% *+!9 ;&</f(x)% F,)I0 @/%0" N X&7$9Y0 678) @/,% F,)I0 @/%0 LA 49&'%0nR remainder term ( (">$ &+@O):

0,001.0,20### xxn

O *%Z0 45 @0@)I0 (A -Q(machine numbers) *!-:#%0 *K![%0 45 \]+C$ _-]!#C+%0(normalized):

ki

d

dddd

i

n

k

...,,3,2

,90

,91;10....(*)121

#

++

++&%

/ R!7677,6 ++$# nk a&'!-.$) BYP 45 \&b/%0 "3 &+>(IBM.

;J"+ 4.!./ @@) LA -!!#$ (>+!y *K![%0 45 Tc"!%: n

kkk dddddy 10....... 2121 &# ""

" @@#%0 (A &Oc-5 02D d%2y *%e% L@@#%0 V@+%0 45 T.!.

5 *+f&#%0 *7.O%0 *K![ F%D \"[",,(*) (floating-point form) - 4$%0" g+-%&' &S% g+-O<f1(y) - g,! @O) L-:#%0 HgJ%0 h&.!D ?k d%2 (>+!" NL-:) ?9-(!.!-7 @/6':

!"# : $%&'((chopping):

?&9-I0 h2/'...21 "" kk dd F,) \"[/,%:

! n

kdddyf 10....1

21&#

(8-1) s¹d9

(9-1) s¹d9

‰Ë_« qBH�« �UM¹d9 ∫‰Ë_« qBH�«1

42

!"#!: $"%&'()(rounding):

!"#$% !1105 "#$ kn !"y #$ %&'(#! )*+!, -.:

! n

kyf 10....1

21"# $$$

/012 ,3" 45.67k = 5 89::;!, %9.6< 8=7e,% >#9 162 8&29:

?)* <@9A+

% &&'".31415 &&'".31416

e &&'".27182 &&'".27183

) ( 2 >BC0!, D*E!, :( F69? G#B< -(relative error bound): !y

yfy 1(

7 @9A+<!,& )*+!, ><!1( 86 %2 >) %16;<C, :0$k %9.6<!, >7 HAI$ -?A(J.

) !( 0 K&6A!, 8L MAN), *, (, + @AO!,& PA*!,& )6Q!, /19#6$ !" A9I< F6C+!,&– @9<A<!, #$– ::;B @1C(!, -<9 R9( S F!T, >7k -1?AU, 86 S>#9 162:

! ! ! ! ! ! ! ! ! ! ! ! .

,

,

,

yfxffyx

yfxffyx

yfxffyx

yfxffyx

)#)

"#*

(#(

,#+

L0 :Q& 8L MANB F+B1C!, /19#6;!, 86 F9#6$ %2 FQ9<,3/1#x 7/5#y

-1?AL FC6E :;B )*+!1B -<9 @1C(!, 8L&(5-digit chopping) %2 >7 :Q&L& S>BC0!, D*E!,& V#*6!, D*E!, F69? F!1(.

) "( 08L MAN410111111.,9.98765,714251. ("### wvu . %2 FQ9<0 :Q&L, /19#6;!, 86 >7 >BC0!, D*E!,& V#*6!, D*E!, @1C( )6 SF9!1<!, F9B1C(!

WX1<0!, #$ @Y+$& SF!1( %2:

! ! vuyvu

wuyuy

*(+

)((

,

,

%&'(#! F9&0;6!, -1?AU, 86 ::;! @9A+<!, Z9A;< >7 >BC0!, D*E!, -:E<C9-&[N6!, ,3[! F#'<6 \A&' #$.

(10-1) s¹d9


43

.

!"#$:

::;!, 8" %1+9p* @YA]+9p ::;!t 86%!&'#()* +,-".*significant (or

figures) digits /012 ,3" St *AI!, V+(9 @!1C A9_ ::$ AB2L >`:

t

p

pp("-

(105

*

)L ( -9? a:6 :Q&Lp* ::;!, F9&0;6 -1?AL F;BAU @YA]+< ><!,:

(i) 1000 (iii) 9990

(ii) 5000 (iv) 10,000

)@ ( H&#$ :( Ab'L @C(,(least upper bound) A,:+6#!*pp( ] K6A9 \:1$& K6A!1B c!*pp( max[ VN<< 16:0$p* )6p F9&0;6 -1?AL F;BAU d!3& S

-9? 86 F69? %2!p F9!1<!,:

P = .1, .5, 100, 1000, 5000, 9990, 10000

-9? %&:Q 86&*pp( p, max 9B*!, e(f ::;! @9A+<!, -&[N6! F#'<6!, F;F9&0;6!, -1?AU, 86.

%,&:!, &L A9:1+6!, M;B @1C( :0$ F9&0;6 -1?AL 8,:+7 @0Q< 8266!, 86A,:+6!, ,3` 1[06 @C(9 ><!, F?5;!, &L F!,:!, 8&01? F_19' \:1$=B d!3& .

8L MAN0 S,3` g9O&<!&:

! 149.36 23 (,(# xxxxf

)L ( F69? @C(,f(x) 16:0$x = 4.71 AL F.5.! /1B1C(!, 8&2< R9(B S -1?) \A6@9A+<!1B aAEL& )*+!1B(>BC0!, D*E!, F69? @C(,& S.

)@ ( hKQ!, %( :$L)L ( F_19' \:1$" :;Bf(x) \A&'!, >7 @<2< R9(B:

! ! ! .149 - x 3 x 6-x xf ,# FQ9<0 )6 8A1?&)L.(

F9!1<!, /1B1C(!, W<10 :Q&L:

(i) BO!1B*. (ii) -1?AL F.5.! @9A+<!1B .>BC0!, D*E!, ::( -..

(11-1) s¹d9

(12-1) s¹d9

‰Ë_« qBH�« �UM¹d9 ∫‰Ë_« qBH�«1

44

)L( 14.1+.0981

)@( (164. + .913) – (143. + 21.0)

)i( .0218 × 179

):( (164. – 143.) + (.913 – 21.0)

j:1;BL -X1? /59*<C6 HK,&<63 S-C4 S-C5-C-C @A?U FC1+6 S . :&:(!, >` 16 >#NC!,& 19#;!,(upper and lower bounds) Jc(*C F(1C6& c6Q( 86 %2!

:,:$U, F9!,&<6 :9!&< 8269:

. / . / ...,811,

271,

91,

31,1

31 0#nn

p

F9A9A2<!, F?5;!, -,:E<C1B:

(*)3

1021 ((

(#nnnppp

Z9A;< )6:

(**)31,1

10## pp

F9!,&<6!, 8L MAN0 h1*EU, &60 >7 @9A+<!, A.L 819B!. /np 86 @C(<C

F?5;!,(*) -1?AL FC6E! @9A+<!, )6 . -9? :Q&Lnp <(8#n -9+!1B 1[0A1?& S

F(9('!,nn

p31# )9AC!, :9,K<!, e(f&)>YCU, &60!, ( \:19K )6 D*E#!n.

%/01(:

V&AN!, F!:1;6! -1;!, %(!,)(* &`: ! nn

nccp 3

1

.231 ,#

R9(21 , cc 899X,:<Bf, 89*AI!, V+(9 H3!, %(!, 8=7 k-]. 86& S899+9+( 89::$ HL

(**) &`: !nnp

31#

8L HL: 0,1 21 ## cc.

(13-1) s¹d9

(14-1) s¹d9


45

:,:$U, F9!,&<6 :9!&< 8269. / . /nn

p31# F+B1C!, F!DC6!, >7 \A&236!,)1-14 (

Z9A;<B10 #p-. S:

,...2,1;31

1##

(npp

nn

@C(,np -9+#!8,...,2,1#n -1?AL FC6E! /1B1C(!, h,AQ" )6 WX1<0!, 8A1?& S!, e(f& F(9('!, WX1<0!, )6& F+B1C!, F!DC6!, WX1<0 )6@9A+<!, D*E! >*E!, &60.

:;B )*+!1B -<< /1B1C(!, 8L MANB3 86 %2 -9? >` 16 S-1?ALQ D, E,

F9!1<!, /1B1C(!, F#C#C h,AQ" :;B: A = 1./3.

B = 3./4. - A

C = 1./2. - A

D = (4 × B – 2 × A) -1

E = 3 × C - 1./2.

Q = E/D

1C( 8269F9:&:(!, @:

! 27279 23 (,(# xxxxp

F#E,:<6!, Fb9'!, -,:E<C1B S \h1N2B(nested form):

! ! ! 27279 (,(# xxxxp

-9? @C(,p(x) *1+0!, :0$: x = 2.95, 2.99, 3.01, 3.05

:;B )l*+!1B -<< /1B1C(!, 8L MANB3 -1l?AL . 1[9#$ %'(< ><!, /1B1Qm, 8A1?&SF*&BO6!, /1B1Qm1B Fb9'!, 86 1[9#$ %'(< ><!,& ! !33(# xxp.

F9:&:(!, F69? @C(, -1?AL F.5.! @9A+<!1B -<< /1B1C(!, 8L MANB:

! 133 23 (,(# xxxxp

(18-1) s¹d9

(17-1) s¹d9

(16-1) s¹d9

‰Ë_« qBH�« �UM¹d9 ∫‰Ë_« qBH�«1

46

-9+!, :0$: x = .9, .99, 1.01, 1.1

Fb9'!, -,:E<C1B F!&[CB 1[9#$ %'(< ><!, F*&BO6!, FB&QU1B WX1<0!, 8A1?& ! !31(# xxp.

%/01(:

F#E,:<6!, Fb9'!, /1B1C(!, >7 -:E<C, %O7L \h1N2 #$ %&'(#!:

! ! ! 133 (,(# xxxxp

F69? @1C( 8269 Z92 8Y9]B111,

(#xx

y F?:B -9+! FBC0!1B Sx \A9B2!,

111x.

F9<T, %,&:!, 86 %2!:

)i( F69? @C(,f(x) F69+! F#B1+6!,x -<< /1B1C(!, %2 8L MANB S\1*;6!, )*+!1B(chopping) :;B3 -1?AL.

)ii( C(6!, F69+!, 8A1?F*&BO6!, FB1Qm1B FB& 4,A9B2 D*E!, 812 ,316! gYO&& S n19BC0.

)iii( F69? @1C(! F?: A.2L F+9A* PA<?,fV:L F+9A*!, j3` :;< ,316! 8Y9B& S.

) L( ! 00495.0, #(#(

xxeexf

xx

) @( ! radiansxxxxxf 01.0,sin

3#(#

) i( ! 0.100,12 #((# xxxxf

) :( ! 002.0,cos #(#(

xxexxf

x

)l`( ! 005.0,sin2

#(# xx

xxexfx

)&( ! 50,11 22 #((,# xxxxf

(19-1) s¹d9

(20-1) s¹d9


47

(21-1) s¹d9

!"# $%&' !xexfx 1"# ()*x = .0195 $"+,-./0 1-- 2/0/%&.' 34 5+60

1/#+4 787. .! 9"-).' 3+/#: ;"<.' =>? @AB- '>/!. CD:: EA:0D!.' $':9.' F 1",. 0%)./0 ,"#( +"G H/!"# H/!:!*x I"&1$$x .'(J. K+L4 ;"< A*4: E f

1",. 0%)./0 M(4 NO/-) P/A*Q CJ<-x $/%&. ;"<.' =>? 1(L-%': ER+";<.'f(x) ()*x = 0.0195.

‰Ë_« qBH�« �UM¹d9 ∫‰Ë_« qBH�«1

w½U¦�« qBH�«

�ôœUF*« qŠ

Solution of Equations

�ôœUF*« —Ëcł

Wł—b�« WOŽUÐd�«Ë WO³OFJ²�« �ôœUF*« º

d½—u¼ WI¹dÞ º

W¹d¹dJ²�« ‚dD�«

N-R Êu��«— ≠ sðuO½ WI¹dÞ

nOBM²�« WI¹dÞ

©WFÐU²²*« �U³¹dI²�«® W²ÐU¦�« WDIM�« WI¹dÞ

W²ÐU¦�« WDIM�« WI¹dÞ w� QD)« d¹bIð º

WOD)« dOž WO½ü« �ôœUF*« º

w½U¦�« qBH�« �UM¹d9 º

51

w½U¦�« qBH�«

�ôœUF*« qŠSolution of Equations

©Roots of Equations) �ôœUF*« —Ëcł

!"#$ %& '(!)*"a" !"#(root) +!# (",-#f(x) = 0 ./ 01!2(zero) +!#$ -#f $3& %(!!4f(a) = 0 . 56!!7#(6 +!!# (",#$ 8!!*)9: ; (!!< %!!4,* = +!!* "#$ >(!!)*65:#$ ?!!@ A!!B. =&

exactly C. D6!!!E6 ; / !!!9,#$ +"!!!E#$/ D!!!*0):#$ F(!!!5(limited capacity). 'G !!!H"I*@ J#3!!!#/?-* (,4 K62*# 03L#$ M*0"::

!"#$ %& '(!)*"a " +!# (",-# 03!Lf(x) = 0 %(!4 $3& ! "#af N!*9" >/(!1: O!5", P/,!E,(given tolerance) .] ;0!!:@ !9: +!B*(6:,#$ %. Q!!9=(interval) %!!, R= !6

)B +5(point).[

R(5/6!7, RS9 +# (",#$ '9 O#& T U: ;0V(6, R(W*2 X C:E: ?:#$ +@/0",#$ 805#$/ +!!2(C >=(!!9 0!!6:": >= (!!",#$ %!!, R$ !!L / !!9, !!< O!!-< =& ; (!!< 8!!G65: = 80!!5 ?!!Y

) RSZ, +Z#(Z#$ +L0 #$/ +*B(Z#$ +L0 #$ >= (",4 .( '!21#$ $3!Y ?!@ O#(!": [$ %3\!6 '/(B:BE/ +!!E$0 +!!*0$04: /. +!!*0*04: R(!!]05 O,!!E: ?!!:#$/ +!!* "#$ 80!!5#$ ^!!"6(iterative methods)

0/3!L (!L*& O!#& ; (!< T U!: 80!5 ?!Y/ _>(!)*65:#$ %, È$/ '(L, (a#/ >= (",#$ '9#+5/6!!7,#$ 0/3!!L#$ b3!!Y O!!#& T U!!: ;0 (!!B %(!!*9. ?!!@/ _+5/6!!7,#$ 0/3!!L#$ %!!, R$ !!L +!!6*0] .

0", 0:1: +*0*04:#$ 805#$ b3Y +*c$ :6$ +*6*0): +,*] +@0x 5/6!7,#$ 03!L-#" _+!# (",-# X$ C:E$ 004I* XZ)8*65: ( +!"6(::, X*!] O!-< '/!29-# >$0!, ; < +*0*04:#$ +)*05#$ +W*2

+!#/6), +!,*] O!#& '2B %. O#& 5/67,#$ 03L#$ O#& D0].(until satisfied) 80!5#$ b3!Y/ _ U!!!: ; !!!9, >(!!!*-,< !!!"I: T– +!!!#$ #$ O!!!-< +!!!2(C 5/0!!!V >!!!9:f +!!!*6*0):#$ X*!!!)#$ O!!!-</

+*c$ :6=$– ; *L >(6*0): O#&(good approximations) +5/67,#$ 0/3L-#.

52

'!!,$/< ; !!< O!!-< !!,:"* >= (!!",#$ '!!9# 80!!5#$ b3!!Y %!!, +!!B*", +!!)*05 0(!!*:C$/!7"6 %**": X. +B*", +# (", 0/3L '4 %**":# d(:9B (B4 $3& (, RSZ, (aB, (!,/ _ 5!)@ (a

+!!!!!*)*)9 0/3!!!!!L#$ >!!!!!B(4 $3&(real) +!!!!!640, X.) +!!!!!* e H)I<complex( +5*!!!!!E6 _(simple) X. ; !!!":,) +1<(!!!7:,multiple(*!!!] (B* !!!# %(!!!4 $3& (!!!,/ _= X. 0/3!!!L-# +!!!*c$ :6$ +!!!*6*0): X _

$34Y/.

+*6*"4:#$ +# (",#$ (B* # %. 01B

023 $%%% cbxaxx

AB. X/-",#$ %,:

i( >B(4 $3&c > 0 D#(E 03L L/* AB\@r.

ii( & >B(4 $3c < 0 DL/, 03L L/* AB\@r.

0/3!!!L#$ `!!!]$/,6 8!!!-":: +!!!,(< >(!!!*0QB ; !!!< !!!L/:/(localization of roots)

# (",6 +2(C#$;(5", + .+*#(:#$ +*0QB#$ >(6Z& %4,* RSZ,:

$% &':

03L Tfz +# (",-#:

! 0...22

11 $%%%%& ''

nnnn azazazzf

50V#$ 8)9:*:

iaz max1%(

+*6*"4:#$ +# (",#$ 03L O-< +*0QB#$ b3Y 50V 8*65:6r %. g:B:EB _:

!cbar ,,max1%(

Cubic and Quartic Equations Wł—b�« WOŽUÐd�«Ë WO³OFJ²�« �ôœUF*«

©Cubic equation) WO³OFJ²�« W�œUF*« ∫ÎôË√


53

!"""#$#$% &'(""") *"""+,& -"""./ !0#1"""2 34 54 6171"""%8 6'"""9: ;""": <"""$# &1"""%&7(finite

interval)92!=7 >0 -.?@2& A#!B02& C'%D9 6'9?.2 ;2!9982& E#F(lower limit) -"./4 -"2G(upward) !"""""#.H2& A"""""#!B02& 74 >(upper limit) *?"""""@4 -"""""2G(downward) . !81"""""0/7

A""#:!I A""+1= '("")2& -""./ *""F%0(sufficient accuracy) *""#J0(remove) "")2& &(""K( ');2!92& 10=2& ;: '0'7K A$#'L= '7()2& A2&JG M7N78 !N#4 'O0& ( *"F%0 ;2!"92!=7 >

!B.% *B@# A#0!P2& A)'12& 38 A21!H8 -./.

AH=&'2& A)'12& A21!H8 !0#12 34 Q'?0:

0234 !!!! dcxbxaxx

A2&JG 3I8# Q#7H92& R&1S9@!= ;=#HI92& 1%2&:

3ayx "

A21!H82& -./ *F%0 ;2!92!=7:

024 !!! sryqyy

T#%:

2

83 abq "

82

3aabcr !"

2563

164

42 abaacds "!"

!"""""B2 A"""""21!H82!: >A"""""#$#$% UV8!"""""H82& 3, &'"""""O074 3&'(""""") 74 A"""""#$#$% '7(""""")$#$%A""#$#$% '7(") !""B2 W#""2 74 3!"# . 3#.8!""/ -""2G A"XJ)92& !""00I8# Y""0Z: *!"% 54 -""./7

A#2!92& 6'7F2& ;: 3##H#='9:

0212 22 #$%&

'( !"#

$%&

'( !! mkyykyy

A""#0!P2& A"")'12& 3""8 3#921!""H8 *""% -""2G A2[""@82& *7""%99 ;2!""92!=7 . \(""K E'""HD97

U'!I#1 A$#'L R@!= !0!#%4 A$#'L2& (Descarte).

(Quartic “fourth degree” equation) Wł—b�« WOŽUÐ— W�œUF*« ∫ÎUO½UŁ

�ôœUF*« qŠ ∫w½U¦�« qBH�«2

54

171"""% 6'"""#PI A"""8#+ 1!""")#] A"""$#'L2& \("""K R1S9"""@9p !B9$9"""^8 74p) A"""L$0 1"""0/ A""2&J] R1S9""@9 !""8I > A"0#H8(removal) A""#'=)2& U_1!""H82& 3"8 '7("")2&(algebraic

equations) ("""X0#% E'""HD97 >– !"""N#4– A"""#=#I'92& A8"""@$2& A"""$#'L R"""@!=(synthetic

division).

!" : #!#$ %&'() *''+,(Evaluating a polynomial)

34 Q'?0: * + 321023 axaxaxaxp !!!

A8#+ 1!)#]p *P8 AL$0 10/x = z !00I8#– A8#+ Q#7H9=x – `9I0 34:

* +

* +* +3210

321

2

32

2

1

3

0

0

azazaza

azazaza

azazazazp

!!!

!#$%&

'( !!

!!!

A#171%2& A8#$2 6'#S,& Aa#F2& \(K7p(z) '0'7"K A"$#'L A#8J'&7S W!@4 ;KK `!@%2A8#$2& \( .A#2!92& A$#'L2!= 6'#S,& Aa#F2& \(K `!@% 3I8#7:

z 3a

2a

1a

0a

* +* +zazaza210

!! * +zaza10

! za0

* +* +3210azazaza !!!* +

210azaza !!

10aza ! 0

a

UV8!""H82& `""9ID9 T""#%310

...,,, aaa 54 -""./ *""F%07 > *7,& 'L""@2& ;"": ;": T"2!P2& 'L"@2& ;": b=!"@2& 1"%2& '"N= ;0!"P2& 'L@2& ;: 1%z *"F%0 !"80#= >

""""%2& <""""8)= T""""2!P2& 'L""""@2& ;"""": 1""""% 54 -""""./ *7,& 3#'L""""@2& ;"""": Y""""2 3#.=!""""$82& 3#1;0!P2&7 . A8#$2& 37I97p(z) T2!P2& 'L@2& ;: !B#./ *F%0 A8#+ 'Sc ;K.

A#2!92& 6'F9S82& 6'7F2& ;: b=@ !8 A=!9I !00I8#7: z

3a

2a

1a

0a

zb2 zb1 zb0

* +3bzp 3b2b

1b 0b

(Horner’s scheme) d½—u¼ WI¹dÞ


55

T#%:

3,2,1;1

00

!

" izbab

ab

iii

0I8#70& A)#902& ;: &(K R#8H9 3d& !A#2!92:

#!#$ %&'() *''+,- &.&!/ 0'12&3!4: Horner’s algorithm for evaluating a polynomial of degree n:

34 Q'?0p A)'1 38 171% 6'#PIn!B9a#F7 >:

* +nn

nn axaxaxaxp !!!! "

"

1

1

10...

A8#+ 1!)#]p AL$0 10/zA#2!92& A#8J'&7S2& <=9& >:

* +n

iii

bzp

nizbab

ab

!

" ,,2,1;1

00

156( (1-2):

34 Q'?0* + 72 3 !! xxxp . A8#+ 1!)#] '0'7K A$#'L b=Lp(8).

5$-3: 2 0 1 7 8

16 128 1032

2 16 129 1039

34 54: p(8) = 1039

'^!=82& Q#7H92!= A)#902& \(K 38 b$%92& 3I8#7:

* + 103978838 3 !!, p

Y""04 e""N&72& 3""8– 2 '0'7""K A""$#'L b""#=L9 1""0/ 171""% 6'""#PI R#""#$9– !01"")7 &(G A8#+ 34nb ) '"Sc ;"K7b !"B98#+ `"@%0 ( 34 54 >'?"F0 nb 3Z":z &'(") 37"I9 A21!H8.2p(x) = 0 171%2& 6'#PI2 &'?F 37I9 54 >p3, C2(7 >:

* + 0 nbzp


56

56(1(2-2):

'7"K A"$#'L R&1S9"@!= U"=P& 34 '0x = 2 A"21!H82& '7(") 3"8 '(")p(x) = 0

T#%:

5$-3: 1 -3 4 2 -10 -4 2

2 -2 4 12 4

1 -1 2 6 2 0

34 54p(2) = 0 34 54 >x = 2 A21!H8.2 '()p(x) = 0.

07$81:

34 U!=PG 3I8#: * +

2622

234 !!!" "

xxxxx

xp

171""%2& 6'""#PI UV8!""H8 T""#%)A""H=&'2& A"")'12& 3""8 (!02& A8""@$2& A""#.8/ 3""8 A"")9T2!P2& 'L@2& ;: A)9!02& R!+',& ;K . 3I27 ;.# !8#: Y#LH0: A)#902& \(K U!=PG !847

!B8#8H9 1H=.

0?34 Q':

* +nn

nn axaxaxaxp !!!! ""

1

1

10...

0'&7.(2-2):

12

2

1

1

0

1

1

10

""""

#$%

&'(

!!!! "

!!!!

""

"

nn

nn bxbxbxb

zx

axaxaxa

zx

xpnn

nn

* + 410243 2345 ""!!" xxxxxxp

©WO³O�d²�« WL�I�«® W¹d³'« �ôœUF*« s� —Ëc'« W�«“≈ ∫ÎUO½UŁ

Removal of roots from algebraic equations (synthetic division)


57

UV8!""""H82& T""""#%sbi ' UV8!""""H82& A""""2_1= -""""LH9sai ' U!""""+VH2& ;"""": !""""8I34 54 > !$=!@ 6'7I(82&:

1,,2,1;1

00

" !

" nizbab

ab

iii

96/&:-3:

* + * +12

2

1

1

0

1

1

10

""""

""

!!!!"

!!!!

nn

nn

nn

nn

bxbxbxbzx

axaxaxa

f7+ UV8!H8 A0'!$8=7x 34 1)0 >A21!H82& \(K ;:'L ;: A?.9S82&: nx : 0

a = 0b - 1"nx :

1a = zbb

01" .- zbab 011 !

2"nx : 2a = zbb 12 " .- zbab 122 !

..... ..... .....

x 1"na = zbb nn 21 "" " .- zbab nnn 211 """ !

56(1(3-2) :

A21!H82&: * + 0410243 2345 ""!!" xxxxxxp

'("") !""B2x = 2 '""Sc '("")7 1" x . A""#=#I'92& A8""@$2!= 3#'("")2& 3#(""K *J4) 54'0'7K A$#'L=.(

5$-3: 1 -3 4 2 -10 -4 2

2 -2 4 12 4

1 -1 2 6 2 0 -1

-1 2 -4 -2

1 -2 4 2 0

34 54: * + * + * + * +24212 23 !!"!" xxxxxxp


58

07$81:

&1S9@!= !B.%2 A0#H8 A2[@8 1&1/G *gB@D9 ;92& U&7LS2& 38 E hF i7 >`@!%2& R A"""#1&19'& j#"""F 74 U!"""+VH= A2["""@82& *"""% A"""$#'L) A"""#1 h7i/recursive (– U!"""+VHI

'0'7"K A"#8J'&7S– 6'#a""F A"/78)8= U!=!""@%2& 3"8 6'"#=I A""#8I E"F7 3"I8# T""#%A#1&19'_& j#F2& 38.

& 3I8# >A"0#H8 AL$0 10/ 171% 6'#PI A$9^8 A8#+ 1!)#] '0'7K A$#'L R&1S9@A#2!92& A#'O02& b#=L9= C2(7:

0'&7.(3-2) :

34 Q'?0:

* +nn

nn axaxaxaxp !!!! ""

1

1

10

1,,2,1;.

1,,2,1;.

1

00

1

00

" !

" !

"

"

nizcbc

bc

nizbab

ab

iii

iii

171%2& 6'#PI A$9^8 A8#+ `!@% 3I8#:p AL$02& 10/x = z A+VH2!=:

* +1" )

nczp

)O%_ 34* +nbzp \'I( b=@ !8I.(

96/&:-3:

A$=!""""@2& A""""#'O02& 3!""""K'== Y#=""""^ 3!""""K'=2&)A""""#'O02-2 ] ( A""""0'!$8 b""""#'L 3""""/ f7+ UV8!H8x A?.9S82&.(

œËbŠ …dO¦� WI²A� rOOIð ∫ÎU¦�UŁ(Evaluating the derivative of a polynomial)


59

5$-3: 1 -3 4 2 -10 -4 2

2 -2 4 12 4

1 -1 2 6 2 0 2

2 2 8 28

1 1 4 14 30

;+$,-3:

* +* + 301084896802

10412125 234

"!!" )

"!!" )

p

xxxxxp

07$81(1) :

-./ *7F%.2 f'S4 6'8 A$#'L2& b#=L9 ''I# 34 e#%F2& 38 W#2* +2p )).

07$81 (2):

8J'&7S2 f'S4 6'8 -2!H9 k& 3(Z= 17H0@ '0'7K A#– 397"#0 Aa#"F A@&'1 1H= U_1!"""H82& *"""%2– 2 '"""#PI R#"""#$92 '0'7"""K A"""#8J'&7S 61"""X!: f'"""06 ;""": !B9$9"""^87 171"""%2&

U_1!"H82& *"% *X!""@82 AH#'"@ lX!"90 -"./ *7"F%2&f(x) = 0 A""2&12& 37"I9 !81"0/f

171% 6'#PI.

56(1(4-2) :

34 Q'?0:

* + 410243 2345 ""!!" xxxxxxp

38 VI '0'7K A$#'L= 1)74* + * +2,2 pp ).


60

A"""""""""""21!H8.2 '(""""""""""") 1!""""""""""")#]f(x)=0 ] T"""""""""""#%f A#8!"""""""""""@98 74 A"""""""""""#'=) A"""""""""""2&1(transcendental) [– A""#'#'I92& b'"""L2& f1""%Z=– A""#X&19=& A"""#=#'$9 A""8#$= 41"""=00x R"""P

*N:4 A#2!998 U!=#'$9 1)70.

A""2&12& !""0.mP i8 &( n&7y=f(x) !""#0!#= 3""/ T""%=0 !""004 -""./ A2[""@82& Ao!#""F 3I8#"": > '7""%8 <""8 -""0%082& <L!""$9x. *1=9""@0 34 ;""K A""#2!92& b'""L2& ;"": A#""@!@,& 6'""I?2&7

'7"%8 <"8 Y"HL!$9 A"L$0 `!"@% *B"@# !="@!08 !8#$9"@8 !LS -0%082!=x . L"S2& &("K A""""L$02!= '""""8# R#$9""""@82&* +

nn yx , Y""""#2G !0.""""F7 `""""#'$9 '"""Sc *""""P89 ;""""92& \!"""")9& !""""847 > R#$9"""@82&A"""?.9S8 61"""#1/ b'"""L= \'!"""#9S& 3I8#""": . VP"""8: !"""9=!P !"""K!)9& 11"""%0 34 !"""00I8#

(fixed direction) <L!""""$.2 !""""#J&78 37""""I# 3[""""I(secant) 3#""""9L$0 *7[""""= '""""8# 5(""""2&* + * +

1100,,, yxyx W!88.2 74(tangent) A"L$0 *7[= '8# 5(2&* +

00, yx . !"00I8# !"8I

&'#a98 !K!)9& '!9S0 34(variable direction) 9# 3[I 5("2& <L!"$2& <8 R#$9@82& b=!L 3#9L$02!"""= '"""8#* + * +

nnnn yxyx ,,, 11 "" A"""L$02!= '"""8# 5("""2& W!"""882& <"""8 74 >* +nn yx ,

]*I""^2& '""O0& .[ J8'2!""= R#$9""@82& L""S2& *""#82 !""0J8' &(Z"":k U_!""89%_& !0#1""2 37""I#: >A#2!92& AH=',&:

) 4( U=!P2& <L!$2&(fixed secant)

01

01

xxyy

k ""

) `( U=!P2& W!882&(fixed tangent)

* +0xfk )

) p( '#a982& <L!$2&(variable secant)

] E"X&J2& <"N72& A"$#'L= A"$#'L2& \("K E'"H97(Regula falsi or false position) 74[LS2&7 A27!%82&[

1

1

"

"""

nn

nn

xxyy

k

) 1( '#a982& W!882&(variable tangent)

] 'H97 397#0 A$#'L= A$#'L2& \(K E- 37@:&'(Newton - Raphson)[

* +nxfk )

Iterative Methods W¹d¹dJ²�« ‚dD�«


61

)p ('#a982& <L!$2& \!)9_& )1 (\!)9_& '#a982& W!882&

EX&J2& <N72& A$#'L 37@:&' 397#0 A$#'L

(Regula falsi) (Newton – Raphson)

6'7F2& ;: A#'#'I9 Aa#F -./ *F%0 !00Z: <=',& U_!%2& 38 54 ;:7: * +

,2,1,0;1

" ! nkxf

xx n

nn

*#""F?92!= C""2( 3#=0""@ !""8I– ;"".# !""8#:– ,& A""$#'L.2 A=""@02!= 6'""#S) 397""#0 A""$#'L–

37@:&' (U_1!H82& *%2 !8&1S9@& b'L2& 'PI4 38 ;K7 >.

397""#0 A"""$#'L R""@!= !""N#4 A""$#'L2& \(""K E'""HD97 . A"""$#'L2& \(""B= A""21!H8 *""%27 6'7""F2& ;"": _74 A""21!H82& <""N0f(x)=0=#'$9 A""8#+ 1"")707 > A""#X&19=& A""#

0x 3""8 '$2!""=

e#%""""F2& '("""")2&/ )3#""""98#+ VP""""8 1"""")70 3[""""I b x , a x A""""2&12& 6'!""""^G '""""#a99f

)4 (\!)9_& U=!P2& <L!$2& )` (\!)9_& U=!P2& W!882&

Newton - Raphson (N-R) method Êu��«— sðuO½ WI¹dÞ


62

!!!!"#$"# %& '(!!!!)*+ ,-!!!!./"# +& 0"1!!!!2 3!!!!"4 0!!!!5+6 %!!!!6 167$!!!!89f %+!!!!.:; <!!!!=>6(?5"# %:!* @1!/A#+ b x , a x !! (1!>B8 C"?!"+ ,0x !DE86"# F!; !ED8 G&[a , b] .(

:"1>>6"# H1*:(D>"# $5+8 IJ ,,,,,, 1321 "nn xxxxx F<: 16.)I2("# (K8#.(

# $ # $

# $# $0

0

0

0tan

xfxf

xfxf

%!&'

&!!% (

# $# $0

0

0

01

xfxf

x

xx

%)!

&)!

LJ6"1*+ %M;: # $# $

# $# $

..........

2

2

23

1

1

12

xfxf

xx

xfxf

xx

%)!

%)!

%M; @16+69+: # $# $

,2,1,0;1

!%)!" nxfxf

xxn

n

nn


63

!"#(5-2) :

"$1/6<" @#(?5 :(N9 I1A(& /*(O $5+&104 !) xx %!6 0(D"1!*x = 2 C!"?+%>+:8 D:(E*.

$%&:

P(+="# F; "$1/6"# QR8:

# $ 0104 !))* xxxf

]%& KST: # $# $# $ 042

0101

0100

+!

,)!

,)!

f

f

f

& %:* (?5 1U" "$1/6"# %& Gx = 1, x = 2 6:D<" 0(A& %+.: %& QA+>8+x = 2.[

:V#$>*T# 6:D"# ?BW8:

0.20!x

"#$"# X?U" %>+:8 Y:= $15:Z+:

# $# $ 14

10

3

4

)!%

))!

xxf

xxxf

# $# $

14103

1410

3

4

3

4

1

)"!

))))!

%)!' "

n

n

n

nnn

n

n

n

xx

xxx

x

xfxf

nxx

# I:D"# 3<9 L=S8 , :"1>>6 H#(6 P$9 Y:="# X?7 [:*E>*+ :"1>":

28555845252.1

855585.1

85578.1

871.1

4

3

2

1

!

!

!

!

x

x

x

x

!!!!S\8! K!!!!D"# %&!%:>:"1!!!!>>6 %:!!!!>6:A %:!!!!* [(1!!!!)<" !!!!D<E6"# 6:1))

nnxx ]*!!!!=&

%!6 (Y=&4105.0 )- %O ,4

34105.080000004747. )-,!) xx ^!A+>8 C"?!"+

:(:(.>"# Y:="# [:*E> %9.


64

!"#(6-2) :

<" (?!!!5 (Y!!!=& %>+!!!:8 !!!D:(E* $!!!5+& !!!"$1/6:xe x sin!) I1!!!A(& !!!J\J" C!!!"?+ :(N9.

$%&:

%::8S86<" F*:(D> I2(*xyey x sin, !! ) FJ#$!S_" !:*:(D>"# !6:D"# %& $!58

F7 P1E/6"# "$1/6<" (?5 (Y=& L*1D> F>"# QE1D>"# ED8" F8:2"#:

# $# $

nx

nx

nn

x

x

xnexne

xx

xexf

xexf

x

cossin

cos

sin

6.0

1

0

))))!

))!%

)!

!

)

)

"

)

)

58853274.0

5885.0

6.0

2

1

0

!

!

!

x

x

x

%O :(:(.>"# Y:="# [:*E> %9 Â+>8+:

3

12105.0 )-,)*& xx

'($)#:

%!!6 FV1!!U8 T $$!!9 1!!U" P1!!E/6"# !!"$1/6"# %M!!; [*1!!2"# I!!2("# %!!6 ]!!R>: 1!!6.%6 0(D"1* QD> F>"# (+?5"#: ,3,2, ...


65

!"#(7-2) :

"$1/6"# (?5 :(N9 I1A(& >2" $5+&:5.0sin2.0 "! xx %>+:8 D:(E*.

$%&:

(!!:Y>> 3!!>6 (!!8" I:!!D"# '!!/* ':+!!/>* 1!!64 (?!!5<" !!:*:(D> !!6:A $1!!5:4 %!!.6: "#$"# P(1N4:

# $ 5.0sin2.0 ))! xxxf

%::8S86<" F*:(D> I2(* +& ,-./"# +& *"12 P(1N4 3"4 *5+6 P(1N4 %6:

5.0sin2.0, "!! xyxy

# QE1D>: %:& (8"%1:8S86".

1*:(D> G+12: (?5"# %& (8 %:>D:(E"# %6 G& a1*>M*+0.6.

# $# $ xxf

xxxf

cos2.01

5.0sin2.0

)!%

))!

# $# $

n

nnn

n

n

nn

xxx

x

xfxf

xx

cos2.015.0sin2.0

1

))))!

%)!"

7

233

6

122

1

105.0

00000004.0,61546816.0

105.00008.0,61546820.0

61629718.0

)

)

-,

!)*&!

-+/)*&!

!

xxx

xxx

x

"#$"# 1U:; %+.> F>"# HT1S"# F;f !D:(E I#$B>!2# %!.6: b!8M; , $+$!S P(:J. %!!!6 L!!!. !!!6:A 01!!!2S" (8(+!!!7# $ # $

nn xfxf %, - %>+!!!:8 Y:!!!= F!!!; %:>$+!!!5+6"#–

F"1>"# L1J6"# %6 %:*>: 16. ,(*.& 9(2* cV1>8"# 3<9 L+=S<" C"?+.

!"#(8-2) :

$5+& :(N9 I1A(& J\J"– %>+:8 D:(E*– "$1/6<" FD:DS"# (?5"#:

# $ 013 !))! xxxp

FV#$>*T# 0:(D>"1* @1V$>*6 3.10 !x


66

$%&: # $# $

# $# $ 0

12

345

012

345

"

%)!%)!

%)!

3.1

3.13.1

0

0

01

1

p

p

xpxp

xx

xpxp

xxn

n

nn

%6 L. 6:A 3<9 d# %?M* L=S82+# $ # $3.1p,3.1p % (8(+7 D:(E*. 1 0 -1 -1 1.3

1.3 1.69 0.897

1 1.3 0.69 -0.103 1.3

1.3 3.38

1 2.6 4.07

# $ # $325.10253071.03.1

07.4103.03.1

07.43.1,103.03.1

1/"!"!'

!%)!

x

pp

*+<E6"# A$"# 3"4 L=8" (B& P(6 %>+:8 Y:= [*E8+.

# $# $ 0

12

345

012

345

%)!%)!325.1

325.1325.1

1

1

12 p

p

xpxp

xx

%!!!6 L!!!. !!!6:A 3!!!<9 P(!!!6"# X?!!!7 L!!!=S8" (8(+!!!7 !!!D:(E I$B>!!!28 (!!!B& P(!!!6+# $ # $325.1,325.1 pp %.

1 0 -1 -1 1.325

1.325 1.755625 1.001203

1 1.325 0.755625 0.001203 1.325

1.325 3.511250

1 2.65 4.266875

# $# $

324718061.1

000281939.0325.1266875.4001203.0325.1

266875.41.325p

0.0012031.325p

2

!

)!)!'

!%

!

x

e>+2 L#+$"# %6 $:$/"# I:A 012S" :"f# H1*21S"# F; P(J.* %>+:8 D:(E I$B>LJ6 E:2*"#:

3, xx


67

!"#(9-2) :

%6 L. 6:A $15:Z %>+:8 D:(E I#$B>21* :(:(.> Y:= $5+&:

)& (a )0 (N a )g (a

1

$%&:

) &( '()8

# $

# $iii

i

iii

xaxx

xax

xx

axxf

ax

ax

"!

))!

!)!

!'

!

"

"

21

2

0

1

2

1

2

2

) 0( '()8

# $

0123

45 ")!

))!

!)!

!

!

"

)"

Ni

ii

Ni

Ni

ii

N

N

N

xaN

Nx

x

Nxax

xx

axxf

ax

ax

1

0

1

11

16$89 @\J6N = 3 – $15:Z G&3 a) :(:(.>"# Y:="# %+.>:

0123

45 "!0

123

45 "!" 231

2312

3 ii

i

ii x

axxax

x

) g( '()8

# $ 01

1

1

2

2

!)!

!'

!

axxf

ax

ax

# $iii

i

i

ii

axxx

xax

xx

121

2

1

1

2

1

"!

))!

"

"


68

!!!!:"1>>6"# !!!!:*:(D>"# I:!!!!D"# 0(1!!!!D> 1!!!!87 0(1!!!!D>"1* F!!!!8/8– 1!!!!U:<9 L!!!!=S8 F!!!!>"# :(:(.>"# Y:="1*– (?5<" E+*R6"# 6:D"# +S8.

%& '(!!!!)8 ,(?!!!!5<" E+*!!!!R6"# !!!!6:D"# F!!!!7 G& !!!!"$1/6<" E+*!!!!R6"# L!!!!S"#0)( !xf.

%& G&:

# $ 0! f

& '()8+%:

# $ 06% f

%& '()8 C"?.n7 :*:(D>"# 6:D"# F; WEB"# LJ6:nx:

)2(

)1(

11 7

7

)!'

)!

"" nn

nn

x

x

W!!EB"# X1!!5># 9(!!2 `$!!6 !!;(/6 0+!!<E6"#n7 9(!!2 `$!!6 G& ,()!!="# +!!S8 P$1!!:h Q!!6 W!!EB"# #?!!7 iA1!!8>n > F!!>"# !!D:(E"# X?!!7 Y:!!= (!!:(.> H#(!!6 $$!!9 L!!J6 .

W!!EB"# 1!!8:E/> !!A\9 3!!<9 L+!!=S"# L+1!!S8 9(!!2"# X?!!7 !!;(/6"+1"n7 W!!EB"# !!"T$* P(N1*6 b" [*12"#

n7 ) D*12"# j1EBO# @1R:& 16*(+

i7 k:Si < n.(

'*+(,(4-2) :

D:(E F;N-R:

2

1 .2 nn f

f7

7

012

345

012

345

" %

%%/

k!!!!!!!:S + ,(?!!!!!!!5<" E+*!!!!!!!R6"# !!!!!!!6:D"# F!!!!!!!7n7 !!!!!!!6:D"# F!!!!!!!; W!!!!!!!EB"# +!!!!!!!7 :*:(D>"#n

x.

-!.+/%&:

%:>D*12"# %:>A\/"# *1>. 188.6:(1) , (2) P(+="# F;:

11 "" "!

"!

nn

nn

x

x

7

7

Convergence of (N-R) method (N-R) WI¹dÞ »—UIð


69

%!!!6 L!!!. %!!!9 ':+/>"1!!!*+1

, "nnxx %>+!!!:8 Y:!!!= F!!!; %:!!!>A\/"# %:>1!!!7 %!!!6

:(:(.>"#:

# $# $nn

nn xfxf

xx %)!"1

3<9 L=S8:

012

345

012

345

" "%"

)"!"n

n

nn f

f

7 7

7 7 1

(*)1

012

345

012

345

" "%"

)!'n

n

nn f

f

7 7

77

%!!6 L!!. C!!)" (+<:1!!> C+!!.)6 I#$B>!!21*+ff %, L+!!S !!A\/"# X?!!7 F!!; !x ,+A F; G& WEB<" :$91=>n73<9 L=S8 ,:

# $ # $ # $ # $ "%%"%"!" 7 7 7 ffff nnn !2

2

# $ # $ # $ # $ "%%"%%"%!"% 7 7 7 ffff nnn !2

2

%& 16*+# $ 0! f ) (?5<" E+*R6"# 6:D"# F7(

%M;:

# $ # $

# $ # $889

:

;;<

="%

%%"%!"%

889

:

;;<

="%

%%"%!"

!"

#$%

!"

#$%

!"

#$%

!"

#$%

&&

'&'&

&&'&''&

f

fff

f

fff

nn

nnn

1

!21

(()

*

++,

-./

//01

(()

*

++,

-./

//0

(()

*

++,

-./

//.1

(()

*

++,

-./

//.

(()

*

++,

-./

//.1

./

.2

!"

#$%

!"

#$%

!"

#$%

!"

#$%

!"

#$%

!"

#$%

0

!"

#$%

!"

#$%

!"

#$%

!"

#$%

!"

#$%

!"

#$%

&&

''

&&

'&&''

&&

'&&''

'&'&

f

f

f

f

f

f

f

f

f

f

f

f

nn

nn

n

nn

nn

n

211

1!2

1

1!2

1

1

!"#$%!& '( )*+$,!#-+(*) ./0 1234:


70

./

//1

(()

*

++,

-./

//001

!"

#$%

!"

#$%

.

!"

#$%

!"

#$%

.

2

1

1

2

211

nn

nnnn

f

f

f

f

'&&

'

&&

''''

1+5& "6663!& 666-#,7- #6664*8,7& &9:666()"6663 ;666-7< +666=+ ( >;666?!& '666( 666@/@,%!& '666( AB$!& ./0 1234 #44:( C%*5&:

2

1 nn f

f'

&&

' !"

#$%

!"

#$%

. /

//3

'( D;#E,!& CF 1#E*+ G E*;?!& D;#E, H"% IJ+, ',!& -+/?%!& AB$!& '=+ '666$*-;, D;#666E, 666!#3!& K9666=(quadratic convergence) . + ""6660 C< L#666J*< '6664$* +666=

.666/0 G 666E*;?!& M*6662! ;666*;7, 16667 N666% L#666-*;E, >0#666J,* 3*36662!& *;666O$!& P#666A;5&

#$!& C7* P! &9F 1A5&1% !"

#$%

!"

#$%

/

//

&&

f

f

2 L&"Q L&;*-7.

!"#$(1) :

M*2 ./0 L#E-#@ #4/23N-R R;67( I6J+* S96!& P6@;!#- 4#$,@T#- *;*;7,!& 666666E*;?!& K9666666= U666666*-?, .7%*+ #666666*/*/3, M*6666662!& K9666666= .666666/0 1666666234 C< V!96666667 #66666644

(analytically)'/* #%7 G:

C< );84nx C<+ G;9Q/! *-*;E, %*A '=

1.nx C< S< G;9Q!& %*A '=: 4 5 01 1.nxf

C< );84+: hxx nn .1.1

8% P&"W,@#-./0 1234 ;+/*#, V+7:

4 5 4 5 4 5 4 5 4 5 .//./.1.6. n

h

nnnnxfxfhxfhxfxf

!21

2

) C51.nx ;9Q!& %*A '=( = 0

H+666A .666/0 S+666,3, '666,!& "+"6663!& 1#666%=:-+2 ;&"666E%/! ;666X7< +<h ) 1#666%=:- S< C% S< #Y*( ',!& "+"3!& ,,, 432 hhh(./0 1234 G:


71

4 5 4 54 54 5

4 54 5nn

nnn

n

n

nn

xfxf

xhxx

xfxf

h

xfhxf

/01.12

/012

/.1

.1

0

M*2 '=+N-R *;*;7,!&.

!"#$(2) :

6666E*;? P&"W,6666@& C66667%*N-R 166667 166663! C6666% 6666*;-Q!& ZT"#6666$%!&(algebraic)

*%#666@,%!& ZT"#666$%!&+(transcendental) #666Y,B%#$% C+6667, '666,!& ZT"#666$%!& V!96667+ G(coefficients) 6666*"E0 #=;+96666Q +<) 6666-7;%complex .( 6666!"#$%!& 6666!#3 '6666( [66664< TF

B%#66$%!& Z&9 66*;-Q!& 66*\&",-& 66%*E- S"66E0 ;966Q .66!F 1+662+!& #66447%* T 66*E*E3!& Z *E*E3.

666E*;? D;#666E, H"666% I666J+, 666AB0 .666/0 U-666@ #666%*( #4/66623N-R . 666*;]4!&+ E*;?!& K9= D;#E,, '7 ;(&+,, C< DQ* ',!& ?+;O!& C*-, *!#,!&.

%&!'(5-2) :

666!&"!& Z6664#7 &9Ff C*,;666% 1666J#8,/! 666/-#A(twice differentiable) Z"666Q++ G C#,%*Aa, b ) );84+a !#W,f(b)C< S< G:

4 5 4 5 0. 7bfaf

) D( !&"/! H;M2 *#Y4 +< .%]0 *#Y4 V#4= `*!f R;,8!& '((a,b)C< S< G:

4 5 4 5baxxf ,0 89:/

) a( f // R;#Ob& #Y!@84 #Y R;,8!& 17 '((a,b).

) "( 4 54 5

abbf

bfab

afaf 07/07/

!"

#$%

!"

#$%

,

E*;? C:(N-R R;,8!& '( *;#*,W& %*A Sc- #4<"- &9F D;#E,, ; <ba,.

!"#$:

- 1+5& ?;O!&)< ( "+Q+ #4! C%J*(existence) R;,8!& '( ;9Q(a,b).


72

- '4#666X!& ?;666O!&)D ( "666*3+ ;9666Q ;9666Q!& &9666= C< #6664! C%666J*(unique) ] ;666]4&P66A; 1766O!&1[ D+66/?%!& ;966Q!& C660 "#66$,-T& P"660 #664! C%66J* #66%7 G) .66!F #66%-;;Wd ;9Q] (PA; 17O!& ;]4&2.[

- _66!#X!& ?;66O!&)a ( 66!&"!& C< '664$*f 66-"3% #66Y4< #66%F(convex) R;66,8!& 1B66Wb) (a, R;66$E% #66Y4< +< #66Y/7(concave) &"!& C< S< GR;66,8!& K966= 1667 1&+66? 66!f

>#66?$4& 66?E4 S< #66Y! `*66!(inflection) )#="6640 '66,!& 66?E4!& '66=+ 01//f (

L#66J*< C%66J* &966=+ G R;66,8!& K966= '66(– '!#66,!& ?;66O!& N66%– C660 "#66$,-T& P"660 ;9Q!&]PA; 17O!& ;]4&3.[

- N666-&;!& ?;666O!&)" ( #6664! C%666J*– U-#666@!& ?;666O!& N666%– 666%*E!& Z6664#7 &9F [6664<66*-*;E,!& nx R;66,8!& '66( N66E,; <ba, & C:66( 66*!#,!& 66%*E!1.nx C66% 66Q,#4!&U66*-?,

R;,8!& '( L#J*< NE,@ *;*;7,!& M*2!&; <ba, ]PA; 17O!& ;]4&4.[

()& *+,)1(

3&+ ;9Q C% ;X7< R;,8!& '( "b) (a,

01/f R;,8!& '(b) (a,


73

()& *+,)2(

D+/?%!& ;9Q!& C0 "0#-,

01/f R;,8!& '(b) (a,

()& *+,)3( ()& *+,)4(

D+/?%!& ;9Q!& C0 "0#-, D+/?%!& ;9Q!& +34 D;#E,

01//f R;,8!& '(b) (a,

C< );84f */*/3, !&"(analytic).

C<+ & '= !"#$%!& ;9Q! ?+-J%!& %*E!&f(x) = 0.

C<+ nx ;9Q/! *-*;E, %*A '=.

C<+ n

' C< _*3- G *-*;E,!& %*E!& K9= '( c?W!& %*A '=:

nnx '& .1

!"#$ %& '$()* #"+!,N-R Error estimate for N-R method


74

C< );84+n' C< _6*3- L#6*(#7 L&;M62 R;*M62ff/ 6-#,;- ;6*M,,(monotonically) '6(

C*- R;,8!& nx,&. C<+4 5 0:/ xf R;,8!& '( #Y@84.

C< );84: 4 5 4 5

4 5xfxf

x /1=

*!#,!& AB$!& ./0 1234 ?@+,%!& %*E!& *;]4 P&"W,@#-: 4 5 4 5 4 5 10; 770/100 >>'='

'==nnn

nnn xxx

C7!+ 4 5 4 5 00 1/1/110 !"

#$%

!"

#$%

!"

#$%

&&&

&='=ff

fx

nn

C:( V!9-+:

4 5

nn

n

nn

n

n

xxff

dxd

xxff

xx

>'>'=

='

01/

1/

10/1

(()

*

++,

- !

"#$

%

()

*+,

-

!"

#$%

/

/

nn

n

xf

fff

xff

>'0///0/

/1

((((

)

*

++++

,

-

()

*+,

-

2

2

/

nn

n

x

xff

>'0////0

/1

(()

*

++,

- !

"#$

% !

"#$

%

()

*+,

-

f/f.ff/1

/

C< );84:

4 54 5

ffK

xfxf

xff

hn

n

n

///1

/6 !"

#$%

/1

/sup


75

nnnn

nn

nn

n

xff

xff

h

xffff

h

xffff

h

>'>'

>'

>'

'

0///

0/0

1

0////0

?

0////0

12

!

"#$

% !

"#$

%

(()

*

++,

- !

"#$

% !

"#$

%

(()

*

++,

- !

"#$

% !

"#$

%

/./1

/./1

/./1

) C< );8-1. 7Kh( Kh

h

.107

C< S<: Kh

hn .107'

_*3: 4 54 5nn

xfxf

h /1

ffK ///1 /sup

* * *

666E*;? 666@&;" "666$- Ce&+N-R )K#666Q,T& ;666*M,%!& `#666%%!& 666E*;? +< ( #666Y-;#E, _6663-+$4 G#Y*( c?W!& ;*"E,+ +#Y-;#E, ;"4! #=;79 U-@ ',!& H;W5& U;?!& )$-! ".

*-.$(10-2) :

Z666-#X!& `#666%%!&+ GK#666Q,T& Z666-#X!& N?#666E!& '666,E*;? C666% 16667 D;#666E, C< Z666-X& '@"4= D;#E, K#Q,T&(geometric convergence) ] '6( C*c6?W!& C*6- AB$!& C< S<

R;+2!& '( C+7, C*,*!#,,% C*,*-*;E, C*,%*A:

nn A'' 3.1

_*3A Z-#X[

*"/0:

% 17! *;*;7,!& M*2!& NJ+ C7%*R;+2!& '( C*,;+79%!& C*,E*;?!& C: 4 5kxf

xx n

nn01.1

_*3k K#Q,T& Z-#X!& 1*%!& +=(constant slope).


76

C< );6668-+& #666% .666/0 E-#666@!& 666AB$!& C666% 1666234( G?+-666J%!& ;9666Q!& +666='/*:

k

f n

nn

!"

#$%

.

.0.1.

'&'&'& 1

4 5 4 5 4 5; < .//./.012 . &'&'&'' fffk

nnnn !2

1 2

1

C< #%-+4 5 01&fC:( G:

.()

*+,

- /01

!"

#$%

. k

fnn

&'' 1

1

c?W!& C< ]3B*+1.n' c6?W!& C6% ;*X7- 1A< C+7*@n' 16*%!& 6%*A Z64#7 &9Fk %*E!& C% -*;A4 5&f /.

/62,% 6!&" #64*?0< #44< );84f !& '6( 6( f;g$% R;6,8; <ba, C< );684+ Gf(a),

f(b) C#,8/,W% C#,;#OF #%Y! . %*A "Q+, ?@+,%!& %*E!& *;]4-(p _*3a *24,!& E*;? C< Ph;p U6*-?,/! I/62,&9F #% !#3 '( R;6,8!& '( ;9Q C% ;X7< V#4= C#7; <ba, ?*6@-,/! );,846@ #644< TF

R;6,8!& K96= '6( "*3+ ;9Q "Q+* [4< . ;;67,%!& >*624,!& .6/0 >*624,!& 6E*;? "6%,$, R;66,8!& C66% 66*\iQ!& Z&;66,8/!; <ba, .66/0 S+66,3* S966!& >6624!& C*66*$, R+66?W 1667 '66(+

;9Q!&p.

(Bisection Method) nOBM²�« WI¹dÞ


77

N6J4 *&"-!& '(bbaa 1111

, C< );684+ G1p '6( ?6@+,%!& 6?E4!& '6=

R;,8!&; <ba,C< S< G:

211

1

bap

.1

Z6664#7 &9:666(4 5 01 1pf C:666(1pp 1 R;#666OF !"""# $ %&' !1pf (""")* +,("""-* ."""/

!1af '0 !

1bf (1"""""""234 . ("""""""5 !"""""""# !

1af , !

1pf +,("""""""-6& (""""""")17 (1"""""""234 !"""""""#

!11

,bpp" 894 :;4<=' >: 1212

, bbpa ##

+,(""""-* ?""""4(5 * ("""")0' !1pf +,(""""-* .""""/ !

1bf (1""""234 !""""# !11, pap" :"""";4<=' >

894:1212 , pbaa ## . +,@37& ABC D<B)E7& F:/ G<HI@ J<E4 KL$ %22 ,ba.

D<7(@7& D<)M,&'N7& .# DO<,I7& F:/ P<NB@ 5)<':

!"# $%&'( $&)*'#+, -&.Bisection Algorithm

D""7J(E)B7 Q""= J(""R<6f(x)=0 +,""@37& .""#[a,b] S""<=f(a), f(b) (@,(""-* ("")17 (@3B@N).

0 T,""34 & U)(""2@7& '""/(tolerance) 0' >V""H W')""2)7&N JJ""C A""XY0 '""/,<,5@7& ?&,)7 VH W')2).

89i = 1.

0 ()7(I D<7(@7& ?&'IN7& :Z34i < N.

! 89 2abap '(#.

! (5 &:* ! 0#pf '0&)'2ab D)<Y 8HI(#p [Y'@'.

! 89i = i + 1.

! (5 &:*f(a).f(p) > 0 89a = p 89# $ %&'b = p.

JJC JEH UB3@ K7 DO<,I7& 0 J<3@ D7(2, 8HI&N [Y'@ KL ?&,<,5@7& ).

0 "5)< I,"- ") ,"L50 J"R'< V"40 \=]<' D"O<,I7& :"<34@ [("O<6 KJN@"2<) $J"HI,""-7& "") :f(p) = 0 '0&)'

2ab )D""<)M,&'N7& .""# ,'5:"")7& .( ,5:""4 .""B< ("")<#'

I',-7& F:/ TEH.


78

(=)(""""2@ ,""""@N&(tolerance) 0*& J""""<7'@ .""""# ,)@""""2&'nppp ,,,

21 0 A""""7*

<7(@7& I',-7& J=0 GO=@<D:

&)' '1nnpp (1)

0,1 +)

' 'n

n

nnpp

pp& (2)

! &)npf (3)

0, +)'n

n

nn aaab & (4)

$/01):

(/,5:4"2 D"<,<,5@ D"O<,I _0 8") ("9<0 G"HI@ 0 "5)< A"7'`& DL]L7& I',-7&QX37& &:/ .#.

""") I,"""- _0 G"""<HI@ J"""4C ,"""1\@ 0 """5)< ?(H'E"""X a("""4/' I',"""-7& F:"""/DOH(27& .?(<7&'@) JR'@ ]L)# , -

np G',37& 0 D<X(N GO=@

1''nnpp A7* b,(O@@

JC(H@@ (1234 D<7&'@)7& ()4<H ,3X7&]D<7&'@)7& QL):

, - .##

/01

n

k knpnp

1

1,

'5@ 0 5))7& ) a7:5' !npf X7& ) DH<,Y (")4<H ,3np &,"<L5 ["B@N@

Cp ] ]L):

! ! ,1110 #2'# pxxf

2(# npn

11

()J4C: ! 110 3 *) ' npfn

57'310')'npp bBI@@1000*n.[

""C D<#(""9* ?("")'BE) _0 'J""H ("")')C'f '0p !""#D""4<(H@) I,""-7&(2) J""Ec@ .H247& dIN7& ,H@N@ (14` G<HI@B7 [(O<* I,- Q9#0(relative error).

234)(11-2) :

D""""7&J7& ! 104 23 '(# xxxf +,""""@37& .""""# I""""O# J""""=&' ,:""""R (""""17$ %2,1 . G""""HI G"O=@< ()J"4C ,"<,5@7& ["Y'@ S"<=H > ,:"R7& &:"/ A"BC Q'"X=B7 [<X4@7& D<)M,&'N

-7&I,: 4105 '3)'n

nn

aab DYJ )9474 D<'4E) K(Y,0.


79

20"#:

! ! 142,51 #'# ff

.7(@7& Q'JR7& K<Y ABC QX=4 [<X4@7& D<)M,&'N G<HI@H':

n na n

b np !

npf

1 1.0 2.0 1.5 2.375

2 1.0 1.5 1.25 -1.79687

3 1.25 1.5 1.375 0.16211

4 1.25 1.375 1.3125 -0.84839

5 1.3125 1.375 1.34375 -0.35098

6 1.34375 1.375 1.359375 -0.09641

7 1.359375 1.375 1.3671875 0.03236

8 1.359375 1.3671875 1.36328125 -0.03215

9 1.36328125 1.3671875 1.365234375 0.000072

10 1.36328125 1.365234375 1.364257813 -0.01605

11 1.364257813 1.365234375 1.364746094 -0.00799

12 1.364746094 1.365234375 1.364990235 -0.00396

JEH V40 \=]412 !# ,<,5@:

364990235.112

#p

_'(2< dIN7& 0 _0:

000488281.0121212

#'4' abpp

0 ()H':

44

12

1212

105106.3 '' 3)35'a

ab

QY`& ABC D<'4E) K(Y,0 DEH,` U<=X b<,O@7(# . ,:"RB7 D=<="X7& D")<O7&'p

./ D<,-C K(Y,0 D<4()L7: p=1.36523001

0 D"""\=])7(H ,<J"""R7& """)' 9p U<="""X7& ,:"""R7& A"""7* b,"""Y0p b"""<,O@7& """) .;(147&12pU<=X7& ,:R7& D)<Y ?)BcC &:* $* &:/ D#,E)7 DO<,I JR'@ $ 57' >.

."1# >+,"<H5 ("H'<C ("17 0 $* DI<"2H' D="9&' ("140 K"e, [<"X4@7& D<)M,&'N' (""""1H,(O@ .""""# &J""""R D"""";<IH) 0 _0N ,&J""""O)7& UH""""X< 0 Q""""HY &J""""R +,""""<H5 UH""""X@ J""""Y

Npp ' (<#(5 &,fX &,<fX(> (I"2'@) &J"<R ("H<,O@ !"# &:"/ ) ,L50'– 0_ D"R<@4


80

.;("147& ,"<,5@7& g<"7' DI2'@)7& ?&,<,5@7& J=0– V"<7* ?"h3h@iBc< 0 'J Q"h)1c< J"Y> 8").;(147& b<,O@7& ) Q9#0 V40 (4<0, ()5 >GH(27& Q(L)7& .#.

[<X4@7& DO<,I 0 ABC"1H M<)@@ D)(/ D<X(N (17 (");&J b,("O@@ ("140 ."/' (Q"""= A"""7* ("""1EH@@ D"""<J<1)@ '0 D"""<;&J@H& +'"""IN5 KJN@"""2@ (""") ("""H7(e ."""1# bH"""27& &:"""17' >

QX37& &:/ D<OH .# (19EH A7* ,<-4 .@7&' +j(35 ,L5`& G,I7&.

%&'( 5'3%!$ -&. !"#(Convergence of Bisection Method)

$&'/ :

0 T,"""34f +,"""@37& ."""# DB"""X@) D"""7&J[a,b] 0 T,"""34' > ! ! 0bf.af ) . &:*O<,I (4OHID D<7(@@) J7'@ (14!# [<X4@7&, -

np ,:R7& bZ,hOc@p D<X(N7& GO=@':

1;2

6'4' nabppnn

(/,H7&: Q57 V40 U9&'16n @@DY]E7& GO=:

!ababnnn '#' '121

a7:5' !nnbap ,".

0 S<=':

! 121 67(# nbap nnn

0 _0np <@IO47& <H D#(2)7& [X@4) .#nnba ,

0 k@4<#:

! !ababpp n

nnn '#'4' '221

4E@ D<,\47& F:/ D4<(H@)D<7&'@)7& 0 .:

, - ,...2,1; #npn

,:R7& A7* b,(O@@p DH@, ) b,(O@ QJE)Hn'2 e]convergenc of rate)(2O[ -n.

(=)("""""2@ ("""""4<ICc0 &:!"""""#0*& <"""""<E@ ("""""445)0np ("""""<,<,5@(iteratively) """""@=A I,-7& GO=@<:

! &)'' abn2


81

'5< .7(@7(H':

&)' ppn

?&,"""<,5@7& JJ""C ,<J"""O@ ("""445)< l,"""N0 D"""<=(4 "")'N D"""YJ A"""7* Q'"""X'B7 KM]"""7& <E) U)(2@ A7* _0 D4<E)&&:5/ >:

!

89:

;<=

'

'

')

)'

ab

ab

N

N

&&

2

2

!

2ln

lnln

lnln2ln

&

&

''*

'')'

89:

;<= ab

N

abN

JJ""""""""C (""""""""4)MB< V""""""""40 _02ln

lnln &'' 89:

;<= ab

&,""""""""<,5@7& """""""") 0 )""""""""94 A""""""""@= ?&)' ppn.

$/01):

?&,<,5@7& JJE7 GH(27& I,-7& DH(@5 5)<N D7$JH?(")@<,(e'B7& g("2m710

g(2`& ) $JHe&:5/ >:

2log

loglog

10

1010 &''*

89:

;<= ab

N

l'"""Y D"""7$JH d"""IN7& U)("""2@ A"""IEc< <"""= b"""240 Df<"""X7& F:"""/ '"""5@ J"""Y'10 >AhIiEc< &:5/ +J(C'.

234)(12-2) :

&,<,5@ K5– (H<,O@– D7J(E)7& Q=7 KMB@:

! 01042

3 #'(> xxxf

V""@YJ n""BH@ Q""= A""7* Q""X4 S""<=H >[<""X4@7& D""O<,IH510'#& D""<&JHB7 DH""247(H a""7:'2,1

11## ba.

20"#: J(R<* Q'(=4N I,-7& GO=@ .@7&:


82

!

6.1610log5

10log

1022

2

52

5

5*2

'2

)#'4'

')

'''

N

N

abpp NN

N

< Q=7& 0 _0 bBI@17 V@YJ b<,O@ A7* Q'X'B7 &,<,5@ 510'.

$/01):

nBH@ DYJ7& ?4(5 &:*310'#& !#106N D")<Y ()4<H >36523475.19 #p _0J'J= .# DO<YJ410' ./ QL) 0 \=]<# ." oIE@ DOH("27& n<X7& F: pJ" h= I"O# &(bound)

JJ""E7& "") ,""<L5H ,""H50 J""=7& &:""/ '""5< +,""<L5 (""<=0 .""#' > D"")M]7& ?&,""<,5@7& JJ""E7b'BI)7& .BE37&.

b""""BI@@ $ (""""14dH M""""<)@@' > ?$J(""""E)7& Q""""=7 D""""<,<,5@7& D""""O<,I7& F:""""/ KJN@""""2@ D"O<,I .# ()"4<H >?(O@-) _0 K<Y b(2=N-R D7J(E)7& Q=7f(x)= 0 b(2=7 q(@=4

D""7&J7& DO@""-) D"")<Yf +'""IN Q""5 J""4C. 8""94 D""@H(L7& D""IO47& D""O<,IH ("") D""7J(E) Q""=7'./' DX(N +,'X .# $'0 D7J(E)7&: !xgx # S<=g .# D7&Jx.

D7J(E) _0 89' 5)< V40 8Y&'7&'f(x)=0 F:"/ ."# +,'5:")7& D"X(N7& +,'"X7&$ JJ""""EH D""""7J(E)7& ]L"""")# >G,""""I7& """") .;(""""14023 #'x ?$J(""""E)7& """") (""""<0 r#(""""5@D<7(@7&:

xxx ('# 23 (i)

552 3 xxx ('# (ii)

22x

x # (iii)

,Ns Q(L)5' : D7J(E)7& !#f(x) = 0 +,'X7& .# D7J(E) _0 r#(5@:

!xfxx ?(#

S<=? &,3X _'(2< $ ?H(L _0.

©WFÐU²²*« �U³¹dI²K� W¹d¹dJ²�« WI¹dD�« Ë√® d¹dJ²K� W²ÐU¦�« WDIM�« WI¹dÞ

Fixed point iteration method (or Iterative method of successive approximations)


83

!"#$ %&!#'()* +!), -()* .!/& +012' 2 134* 531&– 67 !"89)* ,,!-)* *:!; <!( =2')* <(– +!> 4)* ?2&!>)* 6!@x=g(x) +!3A B)* +!'08)* +!012'A !"#$ C,D!1 E!1$A

– F* <:GA H1#I ,-A J28K (L– %2 03 M)N(convergence) %&!#'()* 2:O)* &$8 P ,Q A3 M)N 24R 2 134* C,D1 ,I (81A(divergence).

?2&!!>)* 6!!@ +!!), -( 81,!!) <S <T* U2!!V8x=g(x) . +A!!K 8( +!!(10A S,!!A*0xx

) +A12I+), -()* 2:O <(( +1) 33( &S +#K#K <W&L& P(sequence) +!-A 33( X !A1203 <!(+I9-)* 5*,43K A:

! " ,2,1,0;1

#

nxgxnn

+!1) 33()* Y:") < L *:N Z8S [/*&)* <( ,,, 210 xxx +!1 "8 (limit) $ <G!@ P$ +), -(#) \*2:O <&L1x = g(x) ] <] ^):&! "$$ g .[

!"#(13-2) :

+), -(#) 6010$)* 2:O)* , O1N %&#'()*) \ 0A K ?2&L:()*:( 023 %x +!012'A+3A B)* +'08)*.

*:; , O1N H& $ 2:O)*?2&>)* 6@ +), -()* ./&A:

xxx #% 23 (i)

552 3 xxx #% (ii)

+17*,3A_* +(10)* 2 134 A&2.10 x

]$%&'#: 2:O#) +'&A/()* +(10)* <S 5&#-()* 6;:

259921.123 $.[

&():

)i( nnnxxx #%

#23

1

349.16

293.2

273.0

928.02.122.1

2.1

4

3

2

3

1

0

%

%

%

#%

x

x

x

x

x

%2 03 ,O&1 _ Z8S `$98.


84

)ii( 5

52 3

1nn

n

xxx

#%

#

25992.1

2599.1

2596.1

2544.1

2.1

4

3

2

1

0

x

x

x

x

x

10)* <S `$98[1$>)* 2:O)* &$8 \9-@ %2 033 +1) 33()* 5.

+!18 B)* +a1!>)* Y:!; =1A'3A "1#Q H>$8 63)* +1) 33()* %2 03 %AK (S&)ii(

M!!)&]* +a1!!>)* +!!1) 33( ,!!Q A3&)i( N 21!!b8K@) +!!012' %2 !!03 +!!K*2, ,!!-A F* <:G!!A Z!!1+3A B)* +'08)*.

c'4)* 6'-3 +I9Q , O1d1#n& c'4)* +)_,An&:

<S U2V8$ 2:O#) +'&A/()* +(10)* 6;

11 ###

# '

nn

nn

x

x

&$

&$

+3A B)* +'08)* +012' +a1> 6@ U1&-3) A&:

! "nnxgx

#1

M#Q H>$8:

! "

! " $&$&

&$&$

%# '

# #

#

#

nn

nn

g

g

1

1

<S (A&$ +), -()* =0$3 "8S CS P2:O#) +'&A/()* +(10)* 6;:

! "! "

! " ! "$&$&

$$

gg

g

xgx

nn%# '

'

#1

Convergence of the fixed point method W²ÐU¦�« WDIM�« WI¹dÞ »—UIð


85

! " ! "

! "nnn

nnnnn

g

g

(&&

&$$&((&&

)

#)

#

#

.

,,.

1

1

%2 03)* '2b <&L1 6) 3) A&:

51I <( +(1I HL)n: ! " 1*)n

g (

24R %&#KcA&:

! "

! "

! "nnn g

g

g

(&&

(&&

(&&

)

)

)

# .

.

.

1

112

001

!

! " ! " ! "nn

ggg (((&& ))) '#

1001

..

< L *:G@:

! "

0

1

1.

,,2,1,0,

&&

(

#

#+'

+)

n

n

k

m

nkmg

<S CS:

0&& n

nm+

%2 03)* '2b <&L1 6) 3) A&:

1*m

<&L3 <S CS: ! " 1*) xg

+), -()* H$A "1@ 53"8 63)* ?23V)* 6@ ^):&.

+'K&3()* +(10)* +12`8 =1A'3A&:

nxnn

,#&$$(


86

()*+!,-() .+/( 01234() +516-:

%2 03)* '2b) 6K,8")* M8-()* [/&3 +1) 3)* H Lb]*: 1)( *) xg

+), -()* H$ ?23@ H94:

)S (1*)g .. %2 03 )% (1*)g .. %2 03

)e (1-)g ..,Q A3 ), (1-)g ..,Q A3

+!!!), -()* H!!!$ <S `!!!$98x=g(x) , !!!O1N \ 1!!!K,8; 6!!!8-1 +!!!3A B)* +!!!'08)* +!!!012'A<118$8()* .' 03 +'08:

y = x, y = g(x)

+(10A f,A)* =12' <Q+1A1203 +17*,3A* 0x M8$8()* .'01 \ 1KS2 \ '4 5K28 "8(&y =

g(x) !0@S \ !'4 "8( 5K28 +'08 6@!!'01 \ 1! 5103!K()* '!4)* .y = x "1B*,!$N +!'08 6!@


87

C& !!K1 6!!0@]*)C& !!K1 6!!KS2)* "1B*,!!$N (1x E!!1$ P! "

01xgx M!!#Q H!!>$8 5!!B

2x

<(1x +012') A "KV8 M#Q "A 8#>$ 63)*1x <(0x.

+-A 33()* 510)* M#Q H&>$)* 6@ *:L; 2(3K8& ,,, 543 xxx.

<1#L!!!b)* 6!!!@ `!!!$98&)S( P)% ( 33()* 51!!!0)* <S+!!!1) ,,,, 3210 xxxx %2 !!!033<1!!!18$8()* .' !!!03 +!!!'08 &!!!$8 )(, xgyxy 51!!!0)* Y:!!!; ,!!!Q A33 !!!(81A P +!!!'08 <!!!Q

<1#Lb)* 6@ .' 03)*)e( P),.(

<1#L!!!b)* 6!!!@ `!!!$98 !!!(L)S( P)% ( M!!!8$8(#) g !!!(()* H!!!1( <Sy=g(x) ,!!!8Q .' !!!03)* +!!!'08)+!!!), -()* 2:!!!O ,!!!8Q CS (– @ 6!!!@ !!!")&$& +!!!), -()* H!!!$ ?2!!!3– =!!!0$1

'2b)*1*)g 5103K()* '4)* H1( <( HIS Z#1( <S CS Py = x.

M8$8(#) g (()* H1( (81Ay=g(x) <1#L!b)* 6!@ .' !03)* +'08 ,8Q)e( P), (

+I9-)* =0$11-)g.

!"#(14-2) :

+a1>)* ,Q A3 %AK <W1A)i( a1>)* %2 03& +)ii( H B( 6@(10-2).

&():

H !B( M)N h&O2) A(11-2) +!), -()* H!$)023 %x 2:!O)* E!1$$ 6!@ .!01 ?23V)*. /2,10$ M)&]* +a1>)* <S ,O8:

)i( xxx #% 23

<] '&A/()* 2:O)* <Q +1) 33()* 510)* ,Q A3 M)N C,D3:

! "

! " . /2,11

13 3

01-)2

# )

xxg

xxg

+18 B)* +a1>)* (81A

552 3 xxx #%

<] '&A/()* 2:O)* &$8 +1) 33()* 510)* %2 03 M)N C,D3:

! " ! " 22

53153

51 xxxg % #% )

! "3

10&2531

5311 222 **3*%3*) xxxxg

<S CS! " 1*) xg %&#'()* 2:O)* H&$.


88

7.8() +52,-)7.89( :;9<() 2&() ;= ($,5+. 0> $-?!"() $.,3() Error estimate (or upper bound for the error) in the fixed point method

+1212L3)* +a1>)* 5,43K8 +3A B)* +'08)* +012' 6@:

! "nnxgx

#1

+), -()* H$):

! "xgx

<S 8/2@ *:G@$ +'&A/()* +(10)* 6;2:O#)<S CS P:

! "$$ g

+1A1203)* +(10)* 6@ c'4)* <G@nx \ 1& K( <&L1:

! " ! "

! " ! " ! " ! "

! " ! "11

1

11

11

..%%

%

##

##

)%#)%

%#%

%#%+

%#%

%

nnnnn

nnn

nnn

nnn

nn

gxxgx

xgxgxgg

xxx

xxx

x

(($

$

$

$

$&

)+'K&3()* +(10)* +12`8 =1A'3A ^):&.(

<S 8/2@ *:G@: gm ) max

<S&: m < 1

+1) 3)* +I9-)* c'4)* =0$1@ P=0$3( %2 03)* '2b <S CS:

! "

11

1

1

.

.1.

..

%%

%

%

%+'

%+%

%#+

nnm

mn

nnn

nnnn

xx

xxmm

mxxm

&

&

&&

) <S `$_m < 1.(

!"#(15-2) :

+), -()* H$) +3A B)* +'08)* +012' +a1> <( X*212L3 g(4 =A':

xx sin5.0 #

+1A1203)* +(10) A \ 7,3A(10 x . "1#Q H>$3 63)* +A Od* 6@ c'4)* 2,I 5B.


89

&():

47382.1

341471.1

1

sin5.0

2

1

0

1

# #

x

x

x

xxnn

5555

5

4

3

4972894.1

4971517.1

4953015.1

&$& #**%'

xx

x

x

x

! " xxg

gm

mmxx

mm

sin5.0

max

10001377.0

1 455

#

)

% %

%+&

074.0735781.4971517.1coscosmax

cos)(

44 )

)

xgm

xxg

5

+!!)*,)* <] ^!!):&xcos ?2!!3V)* 6!!@ +1!!>I 83 +!!)*,! "2/,0 5 51!!0)* <S `!!$98& P+-A 33()*ix 2(3K( ,1*i3 6@.

543210 xxxxxx *****

<( HIS "#L&2/5. ! "

! "544

11

, 5

,

xxn

xx nnn

02

0 %%

(

(

2/


90

!"#$% &'( )*+, !54

, xx !"#!" $%&g #'( )*+,!" -.&/-.' )!04x 1!2!3

074.0cos 4 !" xm -4#-5/0 678 -*9.

0000110.00000110.0

0000110.0

074.01074.00001377.0

1.0001377.0

55

5

#$$%&

"

%'!

%(

xx

mm

)

*

:; -./<( -'<%= >&!" ,7-&&*!" /7/?@&!" A/@!" <8<8 :* +=B' -''9!3:

5x+)

1!2!3:

4973004.14972894.1

0000110.055

$$

#$$

)

) xx

+=B*:

?25<! =/=%!" */@!") >4 /?C( A-D?; &8! :1.497300

!"#$%& '()%& (*+' ,+#(! -*./! Convergence acceleration for fixed point iteration

*0122, )34!5 ( (*+'6)37)8!7( Aitken process and Steffensen method

:-EEE9 "20 EEE&7-F!" EEEG@'!" EEE@/?G >EEEH- .)g EEE*/@!" :EEE* -EEE7/?D 1 I?-EEE@& :JEEEH K L-EEEEM/G7 :3EEEE9/ EEEE@/?G!" ..-EEEE&!" :/&@/?EEEEG!" :EEEE* $EEEE9 >EEEEH3 EEEE/!-&&* :/3EEEE9& N?#EEEE' :/&/!

(sequence) ?2E5<! G37EO*!" E*/@!" 3=' I?-@&& >&!" A/@!" :*) I?-E@& :E* P?E8; &7-F!" G@'!" @/?G A/D.

96: : *0122, 34!5(Aitken2, process)

!"#!" :; Q?R'g ?"?*&E8-7 $EO-R&<! E<7-D(continuously differentiable K :;3 EEEE/!-&&*!" ,,,,

3210xxxx- EEEE*/@!-7 ;#EEEE7& >EEEE&!"0

x ?/?9&!-EEEE7 IEEEED-,&!" )EEEE<( #EEEE!3&& AEEEEF EE&7-F!" EEG@'!" EE@/?G7– -EE* EEG@' 3EE=' I?-EE@&&) .S2EE4 :JEEH EEG@'!") EE&7-F EEG@' #EE,&

!"#<!g .:-9 "2 T"3- . 0/ )gH K A/@! U'Jn V?/79!":

- .nn

g *)* !#1


91

!"#$ %& '&1 n

! ()* *+*,-#$ ./n+1 0!"#$ 123 45+!" 456+*7- 89-:+n

!

()* *+*,-#$ ./n .%;/ *"< =>2? 6 >&:

" # " # 10;lim1

$%$%&

'())!

!gg

n

n

n

@+#5--9#$ %A B>7C> ,,, 210 xxx >DC 45+!" =*57--).

@000!7C 80000E>- F000C& (00002:C G57-000H92# @!0000?>-9#$ @0009+7#$ @0000+*IC ($8"-000?56n* %+00006),nx %& J+D6:

" # " # " #nnnnn gxgg !*)!)! .11

%&+&+&

%& '&:

" # " #nnnxggx +%&+

)*) .

1

85E+;6>) 123 BKDC @#85:9#$ LMN %9:

" #" # " #" #" #, -

" #nnn

n

nnnn

nnnnn

nnnn

gxx

x

xgxx

xgxxx

xgxg

*)

*

*

**)

%++

&

%+ +&

%+ +&

%+&%+

1

1

.

.1

1

1

1

1

> @9+) (2:C O %DC P6!#56" #ng *% @!0?>-9#$ @09+7#$ @0+*IC %9 Q*:C 5CC,#> R

%& G57-H92#: " # " # " #

nnn

nn

nn

nxn gxx

xgxgxx

xnr .%&+

+&+

/+

+

+

+

1

1

1

1

J+Dn. %+6 @!7C1, +nn xx . (+07# @60?C#56 F0C& Q*0:C 509,n LM0N %;0/ S*0+6,#$@6?C#$nr .2+ 59 '>5?-:

" # " # " #nnn gggr *). %0%0%&

@!7C#$ %;/ .#5-#56>:

n

nn

nn rxx

xx++

&

1ˆ 1 (1)

J+D:

1

1

++

+&nxn

nn

xxx

nr (2)

*ME2# 456+*7- %>,- %& =E+) %9 *+T,6 BU/&1

, nnxx.

45+C5+6 45U+& $MN V*C %& %,9+> . @)W:#$ %9/)1( %& X-C-?C:


92

" # " #

" #

" # " #nnnnn

nnnnn

nnnnn

xxrxgx

xxrxx

xxrxx

+ &1

! "

!

#

#

ˆˆ

ˆˆ

1ˆ

1

1

! "!nx #$%&'()* +,-) .&/01)* .+%2)* 34:

$ % $ % $ %nnn xxrxgxs #!

5$67! : $ % $ %

1

1

!

nn

nn

xxxgxg

nr

89s(x) :262(-) ;+0%)* 74g(x) $&+%2)* <2=:

$ %& ' $ %& '11,,, nnnn xgxxgx

>??$( @* A??$B& 0??( *C8??9g(x) $??/ DE$??-F(,1 nx G/??H! 0??( *CI "!g(x) D0??/$A%& $/ D0($%&'( D0+,(,1 nx ;+0%)* 89s(x) :?262(-) D*<?J D*<?$J D0?/$A%& 7K$ ! LJ$

g(x) MA&N)* OC4 39 ..?&/01)* .+%2)* 89 3)0&)0/7nx D*<?$J D0?/$A%& 7?K& ! L?J$ ;+0?%-)%2-) D*<J .&/01)* .+( )- :262(g(x) .*C4 GP7$ 3)0&)* >KQ)*7.

R D*A??S271#nx :??)I LA??F! M<0??= 7??K&( ??(nx L??&K2 ! <??$9R* ??(9 Tnx

.????)U</1#nx ????P(090 G$6????H& <????6 V????$)I)AB????H! V????2! WA????&NX$ (* ????( DU<????/ >0(Y&????' $&FEY)*(1) , (2)02$+Y$ *C47 T

.&/01)* .+%2)* .%$A+ LA0%& >$JY&

) .$-(=2) K&Z(


93

$ % $ %n

n

nn

r

rxx

nnn

nnnn

xxx

xxxx

# #!

## !

##

###

11

1

1

ˆ

11

111

)3(ˆ11

1

1

##!" #n

nn

rxx

nn xx

5$6:

)4(1

11

nn

nn

xxxx

nr #

!

.??($F ! S??6E27$ %11

nr )* ??( A???/K! 7??K& V??$-= #7???'%($ %

nr 1 0???K *CI

1*nr.

./'2)* G/H& 0(2$67nr A?$B&& ! ;?F7&2 0?2289 .&/01 D0/$A%&g(x) MA?&N)* 3?9 DE$?-F0???2([& 3???&)* 7???K$ 3)0???&)0/7 Ts(x) ???90K D*<???$J D0???/$A%&? :???262(-) D0$g(x)>???YJ$ *C???47 Tnx

AC???J-) D0???/$A%&( ???( A???$1K/ >???P9!nx . A*<???%()* A???/&Y2 C???\2$67nxx ˆ D*<???$J D*A$<???%&]+,-)n+.

.?????$)0&&()* ! ^0?????/1I ?????K($7 ,ˆ,ˆ21 xx 7?????62 LA0?????%&&( LA0?????%& ?????( _A?????'!

HR* .$)0&&()* .$- ,,, 210 xxx ) "! ^20K `*7'nx ACJ-) D0/$A%& .2$Y(( >?P9! ( !7'! #!nx.(

.??$)0&&( $7??K&) 0??402AKC 3??&)* .??$-(Y)* OC??47 ,ˆ,ˆ21 xx ] .??FEY)0/ :??+Y&(3) [

M0?+Y( .?$)0&&( ?( _A'! LA0%&& ,,, 210 xxx M<0?= :(?'& T D0?$+, D0?/A0%& LA0?%&& .$-(=2) K&Z .3-$ 0( :)I .$('&)* L/' ;JA$7:

$$)0&)* $A0H&,U* 02(<,&'* *CI:

$ %

kkk

kkkk

kkk

xxx

xxxx

xxx

# !

) )!))!)

!)

##

#

#

12

1

2

1

2

.FEY)* 89(3) L&K2 ! K($– ( 0[$9 W$7Y&)* <Y/(4) – 3-$ 0(K:

$ %

11

21

1

1

1

2

1

1

1ˆ

#

#

#

#

#

#!

#!

#

#

nnn

nn

nn

nn

nn

xxxxx

xxxxxx

n

nn

x

xx


94

(*) $ %

,3,2,1,ˆ1

2

2

1 ! ! )

)# nxx

n

n

xx

nn "

.$(??'&)* 3&]??& 0??24 ??(72) . "! :??-= a??$/+&-) .??-/0F .??$-(Y)* OC??4 ! S??6E$7 .?&/01)* .?+%2)* .%$A+/ A$AK&)* ( .J&02 .$)0&&()* OC4 ^20K `*7' D0$+, D0/A0%& .$)0&&(

U #!.

(bA*7, c,-&????& *C????K47 .????$-(= .????$2) .????$)0&&()* 0????2$+=! *CI V????2! 3????9 ????K&Z ,,, 210 xxx :??)I LA0??%&& 3??&)*( .??$)0&&()* L??'62 0??2289 T ,ˆ,ˆ

21 xx #*<,&??'0/ MA$,R* .FEY)*(*).

.??FEY)* :??)I >7??H7)* 0??22K($7(*) .??$0[2)* :??)I `0N$&??'U* :(??'& .+$??'/ MA??KN/(extrapolation to the limit) 3?-$ 0?(K) C8?/ dA?,! MA?( MA?KN)* OC?4 a?$/+& <$Y2?'7

"<<Y)* >(0K&)* = e<0')* >HN)* 39 f*:(

^0(7-Y( 02$<) 0K 0(-K MAKN)* OC4 a$/+& K($ MA97&( 3?9 ]?+,)* g7-?' =0( .$(bA*7, ..$/$A%&)* .FEY)* 02$<) 024 DE1(9:

$ % 1. ,- nn g +(+

= ^0(7-Y( 02$<) 7K$ ! 7<7$ %(g, L&K2 ! 022K($:

$ % $ %

$ % $ %nn

nn

xgx

xgx

,-

,-

#

(((

(((

.

.

1

1

:-= >H62 .('%)0/7:

n

n

n

n

xx

xx

-

# ((

(( 1

1

ACJ)* 39 .)<0Y()* OC4 >6/7( :-= >H62:

$ % $ % $ %

$ %

$ %

$ %

$ %111

11111

2

11

2

11

1111

2

11

2

2.

2..

.2.

.2

##

###

# #

# #

#

#

# ## !

# !#

## !#

!

nnnn

nnnnnnn

nnnnnn

nnnnnn

nnn

xxxx

xxxxxxx

xxxxxx

xxxxxx

xxx

(

((

(((


95

$ %

$ %

$ %

12

2

11

21

11

111112

1

21

2

2.1

#

#

#

###

)

)#

#

#

#

# # #

!

!

!"

n

n

nnn

nn

nnn

nnnnnnn

x

xn

xxx

xxn

xxx

xxxxxxxn

x

x

x(

.$-(= 3472) K&Z 0['N20[J0&2&'* a/' 3&)*.

.???%$A+ :???)I A$???Q2 .???$-(Y)* OC???4 a???$/+& .???$N$K G$???P7&) DU0???1( eA<???2 ! >???/F7 dA????,!– ????'2N$&' .????%$A+ :(????'&– .????+%2)* .????%$A+ LA0????%& >????$JY&) D0????P$! #<,&????'&.&/01)* .5E1?)* aA?+)* ?( >?K/ V&)]?'( >?62 DU0?1( *C?4 <?Y/ eA<2 #1 : .?&/01)* .?+%2)*

'7 K& h*7 '2N$&.

!"#$%! &'$()* ($(+%,- (Steffensen iteration)

0??( .??($F ! 0??2(-= *CI .??&/01)* .??+%2)* .??%$A+/ A??$AK&)* .??)06 3??9kx L??$A%& 3??4 ACJ-)( >P9! .?($%)* ?( A?$1K/kx A(&?'2 ! ^?F7)* ;$$?P& ?( V?2! <?Ki()* ?(9 T

#$????%)* <????$)7& 3????9 ,, 21 ## kk xx . .????+%2)* .????%$A+/ A????$AK&)* !<????/2 ! >????P9R* ????(7 $\<???&/( <???$<J ???( .???&/01)*/.???($%)0kx 3\*<???&/* $???(,&K . .???$(bA*7, :???)I "<i???$ *C???47

a/+2 5$6 '2N$&'– >$-F <Y/ f* C8/ dA2' 0(K– .%$A+/ A$AK&)* ( $&7+, 7+, 0[Y/&& .&/01)* .+%2)* K&I .%$A+ ( M<6*7 M.

3-$ 0($9 '2N$&' .%$A+/ A$AK&)* c,-&$7:

A????????$AK&)* .????????)*< 0????????2$+=! *CIg .????????+%270y #$????????%-) .????????$)0&)* ^*7????????+,)* C????????N229 T ,2,1,0!n ./7-+()* .F<)* .JA< :)I >H2 ! :)I) .?($F :?)I LA0?%&)* :?&6 "!

ACJ-) .)7/%(:(

- >YJ*nyx !0.

- ( >K .($F L'6*:

$ %

$ % .

,

12

01

xgx

xgx

!

!


96

- .($F L'6*:

dxrxd /, 0

1

1 )!)!

- .($F L'6*:

1121 #!# r

dxyn

A??$AK&)* ??( $&7??+, ??( .??$(bA*7,)* OC??4 ??( M<??6*7)* M7??+,)* 7??K&& *C??K47 .??&/01)* .??+%2)* .??%$A+/)L0??'621 , xx (0??([Y/&$ ??K&I .??%$A+/ <??6*7 A??$AK&) L0??'6

1#ny ( .??$\*<&/U* .??($%)* :??-= >7??H6-) .??1E1)* ^*A??$AK&)* OC??4 #*<,&??'* ;??( T$ %0x

.$)0&)* M7+,-).

&./01:

MA$,R* .B$H)* ;P7 K($$ %1#ny .$)0&)* MA7H)* 39: $ %

$ %

02

2

1

12

2

12

01

12

2

221012

2

1

x

x

xxxxxx

x

xxyxxx

xxn

)

)

!

!#!#

#

$ %$ % $ %$ % $ %. /$ %$ % $ % ,2,1,0;

21

2

! !#

# nyggy

nnn

nn

yygygg

ygyggnn

2341(16-2) :

.)<0Y()*: 1.0tan5.1 ! xx

CJ 0[)A 02059216951.0!(

.$)0&)* 5E1)* .$<<Y)* aA+)* ( >K/ ACJ)* *C4 <J7!:

) !( A$AK&)* .)*< #*<,&'0/ .&/01)* .+%2)* .%$A+.

$ %5.1tan1.0 xxg #! .($%)0/ D0\<&/(700 !x

) L( .$-(=2) K&Z.

) j( '2N$&' .%$A+.


97

2/,-:

) !( .&/01)* .+%2)* .%$A+:

$ %

0

,2,1,0;5.1

tan1.0

0

1

!

!#

!!#

x

nx

xgx n

nn

$ %

$ % 1111177.0

0666667.0

5.10666667.0tan1.0

5.10tan1.0

12

01

!!!

!!!

#

#

xgx

xgx

...................

#$?F W?Y/ $?/$ 3)0?&)* >7<J)*7nx 7?62 LA0?%&& .?$)0&&()* ! G?P*77 T( h*7 0K D0\$+/ LA0%&)*.

n nx nx nr ny

0.0 0.0

1 0.0666667 0.2005954 0.6678 0.2005954

2 0.1111771 0.2024021 0.6721 0.2059125

3 0.1410916 0.2039180 0.6774 0.2059217

4 0.1613569 0.2048536 0.6822

5 0.1751813 0.2053721 0.6859

... ................ ................ ...........

10 0.2009941 0.2059055 0.6940

15 0.2051206 0.2059213 0.6955

20 0.2057911 0.2059217 0.6957

25 0.2059004 0.6957

30 0.2059182 0.6958

) L( .$-(=2) K&Z:

$ % ,3,2,1;ˆ

1

2

2

1 ! !

)

)# nxx

n

n

x

xnn

#$%)* 5$6ix .&/01)* .+%2)* .%$A+ #$F .$)0&&( ( 0[2= W7Y2.

$ %

$ %

2005954.0

0.00666667.021111771.00666667.01111771.0

1111771.0

2

2

012

2

12

21ˆ

!

#0

#

!

!xxx

xxxx

#$F $/$ >7<J)*7nx #$%)n .N-&,()*.

L'2)* #$F D0P$! >7<J)* $/$ 0(Knr 5$6: 1

21

#

!nn

n

xxxx

nr


98

DE1(9:

677.0

672.0

667.0

3

2

1

!

!

!

r

r

r

]/ >7%-) .$90K <Y& #$%)* OC47nr V2! VY( k&2&'2 0(( T.&/01 D0/$A%& ^6/H! #$???F >???K)1n 1 8???9 Tnx AC???J-) L???$A%&( ???( >???P9!nx . >7<???J)* #$???F7

*C4 <Ki& a/0')*.

) j( '2N$&' .%$A+:

#$F a/0')* >7<J)* $/$ny ?'2N$&' .?$(bA*7,/ ./7'6()* . ! S?6E271y

34 nx 0['N2 . L0'6) #12y #$%)* j0&62.$)0&)*:

$ % $ %

$ %2059125.0

5.12005954.0tan1.0

2005954.0

2005954.0

2

12

01

10

!

!

#!!!

!!

y

xgx

gxgx

yx

! >7<J)* ( S6E$73y ./7?'6()* .$A?QY)* #0?FAR* >?K) .+7/?P( . *C?K47 .$-(= ( _A'! '2N$&' .%$A+ ! >01()* *C4 >7<J ( GP&$2) T K&Z

.&/01)* .+%2)* .%$A+ ( A$1K/ _A'! 04A7</ OC47.

1234(17-2) :

.)<0Y()*: $ % 020102 23 ! ##! xxxxf

ACJ)* 0[): 368808107.1!(

.$&@* 5E1)* aA+)* ( >K/ ACJ)* *C4 <J7!:

) !( .&/01)* .+%2)* .%$A+ :MA7H)* 39 .)<0Y()* ;P7/:

10220

2 ##!

xxx

A0$&,0/710 !x

) L( .$-(=2) K&Z.

) j( '2N$&' .%$A+.


99

2/,-:

) !( 3)0&)* >7<J)* 39 .2$/()* #$%)* .$)0&&( 02$+Y& .&/01)* .+%2)* .%$A+: n

nx n nx

1 1.538 461 538 13 1.368 817 874

2 1.295 019 157 14 1.368 803 773

3 1.401 825 309 15 1.368 810 031

4 1.354 209 390 16 1.368 807 254

5 1.375 298 092 17 1.368 808 486

6 1.365 929 788 18 1.368 807 940

7 1.370 086 003 19 1.368 808 181

8 1.368 241 023 20 1.368 808 075

9 1.369 059 812 21 1.368 808 122

10 1.368 696 397 22 1.368 808 101

11 1.368 857 688 23 1.368 808 110

12 1.368 786 102 24 1.368 808 107

) L( .$-(=2) K&Z.

.$)0&&()* #$%)* :-= .$-(Y)* OC4 .B$H a$/+& ! S6E2121110 ,, xxx 3+Y$:

$ %368808107.1

0002328.0000071586.03687861021.1ˆ

2

11 -

!x

.????($F 3????4724x 0[????'N2 .$????H #*<,&????'0/9 *C????K47 l????H2 :????)I j0????&62 ????K&I .B+??%9 ^*A??$AK&)* . ^*A0??H&,* :??)I "<i??$ a/0??' ^??F7 3??9 0[(*<,&??'* 0??K 0??(/A7

A1K!.

) j( '2N$&' .%$A+:

.1E1 ^*A$AK& >K <Y/ #0S&20/ .$0[2)* :)I `0N$&'U* .$-(= 347.

:)7R* .1E1)* ^0/$A%&)*210 ,, yyy /01)* .+%2)* .%$A+ ( 04C,]2 ! 022K($ .?&] b?????J)*)! ((#$?????%)* 3?????47210 ,, xxx L?????$&A&)* :?????-= . ?????K&I .B$?????H a?????/+2 #?????1

L0'6)3y: $ %

370813882.12 012

212

23 !#

!

yyyyy

yy

3-??HR* A??$AK&)* @* l2]&??'X$ #??1– b??J)* 3??9 .??&/01)* .??+%2)* .??%$A+ 3??9 0??(K)! (– :-= >7H6-)54 , yy:

$ %

$ % 369203162.1

367918090.1

45

34

!!

!!

ygy

ygy


100

.($F K&I .B$H 02$+Y& #16y: $ %

368808169.12 345

245

56 !#

!

yyyyy

yy

$A$AK&)* 02$+Y& .$)0&)* MA7<)*7:

368808120.1

368808080.1

8

7

!

!

y

y

L7-+()* ACJ-) 3)0&)* L$A%&)* K&I .B$H 02$+Y& 0([2(7:

368808108.19 !y

3??9 +??%9 $&)<0??Y( ??( .??$2m .??=7(J( >??6 :??-= 3??-$ 0??($9 f* C8??/ A??H&%2' $)7[J(y) (x, ?K($ 5?$6/ 0(4AKC2?' $?&-)* $&%$A?+)* ?( >?K #$(Y& K($ K)7 T

??( .??$2m .??=7(J( >??6) 0[(*<,&??'*n 3??9 .??)<0Y(n >7??[J() 5??$6n > 2 .( 0??(!7*0([9 0(['A<2' 0&-)* 0&%$A+):

) !( &7$2 .%$A+. )L (.&/01)* .+%2)* .%$A+.

)5( %6$" &'$()Newton's Method

0(4 0([-6 L7-+()* $&)<0Y()* ! WAN2: f(x , y) = 0

g(x , y) = 0

)&./01: MA7H)* OC4 39 027K& #) I T.$ANH)* MA7H)* OC4 39 $&)<0Y()* ;P2.(

! WA??N27$ %nn yx , 3)0??&)* L??$A%&)* !7 T 3??/$A%& >??6 7??4 7??4$ %11, ## nn yx

5$6:

nnn

nnn

kyy

hyx

#!

#!

#

#

1

1

5$6nn kh , 0([2$$Y& L7-+()* 06$6H&)* 0(4.

$&)*<??)* ??( >??K g??N/f, g >7??6 A7-$0??& .-??'-'&( 3??9$ %nn yx , _0??+&F0/ #??1 T)* :)7R* ./&A)* <7<6 <Y/ .-'-'&((first order terms) :-= >H62:

Simultaneous (nonlinear) equations ©WOD)« dOž® WO½ü« �ôœUF*«


101

$ % $ % $ %

$ % $ %1,.

,.,, 11

nnyn

nnxnnnnn

yxfk

yxfhyxfyxf

#

#!##

5$6:

$ %$ %

,,, nn yx

nnx xf

yxf22

3

$ %$ %nn yx

nny yf

yxf,

,22

3

>1()0/7: $ % $ % $ %

$ % $ %2,.

,.,, 11

nnyn

nnxnnnnn

yxgk

yxghyxgyxg

#

#!##

$ %$ %nn yx

nnx xg

yxg,

,22

3

$ %$ %nn yx

nny yg

yxg,

,22

3

! 02PA9 *C89$ %11, ## nn yx ! "! T .=7(J(-) +7/P( >6 74:

$ %

$ % 0,

0,

11

11

!

!

##

##

nn

nn

yxg

yxf

$&%/0')* $&)<0Y()* 89(1) , (2) :)I U7i&: $ % $ % $ %

$ % $ % $ %nnnnynnnxn

nnnnynnnxn

yxgyxgkyxgh

yxfyxfkyxfh

,,.,.

,,.,.

!#

!#

$)7??[J()* 3??9 $&)<0??Y( ??( .??$+, .??=7(J( OC??47nn kh , 0??([$&($F $??$Y&/7 T)0&)* L$A%&)* :-= >7H6)* 022K($ .=7(J()* OC4 (3: $ %11, ## nn yx.

2341(18-2) :

MA\*<)* :262( ;+0%& .+%2 <J7!:

$ %1222 !# yx

<\*b)* ;+%)* :262( ;(:

$ %2122 ! yx

$&)<0??Y()* >??6) &7??$2 .??%$A+ ??( ^*A??$AK& 5E??1 #*<,&??'0/ g??)C7 T >7R* ;??/A)* 3??9 $&$2@*(1) , (2) $A%&)* >6)0/ D0\<&/(3\*<&/U* 3/:

$ % $ %1,1, 00 !yx


102

2/,-:

&777./01: 0???$+Y()* 0???$262()*– #???'A)* ???( G???P&$ 0???(K– 0???([) +???%2 ;???/A!3-$ 0(K 0[&0$1*<6I $&)<0Y(-) +7/P()* >6)* 3+Y$ ;+0%&:

:-= >H62 D0Y( $&)<0Y()* ;(J/:

2

2247.15.132 22 4-5!5! xxx

:)7R* .)<0Y()* nA+/7(1) .$201)* cF02(2) :-= >H62:

7071.05.012 22 4-5!5! yyy

899:,- 2/,-:

$ %

$ %

nnn

nnn

kyy

hxx

yyg

xxg

yyf

xxf

yxyxg

yxyxf

#!

#!

!22

!22

!22

!22

! !

! #!

#

#

1

1

22

22

2,2,2,2

01,

02,

22

22

122

222

nnnnnn

nnnnnn

yxkyhx

yxkyhx

!#

!#"

:0!n

25.0

25.014

122

022

1,1

0

00

00

00

00

!5

!5!5

!

!#

!!

k

hh

kh

kh

yx


103

75.0

25.1

001

001

!#!

!#!

kyy

hxx

:1!n

$ % $ %

$ % $ %

7083.0

225.1

04167.0,025.0

075.025.115.15.2

125.075.025.125.15.2

112

112

11

22

11

22

11

!#!

!#!

! !5

!# !

! !#

kyy

hxx

kh

kh

kh

:2!n

0011.04167.145.2

0024.04167.145.2

22

22

!

!#

kh

kh

7071.0

2247.1

0012.0,0003.0

223

223

22

!#!

!#!

!!5

kyy

hxx

kh

"<<Y)* >6)* ! "!2x .$AQ= #0FA! .Y/AR +7/P()* >6)* "70'$.

);( !"#$%& '()%& (*+') *+*+,!%&( The (iterative) fixed pint method

!"#$%&'#( )*%"+ $,-%'./0 12/3'#( !"!4%45(:

! ! 0,

0,

"

"

yxg

yxf

6728#( 9::

! !yxGy

yxFx

,

,

"

"

!"#($#( ;/, 327<#( =&* =7>*2F , G 7!7+"#( ?: @: !

!nnn

nnn

yxGy

yxFx

,

,

1

1

"

"

#

#

)!:%+ )A$ )B!A$ )!C($"*D( )!*!7B"#( E!B#( FG%+ (HI )*7%B"' )!#%""' E!A ;#I J$K!.


104

-#$.(19-2) :

!"!GL( !"#$%&'#( M0 )"*%N#( )3BG#( )B!73* $O2-:

! !yxy

yxx

$"

#"

cos

sin

9C($"*D( 1!7B"#%* P%C$"*':

0,0 00 "" yx

-/%&:

)"*%N#( )3BG#( )B!73* M0/# )*2/3'#( 6728#( 9: %"QR%O %"#$%&'#(. !

!nnn

nnn

yxy

yxx

$"

#"%

#

#

cos

sin

1

1

:0"n

10cos

00sin

0,0

1

1

00

""

""

""

y

x

yx

:1"n

! 5403023.01cos

841471.01sin

2

2

"$"

""

y

x

:2"n

9549905.029.0cos

9821883.039.1sin

3

3

""

""

y

x

(HSSS+R2 ..)SSS*2/3'#( )SSSA$#( ;SSS#I MSSS8G - ;SSS#I 7SSS!7+"#( 9SSS: 7(7'"SSS4D( %SSSGG+'! .

9#%"#%+ M2$O 9: %.!/, M80G 9"#( )!#%""'#( E!B#( TU2 %GG+'!2:

n 0 1 2 3 4

nx 0 0 0.841471 0.9821883 0.9336294

ny 0 1 0.5403023 0.9549905 0.9996302

n 5 6 7 ...

nx 0.9350263 0.9351718 0.9350472 ...

ny 0.9978228 0.9980290 0.9980252 ...


105

)V0W':

) -( )!7<, E%A7- )NWN# P%0!08 M0#( $%O!X:

o 7!Y"'/# )*4G#%*x $G, ZA2"#( %GG+'!6x 5: 3

56 105.00001455.0 $&'"$ xx

o 7!Y"'/# )*4G#%* P%U!-2y $G, ZA2"#( %GG+'!6y 5:

356 105.00002062.0 $&'"$ yy

- )NWN# M0#( 2+! [#H*22R )!7<, E%A7:

998.0,935.0 "" yx

) 1( - )&*75 P%0!08 M0#( $%O!X 7)!7<, E%A:

o 7!Y"'/# )*4G#%*y $G, ZA2"#( %GG+'!7y 5:

467 105.00000038.0 $&("$ yy

o 7!Y"'/# )*4G#%* %'-x 5 [#H2 @\7]- F(7!7+"# ^%"0G %GG?::

467 105.00001246.0 $&("$ xx


106

! "#$%!& '()

) -*

!"#$%&! '()(* +,- +./0 "-10xex sin !"*21 3+45 67+! 8#*)*/.

) -)

!"#$%!2 9* "-10x2 ex "*21 3+45 67+! 8#*)*/.

) -+

:));*;%!2 <=#(> =(; "*21 3+45 67+! "-10:

" #2ln

ln

$

!

x y

yx

) -,

!"#$%&! +,- "#-)? :>1); ()+= 6"@>A2xex!

:% B+(!#Cx = 0.5 .D&5 91/*&! C1&=%!2 )+2+E>!2 F2+%!2 ""5 6E:

)0 ( :);G2 :)$H1%! 7"(2 place accuracy)I )B( <H21% >A! 7"(6 place accuracy)I

I6#7+0 >A! 8#*)*/ +,-!2 "-10

) --

%!21 91J2 <C+!2 'K L"1-1%!2 *#A%!2 )+45 6#7+0 $C+J BA*2 L+1/*:)C : 6)(>A%!2 =@!2x – y = 0 D;*;%!21x + ln y = 0 +1*%1y.

) -.

M)C=>&! N+./!2 %)(!2 6#7+0 A%@! "-10:

" # " #xexxg!$ 2

21 2

(1-2) s¹d9

(2-2) s¹d9

(3-2) s¹d9

(4-2) s¹d9

(5-2) s¹d9

(6-2) s¹d9

w½U¦�« qBH�« �UM¹d9


107

!" #$%: ! ! ! ! !./,4,3 xfxgxxxgxxf "#"" $

) "( &'()'( *+ *,- ./0 $,1" )23"$ ) 4x = 1 *,- ./0 $,1" !51 (67 *+x

380.1 %.

) 9( * :-'( *;()<=( )52:7 93>/?'(x!:: < ?'( @/5AB &/A ' ! !xgyxfy "" , C'63 &D,5BB?'( E5,:$AB>' &:$:$1B'( &A:$/'( F()0B-5, &<:<G &:$H4 F5I$" &D,$J) &A:$/

&B,5;'( &/A '( ( &')5D?'( K<'f(x) = g(x).

)i( :$:$1B'( L:G'( M" &'3,A? )DB &:'5B'( &(acceptable) ) N'7 M)OB M"93>/?'( K<'( N'7 9$5AB (P(65?'3 Q&'3,A? $:R )DB 5S:"3

)1( 431 %"# nn xx

)2( 31 4#"# nn xx

)3( 31 4%"# nn xx

)4( 2141nnn xxx #"#

)5( n

n

x

x

nx4

1

#

# "

'( L:G'( M" &'3,A?–E)23 !7– P(65?'3 QF()0B-T' 5S>U+" V)WDXB

)ii( &?:I )52:= &'3,A? &Y:G F)0B-(x . Z5/D?'( L:G'( K1 E 51 (6 [(3 &A:$/ M" \,/ 3" &,-5 ? ]$0" &Y:G M" F)0B-5+ &'3,A? $:R

&?:I )52:= 5S+$DB ]$0"x.

)iii( ?:I *+ ./0'( $)WI &x 5S:>4 E>G< *B'(.

) _( *A:A<'( $62'( &?:I 5?– !::$H4 !:?I$'– &:,:D1B'( &')5D?>'

043 "%% xx

&A:$/ F()0B-5, $62'( (68 K " &:,:1$B'( &?-A'()$ $38 &A:$/( !?3 aFW;– ]$0" &A:$/ M., 3"– P&')5D?'( b6S' K?51'( K<'( )23"

(7-2) s¹d9

w½U¦�« qBH�« �UM¹d9 ∫w½U¦�« qBH�«2

108

&')5D?'( K<' $:$1B'( F()0B-( 93>/?'(x + ln x = 0 58$62 !" F3>D?'(35.0&'.

*B'( L:G'( )23" F; !?3 Q &:'5B'( &:$:$1B'( L:G'( !? K1 9$5AB c<,(F()0B-T' &Y:G KU+"3 Q 5S?()0B-( !1?::

)i( nn xx ln1 %"#

)ii( nxn ex

%# "1

)iii( 21

nxn

n

exx

%

#

#"

!" #$% ! 32 %%" xxxg

) ( &?:I *+ *,- ./0 $,1" 38 5?g !" F>4 (6701.000.4 ("x

) !( $62 )52:7 93>/?'((r) &')5D?>'g(x) = 0 &:$H4 F5I$" &;T;' d5<:<G .

'( E5,:$AB>' &:$:$1B'( &A:$/'( \:,/B K35< - 5 " #$% 3 &D,5BB?) &A:$/&B,5;'( &/A '( (!:B:'5B'( !:B 1??'( !:BY:G'( ])<7 *+ &')5D?'( @U3,:

)1( !22 3%" xx

)2( !21

3 xx #"

P(65?'3 Q&,$5AB? E($:$1B N'7 M)OB M" Q&'3,A? &Y:G )DB !:BY:G'( M.+

93>/?'( $62'( )52:= &'3,A?'( &Y:G'( \,/r .(6 [(3 '( E 51 !5BY:G5?S>U+" F)0B-5+ !:B'3,A? d5,$5AB e$-J( M" Q.

) "( &?:I *+ ./0'( $)Ir 5S:>4 E>G< *B'(.

(8-2) s¹d9

(9-2) s¹d9


109

(10-2) s¹d9

!"#$#%&' ()!*+": xyey x 3,

"+)*#&' ,-: ! " ! "2211 ,,, yxQyxP ##)./0&' 12#'.(

) 3( +4"# 5*"1) 678+9'– 7!:"; 49-'11<=>' ,#"9&' ,?'7$;'1x 6@1& A!$"$=

7$'4 B10C.

) D( 5%"@ 7!:"E D4F)%&'2x 5*"1)&' 6'78+9!G 5"1"1/+&'"5+G!?&' 5H)*#&') " I!G"1*+&' 5*"1)5JG!++%&' .( "+<"=&' % B3 D1!*++ .K-

L&E "+"&!+&' "+"1"1/+&'2xM )i( 31

ix

iex $

)ii( ! "ii xx 3ln1 $

5%"@ 7$'4 B10C 6@1& 7:432x N19>' 5<"=&' 43 O5G1!*+%&' 5<"=&' P"G)+G !+<"=&' I#!/ 'QE R183 5<"= 43 O "+G1!*+% !+<"=&' I#!/ 'QE A!G1!*+

"+7C!G+%.

) S( #%&' "G T14=$%&' 5$!9%&' 7$'4 B10C 6@1& 7:43 A!*G!9 "14/Q%&' ""#$.

IG!?&' 5%"@ ""10C "%@1& 7:43k ,#"9&' 14$%&' 4/" U"$G! "0 y

L#$#%F& A!9!%%

! "0ln10 %& xxekyx

(V4G W&Q4 5")8&' 1"X IY7!J%&' .$& 5+G!?&' 5)*#&' 5*"1) 678+9+ T14=&' ,- AY43 5&7!J%&'

! "xgx (*)

(12-2) s¹d9

(11-2) s¹d9

w½U¦�« qBH�« �UM¹d9 ∫w½U¦�« qBH�«2

110

) 3( !KG1!*+ 4/" L+%4 M 5&7!J%&' ZQ[ .$& 5*"1)&' ZQ[ D1!*+ )10 4[ !%N193.

) D( 1'7*%&' 5-!V\G W&Q4 5*"1)&' D1!*+ 5C19 T7!"]& 5&7!J%&' T14= ."7J+ /%"x' ) U"$' 7JG !%"- 77$" B1!"+8' IG!? (? 5&7!J%&' ,-1) L&E 5%9@ 6

LFC "-1)&''$1 )U"$1&(' (T14=&' ,- 5&7!J%&' ^G=+-:

! " xxgx''

' $$

$

111(**)

T14=&' ,- B3x = G(x) !K"FC 5+G!?&' 5)*#&' 5*"1) P"G)+ /%"-.

_- MT7"7:&' !K+14= ,- 5&7!J%&' .$& 5*"1)&' D1!*+ )10 4[ !%

`- 5G9!#% 5%"@ 1+8' IG!?F&' /%" !% N193 5*"1)&' D1!*+ 4/" U"$G.

) S( 5&7!J%&' .$& 5+G!?&' 5)*#&' 5*"1) PG)03 & &xex 7$'4 1Q: !K& ,+&' % D"1@x = 0.7 5"10C 6!@13 5?a?& A!$"$= 1Q:&' LFC .4=$F& W&Q4:

AY43 : T14=&' ,- 5&7!J%&' (V4G(*).

"#!?!: T14=&' ,- 5&7!J%&' (V4G (**) "F$&' "G 1!@4 O.

5&7!J%&'xex & % D"1@ 1Q: !K&x = 0.5 . A!$"$= 1Q:&' 'Q[ 7:43 b]:&' ,- "+14/Q%&' "+*"1)&' % ./ N!G+!G 5"10C 6!@13 5?a?&)S ( .'c9&' %

PG!9&'12-2 "F$&' "G 1!@4.

678+9' +4"# 5*"1)– 5&7!J%F& 1Q: ""J+& 49-'1x ln x = 1 59%8&5"10C 6!@13.

(14-2) s¹d9

(13-2) s¹d9


111

(15-2) s¹d9

(16-2) s¹d9

(17-2) s¹d9

(20-2) s¹d9

!"####$% &####'$() *+,-!####./0– &####1,/2341 5+(6####7 ,####7"8 "####.9+(012 !z 5/:,####!03 &$:+,!0;+ &3$'1/0iz ! 10

&1,/2341 <7"31+ (671+ 8 =(>%0,012 " #! axax *+,-!./0 <.?@$. &$($(A!1+ &B$C1+2

1 1 ii xax # ! . D''?! 8 <7$ E61+ )(F1+ "G /3a H<(/'!41

I/$%?%31+ $0 J("C?31+ &?/.31+ $$(FK $3L(1 <.?+: y = ln x, y = cos x, y = 0

M$? 2

1 $%% x

1+ &$(FK */L(8 &.3-1 <.?+ $$%?%31+ $0 J("C?31+ &?/.3:

xeyxy # ,cos

M$? 2

0 $&% x

&1,/2341 <7"31+ (671+ </.?0 *"'%. /%%8 =(>%2bxax # ) 3 NA M$?a

, b <7"3 ,,K (&$($(A!1+ &B$C1+ *+,-!./0 21 kk bxax # ! H<(/'!1+ )(F "G /39

$$%?%31+ O)/'! &)'% &$(FK */L(8 &20(P ,7"8:

' (2ln

,2

!

#

xy

ey x

(18-2) s¹d9

(19-2) s¹d9

w½U¦�« qBH�« �UM¹d9 ∫w½U¦�« qBH�«2

112

(23-2) s¹d9

(25-2) s¹d9

(26-2) s¹d9

$0 OL+" ,?+" (67 /Q1 &$1/!1+ &1,/231+1 R2: 1346 !! xxx

&$(FK */L(8 &!.1 (671+ +6G ,7"8.

3 /0$(L +(67 /Q1 &$1/!1+ &1,/231+x = 2:

07976125 234 #!## xxxx

&$(FK */L(8 &20(P $(671+ $6G ,7"8.

&?$?C &$(FK */L(8 &.3-1 &$1/!1+ &1,/231+ N? ,7"8: xx cosln

&3$L SG /3x &1+,1+ N27! S!1+ &07"31+: 2tan

xxy /Q!3$L T1U NC!

HV(BC1+

! 8 A3$ &3$L (BC8 &$(FK */L(8 &.3-1 ,7"8 /Q$1U NCa W'?!$ M$?0 *$L O$371 S1/!1+ )(F1+x H&07"31+ xxa sin)

8 =(>% ' ( ' ( ' (*+,-

./

dxxGd

xGxD /

M$? ' ( 121065 34 #!# xxxxG

) 8( &3$L ,7"8 (%("G &'$() *+,-!./0D(1).

(21-2) s¹d9

(22-2) s¹d9

(24-2) s¹d9


113

(30-2) s¹d9

) ( !"#$% !&' ()&)&*+,$ -+%). (/)0 123) 4$"#$% 4$&)&*+ ( &56 "76)89':, (,"7G(x) = 0 ();$"+2<$ (')=,72 47;"+2'10 x.

(,$":, >'?9,$ (')=,$ @)#0 ""A &BC "6%Dy(x) -' &=,721x $5E F-D G:A: ! " 0sin #$ xyyx

(A%'6':, H# "76)8 -+%). (=)&3 -' 4$"#$% 4$&)&*+ 123:

22

22

yxy

yxx

$

#

-' &=,72(0.8 , 0.4).

H# 7'I, -7+),7+,$ -7+,"79',$(x , y) -' )&B(0.5 , 0.5):

0sin2.0cos7.0

0cos2.0sin7.0

#$

$$

yxy

yxx

(+27J,$ (3=.,$ (=)&3 G$"K+L72 "#$% M&NA GB&, 47#)#0 H#,$ $5O "6%D.

)+)3K,$ &)P -)+).Q$ -)+,"79',$ -' (.%*',$ (A%'6',$-:

! "! "22

22

tan5.0

31

yehy

xehx

x

xy

##

##

7I:# -*')– G)=,h !&)/0,$– &)&*+,$ G$"K+L72.

5KR+ 7'".A (A%'6',$ H#, 476'7.&2 +*$h (),7+,$ G)=,$:

! "10.0,,03.0,02.0,01.0,0

1.001.00

%

h

(27-2) s¹d9

(28-2) s¹d9

(29-2) s¹d9

w½U¦�« qBH�« �UM¹d9 ∫w½U¦�« qBH�«2

114

(32-2) s¹d9

&7)+K72 D"2$: 5.0

1

0

0

y

x

-' H* (')B ST &)/+,$ @20) 7'".A U$&)&*+,$ VWB%D%x, y -' HBD21.0 h F &56,$ (')B X23$ GJ(x , y).

&$"=':, ()2)&=+ (')B "6%D V)0.+,$ (=)&3 G$"K+L723 25 -)'B&, 47#)#0-))%.9' .)"7N&E : &2+A$!

" 253 $ xxf.(

()'*:, ()2)&=+ (')B "6%D3 C (#)#0 Y72+72 Z,5% F()%.9' G7B&D (92&V)0.+,$ (=)&3.

$["# "6%D(bound) H%0#:, V)0.+,$ (=)&3 -' ('\],$ U$&)&*+,$ ""9,^+B" )&=+ >:A410$ (,"79',$ H#,: 013 $$ xx

!&+_,$ ST X=) `)#2[1 , 2] .(6&",$ a5O ^, &56:, 472)&=+ "6%D (B",$ -'.

""A b%*9' 7L# -*') "a" G$"K+L72 Z,5% F('LB ():'A c$&6E -%"2U$&)&*+,$: ! " ,2,1,0;21 $ # iaxxx iii

)D ( >:A ()&)&*+,$ -+%). (/)0 1)23+2 ()&)&*+,$ (/)0,$ a5O >,E H%0%,$ H%7# (2L7.' (,$"f(x) (2L7.' (,"79' H#, MDf(x) = 0.

) ( (')=,72 7;"+2'2.00 x ""9,$ b%*9' L#$a = 4 G7B&D (+L, 47#)#0.

(31-2) s¹d9

(33-2) s¹d9

(34-2) s¹d9


115

-!":

-D d&_."r" ""A &&*' &56m (,"79':, U$&',$ -'f(x) = 0 `)#(multiplicity) m > 1. (,["e9f',$ -+%). (/)0 -D U2J$:

! "! "

,2,1,0;1 &

$ # ixfxf

mxxi

iii

47)9)2&+ HBC$ >:A &7=++)",$ >:A (2L7.' H70+$ 3%&N d&_2 (,$f ( 7'.)2 F 479)2&+ &7=++ < ():0C$ -+%). (/)0.

"7N&E : (27+* -*')f(x) !&%0,$ >:A:

! " ! " ! " ! " 0;. '$ rqxqrxxfm

-!":

!"# $%&'(r )*+*", -+*% &&( ".! /+**"% /+*01m 2!34% *#5) !6 +06! /7&$( 8$9.+ ( :*$;,.+ <#*=, >?#@ AB C)>.&4%.+ ,!#' >?#@:(

! ! ! )1(1;1 "#$%&' mxfxf

mxxgxi

iiii

)>;D$=.+ >.E=%.+ *F'+35-2 ( G H#DI,3. J3@, Km 8!L1% . "%#!>.+&.+ M -$DNO:

! ! !xf

xfx #&(

&'( *01 /$P#M $L. !"#r **"% *#5 *01 Q'".! . >?#@ H#DI, "%# R.$,.$D! >.+&.+ S3( >#3@G+ ,!#'( /$#4#D*, :*$;,, R,.+ >#.$,.+ >?#@.+ RI4,. ) T*UD

8$@,K+ I!*V T4D:(

! ! !xx

xxg ((#$&

! !22 )]([/))(.)()](([

/xfxfxfxf

xfxfx ###$#

#$&

! !

! ! ! !2

.

.2

xfxfxf

xfxfx

##$##

$&)*+

,-.

(35-2) s¹d9

(36-2) s¹d9

w½U¦�« qBH�« �UM¹d9 ∫w½U¦�« qBH�«2

116

WXN.+ ,!#' Y#@ #D *$;' >.E=%.+ )06 RB) >.Z&[4\%.+! C >#3@G+(1) C 7&#&1.+ >.&4%.+!(2) (>'#4% >.+& S3( $L;#DI,D.

>.&$4%.+ M 2!34%: ! 044 24 &'$% xxxf

**"% *01 $L.2&r Q\U\($P[,!))*+*", -+*% &&( ]M (m = 2 . WXN HDI>#_+&,DK+ >%#;.$D /$_&,D% WXN.+ ,!#' Y#@ % 8" % -+*#*",5.10 &x *$ !

J#9@.+ *01.+ <% a_$,'.+.

>.&$4%.+: ! ! 02

1ln2

&'$'& xxxxf

01 $L. &'( *x = 0 . >%#;.$D /$_&,D%25.00 &x -+*% b%c ,!#' >?#@ HDI*01.+ +06 &$1#d . e#" f+0$%.! f*01.+ !9' /$#4#D*, :*$;,, -+*#*",.+ M !&D# 86

f>.$9.+ )06 RB :*$;,.+ g*=,)>F9X% : ! 00 &#f(

>.&$4%.+ M -DN+: ! ! 011ln &'$'& xxxf

&9+! *01 $L."r" 7*,U.+ RB IDP.$D 3] [2, . >,D$N.+ >I;'.+ >?#@ M -DN+!>#*#*",.+: ! 11ln1 ''&' ii xx

*01.+ !9' :*$;,, r >#_+&,D+ >%#` ]ED #_&,D%/ 03,20 1x . -+*#*",.+ &&( *&` S3( 8!@93. >D!3I%.+r /$9#9@>%#h;.$D #h_&,D% 2$ *M >4D*G5.20 &x :=9+! C

r >#_+&,DK+ >%#;.+ )0LD /$_&,D% >`&.+ )0L. !5.20 &x.

#$%&': >;,V%.+ i!3= b*&+ !xf # 7$I4%.+ 7*,U.+ RB3] [2,.

#,*,U.+ ]M[1 , 2] , [2 , 3] f>#.$,.+ >.&$4%3. /+*01 $L#B M %P'

! 015 24 &''$& xxxxf

(37-2) s¹d9

(38-2) s¹d9

(39-2) s¹d9


117

2"! fe#@',.+ >;#*I H#DI, &'( *$D,(K+ RB $60cE' R,.+ >#.$,.+ 7*,U.+ R6 $%!S.O 7*,U.+ 8!I *#@;,. >D!3I%.+ -+*#*",.+ &&(610$f

>.&$4%.+ 89 &'( ! 04 2 &$% xexxf <P! "%# C>,D$N.+ >I;'.+ >;#*ID #,#.$,.+ #,*!@.+ j&9O RB >.&$4%.+:

!

! .4ln

,21

2

2/

xxgx

exgx x

%&

%&

) M( *01.+ S.O QD*$;, !%P% *#*", S.O ]&k, #,?#@.+ ]Mr 7*,U.+ RB/ 0 / 01,5.0, &baf+0$%.! C

) :( S3( 8!@93. 2l3# /+*#*", 2"r >%#;.$D #_&,D% C2$ *M >=%c. /$9#9@10 &x f

) m( &1!Mr :;(! C #,;D$=.+ #,?#@.+ % 8" 2+&c,=$D 2$ *M >=%c. /$9#9@a_$,'.+ S3(.

) M( >.&$4%.+ M -DN+ ! 02cos4 &$$& xxxf &9+! *01 $L.r RB IDP.$D7*,U.+ / 02 0,.

) :( >#.$,.+ >#*#*",.+ >?#@.+ :*$;,, 86).+ >;#*I>,D$N.+ >I;' ( *01.+ !9'r

>%#` ]ED #_&,D%/ 02,00 1x f+0$%.! f

ii xx cos41

21

1 '&'

S3( 8!@93. >%lX.+ -+*#*",.+ &&( *&`r /$_&,D% 2$ *M >=%c. /$9#9@>%#;.$D10 &x&1!M 2N Cr >#_+&,DK+ >%#;.+ i3,D /$_&,D% >`&.+ )0L.10 &x.

) m( nl1.+ 89 &(M): (>#*#*",.+ >?#@.+ /$%&c,=%:

2cos51 $$&' iii xxx

(41-2) s¹d9

(40-2) s¹d9

w½U¦�« qBH�« �UM¹d9 ∫w½U¦�« qBH�«2

118

(43-2) s¹d9

(44-2) s¹d9

!" #$%!&'"( )*+"( ,-./ 0&1%"a" ),-./ 233 a ( 45%6"( 7( 8'.-$4%)%)&'"(:

!

"##$

%&'& 2

231

1

ixaxx ii

$9'$ 4%)%)&'"( 45%6"( :*; <"= >?6?"( >?-/ 4%)%)&'"( 0'?%@ 45%6 A% 4$.-@1 4"( <BCf(x) . ,./( 7D3 20 45%6"( :*E$ 7-F)3 4.18" G-/%/6

4%H( '$I( 41%J"-$ G-H '$1 4%)%)&'"(20 'x.

) 3( 0-& (*= K@3 L$D(r G-M NC-O1 G()*+(double root) 4" -!1B"0)( 'xf 0PQ4"R !1"( 0'?%@ 45%6 -1@%$ SG-%!%$)' ,)-J'' I 0'?%@ 45%6:

( )( ) ( ) ( )

( )xfxf

xxx

xxi

iii *'*+'& ,,

,,1

?/@ G-%!%$)' ,)-J''r.

) ,( 4" -!1"(:

( ) ( ) 0cos1 '&' xexf x

@C )*+ -E"-'x . A$94 (*; <"= >?6?B" 0'?%@ 4J%)9 01 L(?98 41%J"-$ G-H '$1 )*+"(30 'x .'F-Q ,)-J'' 7" 0= TG-%!%$)' 7%J"( ,)-J'' >; U)

G-%!%$)' ,)-J'' 4%)%)&' 45%6 A$9? S4 45%6"( :*; 01 L()%)&'.

0%'%98 )%V 0%'" -!1 01 #"-'"( 7-W@"( -@% ":

( ) ( )( ) ( ) 03.0sin,

03.0cos,

3.0

3.0

'+'

'+'

yeyyxg

yexyxf

x

x

) 3( 0%'1%J"-$ G-H '$1 7-W@"( (*; >/" 0'?%@ 4J%)9 01 0%)%)&' A$9:

1,1 00 '' yx

(42-2) s¹d9


119

) ( !"#$%&' ()*#+ !,#-. /01)2'(quasi-Newton method) 3.4) 5)&'*(#)#&6)&' (#)784&6$:

!"

#$%

!"#

$%

&

!"

#$%

!"

#$%

&

'(

'(

iyixygiyixg

yy

iyixxfiyixf

xx

ii

ii

,

,

,

,

1

1

] 9#:xf

f x ))

* *yg

g y ))

*[

(#);#,&6$ <6=0)$;00 1 yx (( 8>& !,#70 !$*?@ 3AB C*D:A&/67-@ !>.

5&6)&' /6E+&' C:& ()*#+ !,#-. (; <'0:'* <'-#-F) G$.:

+ , 05.321lncos

03.0sin

09.0

22

('&&&

(&&&&

(''&

' z

y

eyx

zyxx

yex

/#,&6$ <6=0)$;0000 ((( zyx.

(#)#.1 -#H (#)&064; (; 5&6)&' /6E+&':

+ ,+ , + , 01sin,

01,

('&&(

('&(

yxxyyxg

xyeyxf x

.*$I;&' C:&' J&x = 0 , y = 1.

) @( + !,#-. (; (#-#-F) G$.(#);#,&6$ <6=0)$; /6E+&' 'KL C:& ()*#:

5.0,2.0 00 (( yx

) ( C'M2&' 5N O-*FK;&' !"#$P%&' ()*#+ !P,#-. (; (#-#-F) G$.)44-2 - (

(#);#,&6$ <6=0)$;5.0,2.0 00 (( yx.

(45-2) s¹d9

(46-2) s¹d9

w½U¦�« qBH�« �UM¹d9 ∫w½U¦�« qBH�«2

120

!&064;&' C:& !)$6>&' !.,+&' !,#-. G#$.) 0+B: + , 0sin2.05.0 (&'( xxxf

3AB 6+AD: 5&6)&' C*0?&' 5N !+#$;&' !4$6));&' Q6$#-,)&': k

kx

0 0.5000 0000

1 0.5958 8511

2 0.6122 4830

3 0.6149 4176

4 0.6153 8219

5 0.6154 5412

6 0.6154 6587

7 0.6154 6779

8 0.6154 6810

9 0.6154 6815

Q6$#-,) /#7 62:& (F)R !,#-. G$. !A S?4;&' !#&6));&'kx.

-#-F)A& !$26+; !&'0 !#&6)&' !&'0&' 04T) !,$62&' !&U2;&' 5N: + , xxg sin2.05.0 &(

(2+V#)2 !#;W-'*1 /'01)26$ !A?4;&' Q6$#-,)&' !#&6)); 2:' XKL -#-F)&' !&'0 C6;4)26$ !#='0)$Y' !;#,&' K1U$*5.00 (x.

!)$6>&' !.,+&' -#-F): ii xx &(& 21

!&'0A& ?*;&' -K?&' 06?#Z: + , 22 ''( xxxf!#&6)&' Q6$#-,)&' !#&6)); [)+#:

99759.1

99036.1

96157.1

84775.1

41421.1

0

5

4

3

2

1

0

(

(

(

(

(

(

x

x

x

x

x

x

-6,)&' C04; 5N (2:)&' E:Y* !#&6));&' XKL C#?4)& (F)R !,#-. /01)2' -K?&' *:+2(-.

(47-2) s¹d9

(48-2) s¹d9

(49-2) s¹d9



WOD)« �ôœUF*« rE½

Systems of Linear Equations

W¹d¹dJ²�« ‚dD�«Ë …dýU³*« ‚dD�« º

…dýU³*« ‚dD�« ∫ôË√

W¹d¹dJ²�« ‚dD�« ∫UO½UŁ

W¹d¹dJ²�« ‚dD�« »—UIð º

W�uHB* wHOD�« dDI�« nB½ n¹dFð º

»—UI²K� WOÝUÝ_« W¹dEM�« º

Y�U¦�« qBH�« �UM¹d9 º

123


WOD)« �ôœUF*« rE½Systems of Linear Equations

!!!"#$%$ #& '(!!)* +!!!, -.!!/$ 0!!!-1 +2!!343 566!!!7-8 +!!9:,3-8 +;(!!!/$ <!!$ =!!9>?@A6(!!7$-8 <!!$ !!9B4 . CD>!!$ +;(!!/$-8 EF!!G <!!$ : #& !!96(7-8 9:!!H(I3-8 @A6(!!7$-8 +!!,

J6#6!,$-8 K#=!I-8 K=!BL !9;2%-8(finite difference methods) M !$9N-8 +;(!/$ +!,# !!!938F-8(eigenvalue problems) 9!!!H(9=-8 O(!!!929I-8 P!!!Q @(!!!*(9L-8 K!!!9Q#3 +;(!!!/$# M

(fitting of data) @(!7L=$-8 =R!S& !N9=BL(least-squares)M 6#6!,-8 @8=!9>? T!9=N3#(polynomial approximation). P!!Q C8=!!9>? 6!!9I9 @(Q#I!!S$-8 @(,DB!!S8 +($73!!/8#

')* +;(/$ +, 9B4-8 @A6(7$-8.

P!!!!Q (UL3?*!!!!/ (!!!!U:, T#!!!!:B$-8 !!!!9B4-8 @A6(!!!!7$-8 <!!!!$ !!!!"#$%$ <& V=!!!!I*J=#S-8:

Ax = y

W9,:

!

"

####

$

%

&

nnnn

n

n

aaa

aaa

aaa

A

!

21

22221

11211

A :@D$(7$-8 Q#IS$ PG

!

"

####

$

%

&

nx

x

x

x!

2

1

x :E6(%91 T#:B$-8 X%3$-8 #G

!

"

####

$

%

&

ny

y

y

y!

2

1

y :PQ $#:7$-8 @L8#>-8 X%3$ #G <$9Y8 Z=B-8

124

nnnnnn

nn

nn

yxaxaxa

yxaxaxa

yxaxaxa

&'''

&'''

&'''

2211

22222121

11212111

B!!NQ 6!!,8# +!!, [(!!*G (!!U9Q <#!!?9 P!!3-8 !!-(,-8 0!!:" (*3!!/8=6 P!!Q =!!S3N*/#x

X!!%3$ +!!?- !!"#$%$:-y <!!$9Y8 Z=!!B-8 P!!Q . !!-(,-8 0!!:" =!!S3N*/ (!!**& P!!*79 8F!!G#!!!" 5#(!!!/9 !!!"#$%$-8 @A6(!!!7$ 66!!!" (!!!U9Q <#!!!?9 P!!!3-8 +!!!9G(%$-8 66) 5#(!!!/9n( 5& M

@D$(!7$-8 Q#IS$ <#?3 ($6*"A !7L=$) C(NL(!/ <9!L$ #!G (!$? .( Q#I!S$-8A P!Q \(!!?7*D- !!:L(] <#!!?3 <& T!!%9 !!-(,-8 EF!!G(invertible) !!"#$%$:- <#!!?9 0!!3,) #&

'()*:- ( X%3$ +?- BLH-(L 6,8# +,y.

Q#IS$-8 <#?3 P?#A -8 KN,39 <& T%9 \(?7*D- :L(]B=^:

( ) 0det *A

W9,det(A) J66,$ PG(determinant) Q#IS$-8A.

!!"#$%$-8 +!!, <!!" =!!L7* <& !!-(,-8 EF!!G P!!Q C(!!H9& (!!**?$9#Ax = y !!-A6L =$8=? J6"(] 0$/9 ($L @866,$-8(Cramer's rule) =!9_ @866!,$-8 '8643!/8 <& A1 M

!!9B4-8 ')*!!-8 +!!, P!!Q !!9:$7-8 !!9,(*-8 <!!$ 6!!9I$ #(!!/39 J6!!,8# J66!!,$ T(!!/, <Y C(!!$#$"– L#7!!S-8 !!%=6 W!!9, <!!$– P!!B4-8 '(!!)*-8 +!!, a!!$ . '643!!/* <!!:Q 8F!!U-#

!!!9B4-8 ')*!!!-8 +!!!, P!!!Q @866!!!,$-8 . =?F*!!!/ (!!!**& A1– b8 <Fc!!!L 6!!!7L (!!!$9Q– !!!N9=B @866,$-8 '9] T(/,-)=4& @A(%$ PQ '9N-8 EFG '8643/A ( 9*L$ N9=B-8 EFG# M

\(/& 0:" 9B4-8 ')*-8 +,- J=^(L$ N9=B.

<99/(/& <9"#* 0-1 9B4-8 ')*-8 +,- 9667-8 K=B-8 '9/N3 C($#$" <?$9#:

!"#$%&' (!)&' (Direct Methods)

*+!+!,-&' (!)&' (Iterative Methods)

. - !"#$%&' (!)&'(Direct Methods)

56d3 P3-8 K=B-8 PG#– 8 O(B4Y8# T9=N3-8 O(B4& T(9_ PQ =4Y– 0!-1 B9!!!/L-8 9L(!!!/,-8 @(!!!9:$7-8 <!!!$ 6#6!!!,$ 66!!!" 6!!!7L B#L!!!H$-8 +!!!,-8 . !!!9,(*-8 <!!!$#


125

WOD)« �ôœUF*« rE½ ∫Y�U¦�« qBH�«

@(!$:?:- J6#6!,$ +8#B.!L +$79 T/(,-8 <Y C8=)* 9:$7-8(finite word length) <c!Q B#L!!!H$-8 +#!!!:,-8 0!!!-1 J6(!!!" 56d!!!3 A J=!!!^(L$-8 K=!!!B-8 . O(!!!B4Y8 <c!!!Q a!!!]8#-8 P!!!Q#

T!!!!9=N3-8 <!!!!$ !!!!%3(*-8(round-off) =8=N3!!!!/A8 '6!!!!"# M(instability) V!!!!7L <86!!!!NQ# M 9#*7$-8 '(]=Y8(loss of significance) !;B(4 (!$L=# !N9]6 =!9_ e;(!3* 0-1 56d3 6]

C(!!!]DB1 J6!!!9I$ =!!!9_# . O(!!!B4Y8 EF!!!G !!!/8=6L K!!!:739 566!!!7-8 +!!!9:,3-8 <!!!$ =!!!9L? O2!!!%#B-8 <" W,L-8# (U9:" T:R3-8 +;(/## (U3.^* 9I9?# TL/ W,L# 0!-1 56d!3 P3-8 K=

(!!U:$%$ +!!9:N3 . ZF!!,-8 !!N9=B P!!G J=!!^(L$-8 +#!!:,:- '643!!/3 P!!3-8 9!!/9;=-8 !!N9=B-8# \#(!!%-(Gauss elimination) (G) !!-f67$-8 !!N9=B-8 #& M : \#(!!% !!N9=B– <86=#!!%

(Gauss – Jordan) . \=6*/#– b8 <FcL– <93N9=B-8 <93(G.

/ - *+!+!,-&' (!)&'(Iterative Methods)

56d!3 !*97$ L!/(*$ !9$2=8#4 g(!L3(L '!> P;86!3L8 T!9=N3L &6!L3 P3-8 K=B-8 PG#+HQ& 7L(33$ @(L9=N3 0-1 . -(,-8 EFG PQ#– !9:$7-8 T=(!N3 a$ 03,– (!$ +!? <c!Q

!!9=9=?3-8 K=!!B-8 EF!!G +!!>$ g(!!L3(L '(!!)*:- P!!L9=N3 +!!, #!!G X!!9-1 +#!!S#-8 +!!$.* . K=!!B-8#-8 ZD34(!!!L Z!!!:343 !!!9=9=?3-8 =8#4 !!!7L3$-8 !!!9$26!!!7$ !!!9,(* <!!!$ Z!!!:343 (!!!U*& ($!!!!? M +

T=(!!!N3-8 .6!!!"(L39 6!!!] K=!!!B-8 V!!!7L# W!!!9,L 696!!!^ OBL!!!L T=(!!!N39 6!!!] =!!!4h8 V7L!!!-8# M 9L(/,-8 9,(*-8 <$ C(9:$" 69I$ =9_ iLS9.

9=9=?3-8 K=B-8 @829$$ 'G&#:

(U3B(/L.

'(!!!)3*8#(uniformity) =!!!%j3 P!!!3-8 @(!!!9:$7-8 C86!!!% L!!!/(*$ (!!!U:7%9 (!!!$$ M

@(L/(,-8 '8643/(L K9LB3:-.

(U39!!!/(/, '6!!!"#(insensitivity) #!!!$*# =(!!!^3*A C(9L!!!/*(growth) O(!!!B4& T9=N3-8(round-off-errors).

C(!!9:$" $643!!/$-8 !!9=9=?3-8 K=!!B-8 '!!G& <!!$# : PL#?(!!% !!N9=B(Jacobi) (J)

\#(% N9=B#– +6!92(Gauses – Seidel) (G – S) . \=6*!/#– b8 <Fc!L– <93(!G<93N9=B-8.

WOD)« �ôœUF*« rE½ ∫Y�U¦�« qBH�«3

126

0-1 (G=#6L 9B4-8 ')*-(L BL3=$-8 @(Q#IS$-8 '/N*3#:

) 0( *1+2, 3#4516%(Dense).

) 7( *8"9 50 !2#:-% 3#4516%(Sparse).

) 0( *1+2,&' 3#4516%&'(Dense Matrices):

!!!" 0!!!:" 5#!!!3,3 P!!!3-8 @(Q#I!!!S$-8 P!!!G 9=I!!!S-8 =!!!S(*7-8 <!!!$ C86!!!% +!!!9:] 66 M L3=#(order) (Q#I!S$-8 EF!G +!>$ C(9L!/* J=9R!S J6(!" <#!?3 @) (!$L=kll +!]& #& .(

K=B-(!!L @(Q#I!!S$-8 EF!!G +!!>$ 0!!:" 5#!!3,3 P!!3-8 +;(!!/$-8 +!!,3 <& J6(!!" .!!I?Y8 <!!$#J=^(L$-8.

) 7( *8";&' 50 !2#:-%&' 3#4516%&'(Sparse Matrices).

9=I!!S-8 =!!9_ =!!S(*7-8 <!!$ C86!!% +!!9:] 66!!" 0!!:" 5#!!3,3 P!!3-8 @(Q#I!!S$-8 P!!G(nonzero elements). @A6(!!!!7$-8 +!!!!, !!!!-#(,$ 6!!!!*" J6(!!!!" @(Q#I!!!!S$-8 EF!!!!G .!!!!^*3#

J6#6!!,$-8 K#=!!I-8 K=!!BL 9:!!H(I3-8(finite-difference methods). 3= <#!!?3 6!!]# +!!>$ !!L C86!!!% J=!!!9L? @(Q#I!!!S$-8 EF!!!G P!!!3-8 !!!9=9=?3-8 K=!!!B-8 '8643!!!/(L +!!!,:- C86!!!% L!!!/(*$ P!!!G# M

Q#IS$:- f U-8 #& J=>(*3$-8 79LB-8 <$ 69I3/3.

'

)0( <5#=& >?@&' *A+!)Gauss (G) Elimination Method

<93#!!!!B4-8 P!!!!Q !!!!9B4-8 @A6(!!!!7$-8 <!!!!$ '(!!!!)* +!!!!,- !!!!N9=B-8 EF!!!!G m4:3!!!!3<939-(3-8:

)i( 0!-1 0!B7$-8 '()*-8 +9#,3"P!>:>$ '(!)* "(triangular system) '(!)*:- nQ(!?$P:!!!!SY8 0!!!!*7$# M"!!!!>:>$ '(!!!!)*P " Q#I!!!!S$ iL!!!!S3 @D$(!!!!7$-8 Q#I!!!!S$ <&

9>:>$.

)ii( N9=BL e3(*-8 P>:>$-8 '()*-8 +,"PI:4-8 V9#73-8 "(back substitution)

!!9I9? CA#& 0-(!!73 b8 <#!!7L i!!H#* <& +!!HI*# +(!!>$L <93#!!B4-8 <93(!!G F!!9I*3B9/L'(" PB4 '()* 5& +,- \#(%- ZF,-8 N9=B 9$2=8#4 8FG 67L =?F* '> M.

(Direct Methods) …dýU³*« ‚dD�« ∫ôË√


127

% B#2(1-3):

!9*h8 !9B4-8 @A6(!7$-8 !"#$%$ +!, 6!%#& \#(%- ZF,-8 N9=B '8643/(L 9-(3-8:

(3)62 3 6

(2)03 4 6

(1)0121520

321

321

321

&''

&''

&''

xxx

xxx

xxx

B@&':

)i( C&5D' 5)E&': nQ(?$ P>:>$ '()* 0-1 0B7$-8 '()*-8 +9#,3.

=!!!!9R3$-8 ZF!!!!,L &6!!!!L81x (7$-8 '8643!!!!/(L !!!!>-(>-8# !!!!9*(>-8 <93-6(!!!!7$-8 <!!!!$6 !!!!-0-#Y8P:9 ($? M:

)(3658-x

23)3()1(

206-

)(2053-x

21)2()1(

206-

)(10121520

32

32

321

+&,'-

+&,'-

+&''

x

x

xxx

!!!-6(7$-8 !!!9*(>-8 !!!-6(7$-8 0!!!:" (!!!*7$%# MP!!!G (!!!$? 0!!!-#Y8 !!!-6(7$-8 (!!!*9NL& <.!!!L [!!!-F# PQ L#=H$ 0-#Y8

206

11

21 ,&, aa

PQ L#=H$ 0-#Y8 -6(7$-8 >-(>-8 -6(7$-8 0:" (*7$%#206

11

31 ,&, aa

3$-8 ZF!!,* <h8# =!!9R2x !!>-(>-8 !!-6(7$-8 <!!$)3( + !!9*(>-8 !!-6(7$-8 '8643!!/(L)2( + MPQ L#=H$ 9*(>-8 -6(7$-8 >-(>-8 -6(7$-8 0:" a$%* <.L [-F#

( )( ) 3

2/12/3

22

32 ,&,,,&+

+,aa

($G ($? 9*(>-8# 0-#Y8 <93-6(7$-8 PNL*#:

)(3651)3()2(3-

)(2053-x

21

)(10121520

3

32

321

++&+'+-

+&,

+&''

x

x

xxx

J=!!94Y8 !!-6(7$-(L &6!!L* 5& MZ!!:4-8 <!!$ &6!!L* e3(!!*-8 P!!>:>$-8 '(!!)*-8 8F!!G +!!,-# $9] (*9B73 P3-83x:

( ) 3051/63 &&x

WOD)« �ôœUF*« rE½ ∫Y�U¦�« qBH�«WOD)« �ôœUF*« rE½ ∫Y�U¦�« qBH�«3

128

$9] (*9B73 9*(>-8 '>2x:

( ) ( ) 3621/30

5302 ,&,-'&x

*9B73 0-#Y8 -6(7$-8 C8=94&# $9] (1x:

( )( ) 920/3012361501 &-,,-,&x

#G +,-8 <& 5&

!

"

###

$

%

,&

!

"

###

$

%

30

36

9

3

2

1

x

x

x

(U!!/I* ZF!!,-8 !!N9=BL (U!!/I* J(!!B7$-8 @A6(!!7$-8 !!"#$%$ +!!, 697*!!/ <h8#(U3D$(7$ #& (UL93=3 PQ @D9673-8 V7L a$ <?-#.

ZF,L &6L*1x 93-6(7$-8 <$0-#Y8 -6(7$-8 '8643/(L >-(>-8# 9*(>-8 <:

)(3658x

23:)3()1(6

)(2053x

21:(2))(16

)(1 053

43

32

32

321

+&,,'+-,

+&,,'+-,

+&''

x

x

xxx

0-#Y8 -6(7$-8 @D$(7$ +? (*$/] (**& ),A(1) 0:"20 +!$(7$ +!7%*-1x

J6!!!,#-8 5#(!!!/9 !!!;Q(?$-8 !!!-6(7$-8 0!!!:" (*:!!!S,Q M)1( + . !!!-6(7$-8 EF!!!G (*$643!!!/8 '!!!>(?$-8 ;Q)1( + ZF,-1x <93-6(7$-8 <$ +? <$(2) , (3) !-6(7$-8 T=!HL [!-F#)1( +

T!!!/(*$ 66!!!" P!!!Q(-6) ZF!!!, T#!!!:B$-8 !!!-6(7$-8 0!!!:" (!!!U7$% '!!!>1x (!!!U*$ 0!!!NL3# M -6(7$-8)1( + PG ($?.

=!!9R3$-8 ZF!!,* <h8#2x <!!?-# !!9*(>-8 !!-6(7$-8 '8643!!/(L !!>-(>-8 !!-6(7$-8 <!!$ +!!HI9– 6!!7L (!!$9Q =!!>?& +9!!SI3L b8 <Fc!!L =*!!/ (!!$?– <93-6(!!7$-8 T!!93=3 =!!o9R* <&

>-(>-8# 9*(>-8– =!$Y8 '2- <1– +!$(7$ +!7%* W!9,L2x $-8 !$9N-8 X!- M =!L?-8 !N:B <& W9,#2/12/3 . <93-6(7$-8 T93=3 =o9R*Q)3(,)2( ++.

658x

23

053

43

32

321

&,,

&''

x

xxx

053x

21

32 &,, x


129

PQ 9*(>-8 -6(7$-8 T=H*(-2/3) +$(7$ +7%-2x J6,#-8 5#(/9 '643/* '> MF!!,- !!%3(*-8 !!-6(7$-8 (!!*GZ2x !!>-(>-8 !!-6(7$-8 <!!$ !!9*(>-8 !!-6(7$-8 T=!!HL [!!-F# M

) !!%3(*-8 ( P!!Q1/2 !!>-(>-8 0!!:" (!!U7$%#) !!9*(>-8 !!-6(7$-8# 0!!-#Y8 !!-6(7$-8 O(!!NL a!!$($G ($? %3(*-8.(

2151

41516x

053

43

3

32

321

,&,

,&'

&''

x

x

xxx

P!!!Q J=!!!94Y8 !!!-6(7$-8 T=!!!H*(-15) +!!!$(7$ +!!!7%-3x J6!!!,#-8 5#(!!!/9 . [-F!!!L# 9-(3-8 J=#S-8 0-1 WD>-8 @A6(7$-(L +S*:

30

41516x

053

43

3

32

321

&

,&'

&''

x

x

xxx

@A6(!!!!7$-8 <!!!!$ !!!!"#$%$-8 EF!!!!G#– C(!!!!9>:>$ C(!!!!$()* +!!!!>$3 P!!!!G#– !!!!"#$%$-8 nQ(!!!!?3 J(B7$-8 9:SY8)X/I* +,-8 (U- 5& (WD>-8 @(9:$7-8 <$ D? <Y [-F#:

) &( @L(> PQ -6(7$ T=H.

) T( -6(7$ T=H=4& -6(7$ a$ (U7$%# @L(> PQ .

) p( V7L-8 ($U9H7L <(?$ <93-6(7$ +96L3.

) 9:!!!SY8 !!!"#$%$-8 0!!!:" (!!!G(*9=%& P!!!3-8 @(!!!9:$7-8 P!!!G# ( +!!!, <!!!$ =!!!9R3 A "#$%$-8.

)i( *+:#2&' 5)E&': PI:4-8 V9#73-8 N9=BL e3(*-8 P>:>$-8 '()*-8 +,.

P!!!>:>$-8 '(!!)*-8 +!!!, &6!!L*– @8=!!9R3$-8 '9!!!] 6(!!%91 5&321 ,, xxx - '8643!!!/(L J=94Y8 -6(7$-8) >-(>-8 5& ( $9] PB73 P3-8#3x: 303 &x

J=94Y8 +L] -6(7$-8 '>) 9*(>-8 5& ( $9] 0:" +S,* (U*$#2x:

363242 ,&,,&x

0-#Y8 -6(7$-8 C8=94&# $9] PB731x: 918271 &,&x

X/I* KL(/-8 +,-8 0:" (*:S, 8F?G#.


130

&a +o6!!!qL 5& M@A6(!!!7$-8 T!!!93=3 =!!!o9_(permute) !!!-6(7$-8iLS9 W9,L =4& -6(7$ a$ 0-#Y8011 *a.[

PQ L#=H$ 0-#Y8 -6(7$-8 EFG ZH&:

13121 ,,, naaa ,,,

']= -6(7$-8 0-1: 2 , 3 , ... , n

ZF!!!!,- [!!!!-F# M T!!!!93=3-8 0!!!!:"1x 0!!!!-1 !!!!9*(>-8 <!!!!$ @A6(!!!!7$-8 +!!!!? <!!!!$J=94Y8.

696%-8 +$(7$-8 0:" 9*(>-8 -6(7$-8 '/]822a+ . PQ 6#%#$-8 =S*7-8 8FG# C(96!3# #& C(!92?3=$ C8=!S*" 0$/9 X9:" -6(7$-8 '/N* 5F-8# P/9;=-8 =BN-8

(pivot element)] .@A6(7$-8 T93=3 =o9RQ C8=IS 5#(/9 <(? <1 C(H9&#.[

Gauss Elimination Algorithm ”ËU' ·c(« WI¹dÞ WO�“—«uš

0,0 ** yA

<939/9;= <93#B4 <$ 9$2=8#4-8 <#?33:

C&5D' 5)E&' :FG2H2% I#GJ: CG&K CG)L%&' I#GJ:&' B+5@- (Triangular System)

<9L#:/& 6,.L 8FG '39 <& <?$9#:

)0( !++L-&' M% >?@&' (Elimination with Normalization):

0!!:" 0!!-#Y8 !!-6(7$-8 '!!/]811a . #& C(!!92?3=$ C8=!!S*" 0$!!/9 =!!S*7-8 8F!!G# C(963#(pivot element).

]<(!!!? 8F1011

!!!!"#$" !!!! %$ &' (!!!!)*"+,ix -!!!!./ (!!!!01#$"02 3!!!!4i /!!!!))$502 (!!!!)*"+ 6"!!!!75 81!!!!9,02 :,#!!!!7)

(normalization) . !!!"#$" !!!$'0 ;<1#!!$"02 )1!!!=5 (!!)*"+,ix -!!!./ (!!!01#$"02 3!!4i 2>

3!?@'02 @#A5/<2 ()*"+ 6"75 (B*C" ("). /=AD(partial pivoting). ED EF2 G/!HI J#!"#+ J#!"#KI #I)1!0. Ax = y L!)9 M !NH02 2>!O P!*C" 3!4 Q!H)/$5 R=!7 :>!02,A 3!O

, ;S"#$"02 (4,HN"x , T1#!')U V,!*C"02 Q!'5"02 ,Oy 2 Q!'5" ,!O (!",*$"02 ;!=2,W0)X2 Y/C02 34E" ED G/HI, :


131

!"#2x $%% &'()*+, -.*')/0# 1'23 &'42$42# 5'% 678$'9%2# 5'% :; 5% &'<4

!" &()*+21x=> &,?*@% .8)8A2# &)B$42# &28$9%2# &>$@C, DE -:

24232 ,,, naaa ! ! !

FG* &28$9%2# 123: 3, 4, ..., n

H)I*I2# 1<J)&K"L% :ija

11

1

11

31

11

21 ,,, aa

aa

aa n!!!

F'G* &'28$9%2# 1'23 2, 3, ..., n !'"2 M'2!? -H')I*I2# 1'<J1x 678$'9%2# 5'% :'; 5'%

42# 5%.*)/0# 123 &)B$.

!"#2x &()*+2 &<4$%% &()*+, .*)/0# 123 &42$42# 5% 678$9%2# 5% :; 5%

!"1x=> &,?*@% .8)8A2# &)B$42# &28$9%2# &>$@C, DE -

22

2

22

42

22

32 ,,,aa

aa

aa n

!

!

!

FG* &28$9%2# 123 :3, 4, ..., n H)I*I2# 1<J.

*%IN#*O$B92# !" &)<%J =>143 ,,, !nxxx H)I*I2# 1<J.

.*?O2# => =4<4% F$KB 1<J :O"B &)$PB2# =>:

nnnn

nnnnnnn

nn

nn

zxc

zxcxc

zxcxcxc

zxcxcxc

"

"#

"###

"###

!!!!!

,

1,111,1

22323222

11212111

!"#$:

&2$"2# =>)E) (*'))9I2# Q'% !'"2# (&')*+(2# 6L%$'9%2#iic DE .8'"?2# D?$'NI

5E:

nicii ,,2,1;1 ""

81)1'02 ;<1#$"02 34 ;S"#$"02 3O.(

/!!!N#I$02 Y>!!!9 (!!!)*"+ 3!!!4 /"5!!!72143 ,,, n

xxx /1!!!. MV!!!)5/502 6!!!*+(+#C57<2.

) ( ! "#$ %&'()*+,, (Elimination without Normalization):

34 (=,/Z" 60,X2 (01#$"02 YZD


132

%&'()* +,-.)* :/0-,)* 1.2.$)* 3-!,)* 4") 152()* 6+&70)*

(Back Substitution)

#!';R !"#$%&'( )#*+,-.-+ /(.#,0-&'( 1,2 345 6789 :;4<'( =,"%-')+: ) >#&?.,<@( A'B)%&'((

1,1

11

1

,

,

/

!

!

nn

nnnn

n

nnnn

cxcz

x

czx

)?.,<@( 6+2 A'B)%&'( >&(

ii

n

ik

kxkii

i c

cz

x"

! #! 1

,

68'( A,&C.("< D,<4- )99$&," :E4E&'( 1)F94' :;4<'( =,"%-')+ Cx = z (*)

G###,8C A###,"45 A###,E4E& AH";###7&$ %nn& I(.;###7 J")###K- L A###,.MN'( )O.###7)95 6###$A,')-'( /)2P%')+:

1,,1,

;1

!

"

! #!

nni

c

cz

xii

n

ik

kxiki

i

)&B95 Q9R AF8P& S&i = n T"#&U&'( >VH"#!

n

ik 1

W+#7,"#!

n

nk 1

B"B#8 B#U"- L Q#9R JR

I(.;7 :M%, :')-')+" T"&U&'( (XO :H) I)8PM7(.(

!"#$ %&'$( )*+,- ./ )+0"1'$( 2"+345$( 667 Number of Operations in Gauss Elimination Method

>#& >"$& :M< 1)F9 68' !")U' YX8'( AN,.M )9&B<-K( (XZn :#H A#'B)%&n )+EZ >$&, Q9VH 6"*U&>R /:

[.\'( /),4&5 BB5 /A&KN'(: 3

3 23 nnn #

S&U'( /),4&5 BB5" /].M'(: 6

532 23 nnn #


133

A&######KN'(" [.######\'( /)######,4&5 BB######5 >R JR)].######M'(" S######&U'( ^'X######$" ( J")######K,)+,.N-3/3n.

.89#$( 9":;,<"0 %&'3$ !"# )*+,- (Gauss Elimination with Partial Pivoting)

L B#2 A#,&C.("<'( _XO /("M< >& ?"M< JR :H JB-"'( "R JC$-.&'( .79%'(;7 J")K,Q#,45 A&#KN'( >#$&, :')-')+" I(. B#2 :')#-')+" I(B#U I(.,0#7 >"#$, B#2 Q#9R LZ

?.####,+$ `)####M<R 3####'Z JBa####, .$ b####c9, ?B)####5 I(B####U .,0####7'( 6####&)%&'(">,BB####5 >,####+ d.)####;

I)+,.N- >,,")K-& /LB)#%&'( 6,B#+-+ `)M<@( _XO 6E& :HP- >$&," (permutations)

)###*+,-.- .###,,0-+ JR JC###$-.&'( .###79%'( W+###7, G###,8+ [###,-.-'( (X###O 6###&5 6###\;,H –

Q###,45 1###KN9K JX###'(– .###79%'( >"###$, G###,8+ /LB)###%&'( [###,-.- B###,%9 ?B)###5" I(.###,+$A##N4M& A##&,2 .##+$R "X 6##&)%&'( "##O JC##$-.&'( . S##+-9 :##-'( !")##U A##N,.M 3e&## fKg- ?B)##5"

`(.Uh( (XO )*,H : :iCU'( C)$-.L( S& YX84' !")U AN,.M(partial pivoting).

A,')-'( AU,-9'( 3'Z JBa- I)N+)K A8".c&'( :;4<'( =,"%-'( AN,.M:

)+,=>:

/##9)$ (XZC A##,2"H A##,E4E& AH";##7&(upper triangular matrix) 6##$ /##9)$")O.#####7)95 AH";#####7&'( >V#####H I(.;#####7 J")#####K- L A#####,.MN'(C !)#####$%9P' A#####4+)2 >"#####$-

(invertible) !"$%& )*' JR1 C.

?"@,0$(:

:E4E&'( 1)F9'( >R >,+- ?."$X&'( :;4<'( =,"%-'( AN,.M >R S2("'(

(*)zCx !

) I)N+)#K 6,#7;-')+ ["-$&'( (' B#8(" 6#8 .#E$@( 3#45 Q#' Q#U-& 6#$z 3#fM%g& . >V#H :')#-')+"C A,')-'( A&"4%&'( A,.F9'( 345 I )9+ !)$%9P' A4+)2 >"$- >R [U,:

)+,=>:

>R )9\.H (XZA A%+.& AH";7&nn&AiH)$-& )*4$ A,')-'( /(.)+%'( >VH :

) R( !9)U-&'( 1)F9'(Ax=0 (homogeneous system) QH)-'( 68'( LZ Q' !,' x= 0.


134

) ( !"#$ %&'y ()*+', -./ 0-$12, 345', 6/ 758$Ax = y %9 !'.

) :( ;/<=>$',A ?)&8+@' ;AB)C.

!"#$', -./ D'E'<x 7F$GH1 ;IB)G', 6=AJ', K1<8#', ;1$L4,<JB 7M58$', : %9 6NAN$', ()*+',[the solution of (*)] (*).

?1'< : 6NAN$', ()*+A' O@9[a solution of (*)] (*).

) ( !"#$ %&'()– *+,-. /01("$

Gauss-Jordan (G-J) Elimination Method

51PGB %1Q8# R$ 3E9A' ?<)" ;I145 )SG=+ 6T ;I145', UET Q8# . 6P/ O@NP$/ VLPP&#4$', 4PP>+8', 7PPAW )PP$ ;PP'Q)8$ (PPGI+ -X QPP8B 4PP118#', RPP$ 3EPP9A' ?<)PP" ;PPI145

1P8$ 4P1Y#$ 3EP9' ;P'Q)8$', UEPT (,QJ#G)B (<I+ )++./ %=PGX 6P#', Z[Q)P8$', %P& -P$ -;PP'Q)8$', UEPPT . ?<)PP" ;PPI145 6PP/ )PP$X– ,EPPT 3EPP9 ;PP1A$W \,4".PPB (<PPI+ )PP++./ -,Q4<PP"

;'Q)8$', UET %=GX 6#', 7'] ;/)^_)B </ 6#', Z[Q)8$', %& -$ 41Y#$',.

?<)" ;I145 ;1$L4,<J/ D'E'<– )SPG=+ ?<)" ;I145 ;1$L4,<J 6T -,Q4<"1PGB', %1QP8#', ,EPT R$ 5) 4P>+8', %=PGX 6P#',< <P/ 6P#', Z[Q)P8$', %P& -P$ 3EP9',

VQPP#<', .( ;PP145C Z@$)PP8$ ;/<=PP>$ 7PPAW O,4PP1JX %PP>9+ ;PP'Q8$', ;PPI145', UEPPT 6PP/<(diagonal coefficient matrix) aQ9<', ;/<=>$ 6T %B(unit matrix) . )P++./ D'EPB<

G9 VX -<QB Z[Q)8$', ;W<$"$' 6b)S+', %9', 7AW a4c)B$ %>9+ @/ 0d4JX Z)B)6=AJ', K1<8#', ;I145 `1B5#' :)#9+.

2 3#4(2-3):

?<)PPP" ;PPPI145 (,QJ#PPPG)B– , ;PPPW<$"$ %PPP9 QPPP"<X 3EPPP9A' -,Q4<PPP" Z[Q)PPP8$'%)N$ 6/ a)58$', ;15J',1-3 .

3,.0:

%)PN$ %P9 6P/ )P$& %9', XQB+1-3 3EP9+ eP19 01x !"#$%&' (")&*$!+,&' (!, -!. (!, !!%&$%&'/. .01!!2 !!"3,4 5!!&6 -!!7# $,*!!#4/2x (!!.&/ 8!!9: !!%&$%&' !!&*$+,&' (!!, 01!!2;" <!!:

=$!!>"? 5!!&/@' (!!,) =$!!>"? !!"#$%&' !!&*$+,&' A'*B)!!C$D E!!&1/ .( FG*$!!+,&' (!!, =G*!!D: E&1!!D/ "&$)&')-$%, -2 H$#%? $I"34 $#372 J)&' 1-3.(

ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�« 3ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�«

135

2151

41516

053

43

3

32

321

!

!"

!""

x

xx

xxx

!"#$%& '() *+,- ./%"0%& 123%& 4)567, 1/,2x 897:; <'7%=>& .7%!"#$%& ?7$ @A ./-"3%&3/4 - '%=>& '() "B#$C=.(

2151

41516

351

3

32

31

!

!"

!

x

xx

xx

@777A D9777/E>& .777%!"#$%& 89777:- F7773(-15) *777$"#$ *777#C%3x F!E0777G-= <D!777,=%& H="777G/56777,% .777C0"-%& .777%!"#$%&3x 89777:; I777%6= <.777/-"3%&= '777%=>& ?/0%!"777#$%& ?777$ *777J ?777$

.773%"3%&).77C0"-%& ( @77A1/5 @77A .773%"3%& 8977:= <'77%=>& '77() "77B#$C=-16/15 "77B#$C=&6JK <./-"3%& '():

30

36

9

3

2

1

!

!

!

x

x

x

.77C",%& ?=! <'77L#$%& F"77M-(% 8=77(L$%& *77,%& @77L#0 12377%& 4 !"77#$%& N677K=HO P&9CQ '%Q R9EO 4";"G,.

"7$J 4 !"7#$%& ?7$ !7; 4"A=S7+$%& .7% !; "7B(J *7,%& 4&=LE .;"0J "--J$/= @777(/) 56777,%& .777T/9L F&!E0777G"; *777,%& !777-) N6777K .777;"0J%& .777T/9L U"777;0& "777:/O "777--J$/=

V="C%:(

6

0

0

58

230

53

210

53

431

6

0

0

236

356

121520

321

#

yxxx


136

0

4

0

53

210

151610

53

431

32

0

6

0

53

210

58

230

53

431

$#

#

$

2

4

3

15100

151610

5101

43,

21

#

$

30

4

3

100151610

5101

151

30

36

9

30

36

9

100

010

001

1516,

51

3

2

1

!

!

!

# $

x

x

x

!"#$:

?O "-777777:9A &6Qn *777777/K"C$%& !!777777) W777777GS- =777777K= 4 !"777777#$%& !!777777) *7777773$/) HO4&977/X0$%& ( 4"77/($#%& !!77) ?Y77A(number of operations) V="77C .77T/9L @77A– "77$J

Z"T;"777G "777-/O9– =777,-; 9!777T[/3/3n V="777C .777T/9L @777A 4"777/($#%& !!777) 9!777T/ "777$-/; <–

=77,-; ?&!9=77C2/3n 9L ?Y77A I%677%= <8;77G%& &677B% Z"77/($) *77:S0 V="77C .77T/ ?77J$/= <@(/ "$ 4";3Q:

%&'( )*+, -. */'0"12 3'*4$512 667– 826+&(:

89:%& 4"/($) !!) /.$GT%&: 22

23 nnn "

\$C%& 4"/($) !!) /]9L%&: 22

3 nn

77) ?O HO .$77GT%&= 8977:%& 4"77/($) !!)]977L%&= \77$C%& I%677J= ( Z"77;/9T0 H="77G/2/3n.

9:"12 )*+, ;47 3')*/,< Applications on Elimination Method

)= ( 3266"$12 >'0" (Evaluating Determinants)

5677,%& .77/$ 9&=E 4&=77LE ?77$ D=77LE !77C=0 W77-& F77(#- 4&!!77,$%& _&=77E ?77$77C% V=") V="77C =O– ?&!9=77C ( P"-3077G"; 42$"77#$%& .A=S77+$ D!!77,$ .77$/` ?77$ 977/X0


137

?/S77+ */!77;0 .77/($)= <H 77J09$ 977+-) *77J '77() .$77GT%& .77/($))?/!=77$) =O .( 177/,= H="77G0 .77C0"-%& .A=S77+$%& D!!77,$ .77$/` ?O1 ) M77, – 5677,%& @770T/9L ?77$ HO @77A–

H9S+ @(SG 1(3$ "B% .C0"-%& .A=S+$%& ?O@A !",a= < @G/b9%& 9LT%&( .7$/` ?=7J0A < .777/ J09$%& 9777+"-#%& 89777: *777+",% ./="777G$ I%6777; ./(777+>& D!!777,$%&(product of

pivots)< */!777;0 4"777/($) ?777$ H!9777A !!777) H9777C[O !777` ?"777J &6Q c0"777-%& D9"777dQ 9777//X0 \777$D!$)>&= 5=S+%&.

% & % & n

kpppA $$$ ! 21det

1/, :k ?/!=$) =O ?/S+ */!;0 4&9$ !!) *3$/.

npp ,,1 :./ J09$%& 9+"-#%&.

!"#$:

.A=S+$ D!!,$ .$/` 8"G,nn$ !!7#% e"70,/ 567,(% V="C .T/9L U";0";=,-; 9!T/ 4"/($#%& =O 4&=LE%& ?$3/3n !=!, '%Q !!#%& &6K \S09/ "$-/; <n! !-)

%& 8"G,% D!"0#$%& f9L%& U";0&4&!!,$.

"3$ 6Eg- ?O *; =567,%& .T/9L; 4&!!,$%& 8"G,% Z @-"73%& f7/;L0%& V9!7- <()4"A=S+$%& VJ) =K= .T/9L%& N6K ' Z"#$ ?/T/;L0%& '() Z "3$ 9J6- F3 <.

)> ( .&?@$ %&A5$ 6'(*B (Matrix Inversion)

?$ Z2J ?O h9S-A, Z, I .#;9$ .A=S+$nn$ ?O= <A.Z = I

?O HOZ .A=S+$%& V=J#$ @KA: 1 ! AZ

.A=S+$%& ?O HOA D!9S-$ 9/i(non singular).

) ?O h9S-A "BG=J#$ !"C/Q 8=(L$%& .A=S+$%& @K.(

.`2#%&AZ = I ./%"0%& D9=+%"; */+S0%"; 80J0 ?O ?J$/:

''''

(

)

****

+

,

!

''''

(

)

****

+

,

''''

(

)

****

+

,

1

1

1

0

0

21

22221

11211

21

22221

11211

!""

""

nnnn

n

n

nnnn

n

n

zzz

zzz

zzz

aaa

aaa

aaa

6K ?YA 4"A=S+$%& _&=E ?$=./%"0%& 4"`2#%& .)=$C$ "-/L#0 .`2#%& N:


138

''''

(

)

****

+

,

!

''''

(

)

****

+

,

''''

(

)

****

+

,

0

0

1

1

21

11

""

nz

z

z

A

''''''

(

)

******

+

,

!

''''''

(

)

******

+

,

''''''

(

)

******

+

,

0

0

1

0

2

22

12

""

"

nz

z

z

A

.........................

''''''

(

)

******

+

,

!

''''''

(

)

******

+

,

''''''

(

)

******

+

,

1

0

0

0

2

1

"

"

"

nn

n

n

z

z

z

A

4"777`2#%& N6777K ?777$ .777`2) *777J=) "K!!777)=n ( 4 !"777#$%& ?777$ .777)=$C$ *7773$0 ./LE%&) "K!!)=n .%!"#$.(

"-#:= &6Q "--O HO1 ! AZ ?YA <AZ = I

=O njeAz jj ,,2,1; !!

1/,jz : F 9 !=$#%& *3$0j .A=S+$%& @AZ.

je : F 9 !=$#%& *3$0j .A=S+$%& @AI.

?O HO:

) F 9 5+%& j........ (-.

''''''''

(

)

********

+

,

!

'''''''

(

)

*******

+

,

!

0

0

1

0

0

,

2

1

""

"j

nj

j

j

j e

z

z

z

z

D!77777$)>& ?Y7777A &67777JK=.A=S7777+$%& @7777AZA ! 1 7777(, @77777K ?77777$ 4"7777)=$C$% *=.7777/LE%& 4 !"7777#$%& D!7777$)O H="7777G0 4"7777)=$C$%& N67777K @7777A '7777-$/%& 5&97777L>& 17777/, <


139

D!777,=%& .A=S777+$I 4"777)=$C$%& N6777K ?777$ .777)=$C$ *777J @777A 42$"777#$%& .A=S777+$= < .A=S+$%& H="G0A )"BG=J#$ !"C/Q 8=(L$%&.(

V="C% 56,%& .T/9L @A=)"C =O V=– ?&!9=C ( D!7) \7$ *$"#0- ?O "--J$/WGS- 4 =%& @A '-$/ 5&9LO . F"7M-%& '7() V="7C% 567,%& .7T/9L "-T;L &6YAAZ =

I D!777,=%& .A=S777+$ D!777$)O *777$"#- 1777/,; & @77A 4"77BC0$ D!77) HO ( .A=S77+$%& D!77$)O=Z "77B-O '77()

7(,%& 4"7BC0$ .7(;"T$%& *=(corresponding solution vectors) x I%67; *7+,- "7--YA <– N677777777777K *=77777777777(,%& 4"77777777777BC0$ ?77777777777$– 8=77777777777(L$%& .77777777777G=J#$%& .A=S77777777777+$%& '77777777777()

"K!"C/Q1 / AZ.A=S+$%& I(0 D!$)O @K 4"BC0$%& N6K ?=J0 1/, <.

$ C'D(3-3):

?O h9S- '''

(

)

***

+

,

!'''

(

)

***

+

,

!

6

1

9

,

583

691

472

yA

)i( F&!E077G"; !77C=O V="77C .77T/9L– @77LE%& F"77M-%& *77, ?&!9=77CAx = y ) ?=!4";"G,%& @A ./!"/0) & 9=GJ%& F&!E0G";= < D!$)O =O 5=S+ HO */!;0.(

)ii( .7777G=J#$%& .A=S7777+$%& 97777+"-) 87777G,&1 A F"7777M-%& *7777,;AZ=I F&!E07777G";= <@b C%& ^"J09 &.

)iii( .$/` !C=O|A|.

C"12:

(i) <(ii) =S7777+$ ?> Z&97777M- 42$"7777#$%& .AA *7777, ?7777$ *7777J !"7777C/j F!E07777G0G @7LE%& F"7M-%&Ax = y .7G=J#$%& .A=S7+$%&= <1 A ?/b 7C%& *7,- I%67(A <(i) <(ii)

Z"#$ *&kG%& ?$.

100

010

001

6

1

9

583

691

47221

$


140

1023

0121

0021

239

27

29

11250

82250

2271

252

$

151

57

0252

251

0257

259

594257

2588

54700

251610

25601

475

$

47/547/147/7

235/16235/22235/13

235/6235/67235/93

2

1

4

100

010

001

'''

(

)

***

+

,

!'''

(

)

***

+

,

!#

25535

162213

66793

2351,

2

1

41Ax

)iii( ?/!=77$) HO =O ?/S77+ HO */!77;0 F077/ F77% W77-> Z&977M-(k = 0)?=77J0A < : .77$/` D!!,$%& =./ J09$%& 9+"-#%& 89: *+",.

% & 2355

472252 !$$!/ AADet

.&?@$ !"#$ %&'() *+!,-$./ *(+&01./ 2&(,$#./ %%3

!"# $%&'( ")*+,- .+"/0- 1"230- 45– 67*3 8"6#&9 :;6<0- 465 =-8'!# ! $5!>?30-A @86<!0- $5!>6?3 ABC $&"D)0- 45 '/&E- :'(0- 45 ")B?< =F+ G0;! H

I$66/!7*30- $5!>66?30- !66I =663&E- :'66(0- =J K,66)&5 H1 A . L"66+,- =667330- =663 ="667! !"66# $66%&'( =663 MN866+ :;66<B0 !"66# $66%&'(– :'66(0- 4665 166?<) =F66+ =-8'!66#

!>666?3 A666BC $666&"D)0- 46665 '666/&E- $666&O!5 $666&2B23 $5U P&!666*,0- $666&3Q'-!R .666+() S6662 88C 4>BR0-n T-'30- =3.

U)J T"+2V =73&!:

) J( !"# $%&'( ")*+,- -;V– !7*30- 8"#&9 =-8'!#1 A:

W'X0- T"&B3C 88C =Y5 /$3/%0-:

nn21

23 3

Z3#0- T"&B3C 88C! /['(0-: nnn212

23 23 !


141

) W( !7*30- 8"#&9 :;<B0 !"# $%&'( ")*+,- -;V "3)&+1 A:

W'X0- T"&B3C 88C =Y5 /$3/%0-:

nn31

34 3

Z3#0- T"&B3C 88C! /['(0-:

nnn61

23

34 23 !

!667*30- 8"66#&9 =-8'!66# !"66# $66%&'( 4665 U66)J \J1 A 866C =!667& T"66&B3C 8 $366/%0-! W'66X0-)['66(0-! Z663#0- T"66&B3C 8866C G0;667! ( M"66+&'%, M"&!"66/33

23 n "663)&+ H

!I :;<B0 !"# $%&'( 45 1+"%30- 88*0-3

34 n.

)4( 5(,16)5&$37 !8 (*9!:;$ :Factorization of a matrix

$5!>66?3 166&B<, A660V ]"66,<) ="66&<E- P66*+ 4665 $66*+'3A W'66X 166?"< A660V=&,5!>?3 : $&B>6/ $6&2B23 $5!>6?3 "3I-8<V(lower triangular matrix) L '6RE-! H

$&O!5 $&2B23 $5!>?3(upper triangular matrix) U.

=J \J: A = LU

_;66I 166&B<,0- $66&B3C A366/, M"66)"&<J! : 4662B230- .66&'>,0- !J 166&B<,0-(triangular

decomposition) . $5!>?30- '?")C ABC (!'` G")I =!7, M")"&<J!L '6?")C !J $5!>?30-U.

")&860 =J P'6>) H16&B<,0- -;6I 163C 45 :;<0- $%&'( =3 @8">,/N- $&>&7 $5'*30!n

45 $08"*3n 1!D#3: Ax = y

a&<A $5!>?3nn" . $5!>?30- b+?, H:;<0- $&B3C =3 @8<-! @!(R 8*+A:

####

$

%

&&&&

'

(

))

))*

nnn

n

n

aa

aa

aaa

A

2

222

11211

1

0

0

!

!

Tc&8+, \J 8#!, N U)J P'>+ G0;! . $5!>?30- ABC ")B?< 8%5 SB*) "37!1A =3 $5!>6666?30-A 466665 M"+!'6666X3 1!E- :66666?0- ['6666(+ H+ ,111 / aai S6666O' :6666?0- =66663i .


142

$5!66>?30- @8"*,66/- "6))73&!A $5!>6?30- =6631A 4665 M"+!'66X3 1!E- :6?0- $5"66XY++ ,111 / aai SO' :?0- A0Vi a&<ni ,,2,1 !* W'X0- $&B3C d5"7& -;I!:

11ALA *

a&<:

######

$

%

&&&&&&

'

(

*

1001

101

11

001

1

31

21

1

!

""

n

L

niaaii ,,3,2,/1 1111 !**

8&8#0- =3&E- :'(0- =Y5 1230"+!1y $3&8%0- U,3&%+ (+,'&y $Oc*0"+:

11yLy *

466I :;66<0- $66&B3C 4665 $66&)"20- @!66(R0- =Y665 Tc&866+, \J 866#!, N U66)J P'66>+!$5!>?30- =&!7,

######

$

%

&&&&&&

'

(

))

))

)

))

))

)

*

nn

n

n

n

n a

a

a

a

a

a

a

aa

A 3

2

1

3

33

22

1211

2

00

00

0

!

!

4)"6620- :66?0- ['66(++ ,2212 / aa )) S66O' :66?0- =663i S&66%B0 G660;!ni ...,,4,3* . @'663! @8"*,/- "))73& 'RJ1A $&/7C $&B3*+: 221 ALA *

a&<:

+ , niaa

L

ii

n

...,,4,3,/1

10010

0

0

11

10

001

2222

2

32

2

*))*

########

$

%

&&&&&&&&

'

(

*

!

"

"


143

=Y5 -;7I!: 22111 ALLALA **

1230"+!: 22111 yLLyLy **

a&<2y =&,!(R 8*+ =3&E- :'(0- !I.

N U)J P'>+!W,7) =J "))73& .+/ "3 '&'7,+5 H Tc&8+, \J 8#!, :

zLLLy

ULLLA

n

n

121

121

*

*

!

!

a&<:

U: $&O!5 $&2B23 $5!>?3 'Re.

z : =3&J :'( 'Re.

jL:$&B>/0- $&2B230- $5!>?30-.

###########

$

%

&&&&&&&&&&&

'

(

*!

10100

0

11

10

0

1

,

,1

0

!!!

"

"

"

"

jn

jjjL

ijl: :fC"666X,3(multiple) 666O' :666?0- Sj S666O' :666?0- =6663 ['666(& \;6660-!i

1-Q,RN- $&B3C g")2J.

=Y5 G0;+!:

######

$

%

&&&&&&

'

(

*

1111

111

11

1

1,21

3231

21

121

0

nnnn

nLLL

!

) Mc23 (= L

a&<L $&B>/ $&2B23 $5!>?3.


144

-

6

0

0

236

346

121520

+ ,+ ,

-

6

0

0

58

230

206

53

210

206

121520

+ , 6

0

0

51003

53

210

121520

###

$

%

&&&

'

(

*

####

$

%

&&&&

'

(

*

12/12/30

010

001

,

10206

01206

001

21 LL

!"#$%& "'( )*!+,&-. )%+/ .01 2%3 !4% .0/56(non singular) 78#9% :#- .0 ;.6)<'!=8 >?@AB C:

A = LU ,

y = Lz

D'9L E F'GHG& FB6$I&!+9J +KH/ F'"LM-. +5"I+%N F'H$.

6U F'O6B F'GHG& FB6$I&.

FB6$I&-. @1 P6M% 0Q%'96A )HRH S9 !O))HT& UNS3 63 (@''GHG& @'H&+N V-1 L , U.

!"#(4-3):

& P##9 @##& P+##G& W##B F##'LX-. )*!+##,&-. F##N6&41-3 Y6+##4- Z0##9-. F##M'"L= )*!+###,&H- )<'!###=8 >3 @6!###=)-. [+###/8"*. @6!###= >3W###Q[4 ( FB6$###I& P###'H98 !###463 @''GHG& @'H&+N V-1 )<&+,&-.L , U.

!$%&:

WQ[4-. [+/8"*. @6!= Z09-. FM'"L= P9-. @& V-6?. 6LX-. \'XH8 +%%/&'WH' +&'B:


145

####

$

%

&&&&

'

(

*

###

$

%

&&&

'

(

####

$

%

&&&&

'

(

**

13103

01103

001

130

010

001

10103

01103

001

21LLL

=J 8#) "D&0V ")B?! TN8"*3 $C!3#3 'Re =3!:

###

$

%

&&&

'

(

*

####

$

%

&&&&

'

(

*

6

0

0

,

5100

53

210

121520

zU

=J =3 .%<,0- 1D/0- =3!LU = Aa&< H

A

LU

*###

$

%

&&&

'

(

*

####

$

%

&&&&

'

(

####

$

%

&&&&

'

(

*

236

346

121520

5100

53

210

121520

13103

01103

001

=Y5 G0;7!: yLz *###

$

%

&&&

'

(

*###

$

%

&&&

'

(

####

$

%

&&&&

'

(

*

6

0

0

6

0

0

13103

01103

001

5(,16./LU !"#$ %&'$( )*+,-(!. /(0.12"3 )4!56+$

!" #$%&'( )&*!+ ,-#&*$!+ . &/%* #&$$01 #&2 %1#3 4&!" 56(7&'1 8& %9*! #&2L, U

5 !#*!+ 5 :;$!+ )6 <=9%! >(#? 5 1@:+(/ A+B/*C#D:

)*-78:

A#&&&;$!+ 4&&&%3 >(#&&&?! <=&&&9!+ 5&&& 1@:+(/ =&&& 7$* E&&&011!+ E&&&1 E#&&&0 +="yAx

<(7&'%! 8 BD* FG E(BD 56(7&'1!+ 8& %9* E&011!+ E&1 H&$I6 JA K:&L 8&'#9 4&!"E M%M1 E %1#3 :A = LUN 9 J


146

! "ijL 1# 5 %7C 5 M%M1 56(7'1.

! "ijuU # 5 (%3 5 M%M1 56(7'1.

5 !#*!+ O#PQR!#D E *6(7'1!+ E *#S :'#$3 4TURV*(:

EG W:7$A 5RD:1 56(7'1nn$ . A P E1 51 P 80%6nj ,,2,1 :

njji

jiau

ij

ij,,2,1

,,2,1

0

%%

njji

ji

ji

mij

ij

,,2,1

1,,2,1

1

0

1

%%

&

N 9:

ija A&&&P: <&&&'!+ )&&&6 :&&&'$R!+ (&&&Si A&&&P: B(&&&1R!+(j )&&&*!+ X:&&& /Y+ 56(7&&&'1!+ )&&&6<=9%! >(#? 5Z :UD #2 %3 8'9$.

ijm K:#&&&&&&L!+ (&&&&&&S(multiplier) <=&&&&&&9!+ X(&&&&&&U/ )&&&&&&6 AB/*&&&&&&C+ F=&&&&&&!+ BB&&&&&&R!+ FG! " ijiji EEmE '$& 5&&!B#R1!+ )&&6 K:&& VL F=&&!+ BB&&R!+ FG J)<&&'!+ ( A&&P:j

5&&&!B#R1!+ E&&&1 ,*#&&&$!+ [:&&&UV !)<&&&'!+ ( A&&&P:i 8&&&91 5&&&?*#$!+ 5&&&!B#R1!+ 8&&&9* A&&&M J AP: 5!B#R1!+i.

))7'9+: @1:!#&D K:#&L%! #&$S #$:\Gijm E&1 ] B&Dij1 4&%3 5&!^B%! #&$S AB/*&C+ F=&!+B9!+ 56(7'1!+ )6 A#R!+L.(

)6 ]QM16:

+ :"; (4-3))_D#C!+ 8#M1!+ (S(:(

((((

)

*

++++

,

-

'

13103

01103

001

31

103

2061

103

2061

3232

3131

2121

L

m

m

m


147

Q B&&D* FG O&&1* +=" H&&$G ;&&9Q ( 8+@&&*/^+ 5&& %13 `#&&$MG O)<=&&9!+( 8B&&D #&&$$I6 J 56(7&&&'1%! E &&& M%M1!+ E %1#&&&R!+ 4&&&%3 8&&&'9$ EGA E &&& M%M1!+ E %1#&&&R!+ 4&&&%3 8&&&'9$ J

56(7&&&'1%!A*&&&'9$ )&&&*!+ )&&&S( J <(7&&&' 4&&&%3 5&&&%D#Z1!+ OQ B&&&D*!+ 8&&&1RD #&&&2 %3 8 56(7'1!+ XB13G(A.

<*14!56+$( =$> )4!56+ :*?'1 )*+@A L , U

)6 4$1 <+:UG XB3 a1 >(#?! <=9!+ 5Z :U )6 81#R*$ EG E01 H$G A%R$ H&&&&C7$ O&&&&P(!+) OQ1#&&&&R1!+ 56(7&&&&'1 #&&&&2! O^B#&&&&R1!+ E&&&&1 O#&&&&3(1?1 XB&&&&3 8&&&&9$ FG

#2C7$ .( M0 E# 9G )6 E0!(5& +BD!+ =&$1 #$ B&! X:6(&*1 4$1 !+ <+:UY+ 80 E(0* ^ X: .

E *3(1?1!+ 89$ EG ]QM1 B :$ BZ6:

2211 , bAxbAx

N 92b )6 #1 5!+B1x .+ E1 (BD BP #$S(5 +BD!+ E1 <=9!+ B R$ EG F:(:L! #&11 J1R!+ E1 ]+: D0 ]+BB3 #$7%0 O# % . +=S K$?*$ < 0 Eb+ c:$C(): D0!+ BBR!+.(

56(7'1!+ 8 %9* #$RU*C+ +="A E * M%M1!+ E *6(7'1!+ 4!"L , U: A = LU

A#;$!+ EI6: Ax = y

A#;$%! ]#-6#01 E(0 : LUx = y

E U CD!+ E M%M1!+ E 1#;$!+ 4!" _:7* +=S(: Lt = y , Ux = t

A#&&&;$!+ EG d&&&L+((Lt = y H&&&%9 E&&&01 )1#&&&1Y+ W (R*!#&&&D(forward

substitution) A#;$!+( JUx = t )7%/!+ W (R*!#D H%9 E01 .

#&&&&$1%3 +=" +=&&&0S(L, U A#&&&;$!+ 8&&&&9$ EG #&&&&$$01GAx=y O#&&&& %1R!+ E&&&&1 BB&&&&RDF(#&&C 222/12 nn $ E&&1 ] B&&D3/3n 5&&Z :U =&& 7$*! K(&&%U1!+ BB&&R!+ (&&S( J5&& %13

(#?! <=9!+ 56(7'1!+ 4%3 >A.

#2&&C7$ OQ1#&&R1!+ 56(7&&'1 #&&2! )&&*!+ A;$&&!+ E&&1 :&& D0 BB&&3 8&&9 #&&$$01 e!=&&D(O# %1R!+ E1 8PG BBRD( 5!(2CD.


148

E && M%M1!+ #&&2 %1#3 4&&!" 56(7&&'1 8&& %9* )&&6 5 &&C#CG 5&& :;$ 4&&!" Eb+ : &&\$( .

`+:Z*C^#D 5 :;$!+ 5$S:D E01 ((induction).

)*-78 LU:

EG W:&&&7$A &&&RD:1 56(7&&&'15nn$ .EG W:&&&7$(kA 56(7&&&'1!+ )&&&Skk $

8(G aU#Z* E1 5$(01!+k )6 B(13( <'A .E#0 +=":

! " 1,,2,1,0det & . nkAk

XB 9( 5 %7C 5 M%M1 56(7'1 B?(* H$I6 ! "ijL 1

niii ,,2,1,11 N 9D

XB 9( 5 P(6 5 M%M1 56(7'1( ! "ijuU

H$G N 9D LU = A

<*14!56+?$ )5?1.+ -!6 L, U:

EG #$L:6 +="A 56(7'1nn$ 5!B#R1!+ EI6: A = LU

8 %9*! 51B/*C1!+A E M%M1 E %1#3 4!"

! "! " njiuU

nijL

ij

ij

///

///

1;

1;1

8&&&M1*2n 8&&& S#?1!+ )&&&6 5&&&!B#R1ijij u,1 #SBB&&&3(nn %2 . ]#U(:&&&\ a&&&L$ EG E&&&01 (:&&&/b+ WRD&&&!+ E &&&R$( 8&&& S#?1!+ f=&&&S W&&&RD 4&&&%3 5&&&'#/ . NQ&&&M )&&&% #&&&1 6 :0=&&&$(

5&&&7%*/1 _:&&&U NQ&&&M 4&&&!" FBg&&&* U(:&&&\!+ f=&&&S E&&&1 O#&&&3(1?1– 5&&& 1SY+ W&&&RD #&&&2! 5 %1R!+– 8 S#?1!+ f=S E R*!:

)A ( )B*-C):1*$!0 ((Doolittle's Method)

U:*\$ #2 6(: niij ,,2,1;11

56(7'1 )6 5 :UZ!+ :'#$R!+ EG FGL B#9h #2%0) . 5&Z :U E1 H %3 8'9$ #1 +=S( ]#ZD#C #$ G: #10 <=9%! >(#?.(

)ii( )B*-C)D!-E ((Crout's Method)

56(7'1 )6 5 :UZ!+ :'#$R!+ E(0* EG U:*\* f=S(U %0 ]+B#9h #2.

niuii ,,2,1;1


149

)iii( )B*-C)FE2$!G1 ((Choleski's Method)

E *6(7'1!+ F:UP :'#$3 _DU$* EG #2U(:\(L, UEG FG J:

niuiiii ,,2,1;1

D"4!56+?$ -G"3+$( :*?'1$())*;?;+$( "H?+(!I =$>(

Direct Factorization of Matrices

4&&&&&%3 8(&&&&&'9!+ #&&&&&$$01 <&&&&& 0 #&&&&&$ G: E &&&&& M%M1!+ E %1#&&&&&R!+ UL, 56(7&&&&&'1!A

>(#&&&?! <=&&9!+ 5&& 1@:+(/ A+B/*&&C#D . E %1#&&&R!+ E =&&S 4&&%3 8(&&&'9!+ e!=&&0 #&&$$01 ( 5PQR%! 5%D#Z1!+ 5U CD!+ O^B#R1%! :\#D1!+ 89!#DA = LU

)!#*!+ 8#M1!+ E1 e!= E D* #10:

+ :";(5-3):

56(7'1!+ 8%9:

(((

)

*

+++

,

-

&&

&

&

221

317

432

A

E M%M1!+ #2 %1#3 4!"L, U N 9

3,2,1,11

,

i

LUA

ii

)8* !(B 5Z :UD 8 %9*!+ FG.(

:'$(:

5PQR!+A = LU i6#0* X#UR1!+ U(:\!+ O9*:

(((

)

*

+++

,

-

(((

)

*

+++

,

-

(((

)

*

+++

,

-

&&

&

&

33

2322

131211

3231

21

00

0.

111

011

001

221

317

432

u

uu

uuu

8 S#?1 5RC* )6 5 !#*!+ aC*!+ O^B#R1!+ #$ UR* f=S:

13

12

11

4

3

2

u

u

u

&


150

! "

! "1911

3333191

21

3323321331

191

3232219

21

22321231

21

31311131

232327

231321

219

222227

221221

27

21211121

114

112

113112

1211

11413

311

2117

'%&$%$&

%% &

'%&$& %

& ' &

& '%$ %

'%&$ % &

'

uu

uuu

uu

lu

uuuu

uuuu

lu

(((

)

*

+++

,

-

&

&

(((

)

*

+++

,

-

&

'

19

11

2

19

19

1

2

1

2

7

00

110

432

,

1

01

001

UL

#&2% %9* 5&Z :U e&!= B&RD :0=&$ A&M J O#6(7&'1!+ E&1 ]#&'#/ ]#3($ <j:R$ Eb+(5 M%M1!+ #2%1+(3 4!".

%*-J1:

8#Z )4!56+$A #2$"0*0'1$"3 )3#!+ (positive definite matrix) O$#0 +="

) G( 5%M#1*1 56(7'1 (symmetric matrix).

) K( 51 PAxx t F:7' : k H?*1 FY 5D?(1x.

5&&&&Z :U _&&&& DU* B&&&&$3))0&&&&C!(\* ( EG ,*&&&&$ B&&&& B9*!#D 5&&&&D?(1 56(7&&&&'1 4&&&&%3 5 :UZ!+ :'#$R!+) )&6L (GU (?(1 #&2%05&D 5& (%R!+ 5& M%M1!+ 56(7&'1!+ EG( JU )&S

:l(B1(transpose) 5 %7C!+ 5 M%M1!+ 56(7'1!+LEG FG J:

tLLLUA

5 D#&&C9!+ O+(&&&U/!+ BB&&3 8&&&Z e!=&&D(– <&&&'$!+ 4&&!" ]#&&&D :Z*– 56(7&&&'1 8&&& %9* B&&$3 5&Z :U!+ f=S . /%* E01 ( J )0C!(\* 5Z :UD #2 %1#3 4!" B B9*!#D 5D?(1 8& %9*!

)% #1 6 O#6(7'1!+ E1 m($!+ +=S:

tLLLUA

b¹b×²�UÐ W³łu*« �U�uHB*« qOK×²� ©wJ��uAð® WO�“—«uš

Factorization of Positive Definite Matrices (Choleski's Algorithm)


151

5 %7&&C!+ 5&& M%M1!+ 56(7&&'1!+ :&&'#$3 K#&&C9 E&&01 L K&& *:*!#D ]+B(&&13 ]+B(&&135 !#*!+ n '!+ A+B/*C#D:

11111 a

0123

45 & 6

&

1

111

111

j

kjkikij

jjij a )5 :UZ!+ : k :'#$R%!(

6&

&

1

1

211i

kikiiii a )#$R%! +B3 5 :UZ!+ :'111(

+ :";(6-3):

5&&& !#*!+ 56(7&&&'1!+ 8&&&%9 )0&&&C!(\* 5&&&Z :U A+B/*&&&C#DA )&&&6 B&&& B9*!#D 5&&&D?(1!+X:('!+ :tLLA .

(((

)

*

+++

,

-

&

&

5.375.21

75.225.41

114

A

!"#:

!

"

###

$

%

!

"

###

$

%

&

!

"

###

$

%

'

'

33

3222

312111

333231

2221

11

100

110

111

111

011

001

5.375.21

75.225.41

114

5.021

11

5.021

11

241

11

3131

11

2121

1111

&&&

'&'&&

&&&

a

a

a

( )

( ) ( )( )

( ) ( ) 15.15.05.311

5.15.05.075.22111

111

25.025.41

22

322313333

21313222

32

22212222

&''&''&

&''&'&

&''&'&

la

a

la

!"#$:

!

"

###

$

% '

&

!

"

###

$

%

'&

100

5.120

5.05.02

,

15.15.0

025.0

002

UL

* * *


152

$%!&':

%%%& '%%%()*&+ '%%%,-.&/& '(!0%%%1& ).%%%2/3" 4%%%$&56 78 9%%%59:/+.2 '%%%2;!& <%%%=.$ "#> . ? @%%%%( '(!0%%%%1&+" A%%%%5,:/ 4%%%%$&8 4B%%%%( ?.%%%%C5,D E#%%%% @$%%%%F+!G/ '%%%%5&H)"!3 I%%%%52J/2 K%%%%+#!

L)!1+"tLLA &

&

@J3+" 7.M=+" A:= 7-

yxLLt &

455-,-&+" 45&.M=+" A: @=*5 N#+"!:

zxL

yLz

t &

&

O%J*&+" @%J3+" 7.M=+" A:+ 'P5)J+" E# !Lx = y 4%& 99%D Q")%;> O%+> N9R%/%%5,&*+" 99%%D S%%1= T.%%25)P/ N!.%%F5 '52.%%F:+" <.%%5,&*+" S#%%:+" '%%P5)J @%%( '%%2!,J&+" <.

@$%%F+!G/ '%%5&H)"!3 7"93/%%F.2 @%%J3+" 7.%%M=+" A%%: 48 <.%%2-> 4%%$&5 U%%5: ? V!.%%;+W,J/5:

' 99D 9.;5>n '5*52)/+" )!#;+" 4&.

' W)X <.5,&D /. 99D '&FY: 6

29 23 nnn **

' Z&; <.5,&D /. 99D [)J: 6

76 23 nnn '*

4%%%%& @$%%%%F+!G/ '%%%%P5)J LH%%%%5&! <.%%%%5,&*+" 99%%%%D O%%%%,D 9%%%%&/*/ '52.%%%%F:+" '%%%%5:.=+" '5*52)/+" )!#;+" 75Y 9.;5\ '2!,J&+") . 99%D @/+"n ( "#% <.%5,&*+" 99%D 4] T")%M=!

@%J3 A%&.D(linear factor) @%(n '%5!=*&+" '%&5P+" '%5:.= 4%& ^Y.=/5%F _%=B( ?(in

significance) L9.5H Z&n.

!" #!$% &'()*+,+-%.&/ 0/1,! 2 34 , 54,6-%.7 7849 :%4, ;$ <(= >- 0/ !" 0,?,/@%.7 @!8 .7 5A7+ @,+&B".7 5$ #!$% #=) :C@ 0.DE".7 @>(727-3.(

!"#$%& '()*+,$- .,/-&"01 2-#, .034#, 56

;4444FA.7 :&4444>(.7 54444- 3!44446F".7 #= G@4444H(Ax=y I4444,-A 04444/ !" 0*!H4444J"+44444,+-%.&/. $44444E.!K% 044444,"L@7!A :+A%44444E( #= 544444MH, 044444.&-.7 N8444449 ;44444* 544444,6-%. OP!= ;

0*!HJ".7A Q@!J.7 ;*

tLLA


153

@%%-,-&+" 7.%%M=+" A%%: 4%%$&5!Lz=y 5!*/+.%%2 4] @&.%%&]" `L '%%5-,-& '(!0%%1&'5,0%%%%F@%%%%-,-&+" 7.%%%%M=+" A%%%%:! ?zxLt & 4] @%%%%0,3+" `5!*/+.%%%%2tL '%%%%5-,-& '(!0%%%%1&'5!,D@,5 .&$ ?:

()*+: @,0F+" @-,-&+" 7.M=+" A:Lz = y.

ni

zyz

yz

i

jjiji

iii

,,3,2

;111

1/

1

1

1111

&

!"

#$% '&

&

+'

&

(,-.,/: N!,*+" @-,-&+" 7.M=+" A:zxLt &.

1,,2,1

;111

1/

1

''&

!"

#$% '&

&

+*&

nni

xzx

zx

n

ijjjii

iii

nnnn

$%!&':

N!%%%%%,*+" @%%%%-,-&+" 7.%%%%M=+" A%%%%%: 9%%%%=D T.%%%%&!&DUx=z '%%%%%5+./+" <.%%%%Ya*+" 793/%%%%F=)@0,3+" `5!*/+":(

;1

/

1 !"

#$% '&

&

+*&

n

ijjijiiii

nnnn

xuzux

uzx

'5+.:+" .=/+bF& @(!tLU , 4B( K+#+! ?jiiju 1&.

* * *

%%3,= @%%,5 .%%&5( O%%+> <.(!0%%1&+" A%%5,:/2 '%%1.3+"! '%%&!,*&+" cd.%%/=+" `%%*2 ^'5-,-& A&"!D.

$-/0/'"# ,10'#*2 3"4 5,6*78'"# -0!9 *! :;,9.

<a59%%2/ N8 A%%&D 4!9%%2 V!.%%;+ S#%%:+" '%%5&H)"!3 #%%50=/ 4%%$&&+" 4%%& 4.%%$ "#> '(!0%%%%1& S!0%%%%1 O%%%%,DA '(!0%%%%1&+" A%%%%5,:/ 4%%%%$&5 _%%%%=B( ?A .%%%%C5,&.D O%%%%+>

455-,-&+"L , U5:2 ? 48 UA = LU.


154

.& '(!01& 4!$/ 48 4$&&+" 4&A L9)0=&(singular) A%5,:/ .C+!L-U !8 ? L9)0=& )5e 4!$/(non singular) A5,:/ .C+ V5+!L-U.

'(!0%%%%1&+" <%%%%=.$ "#>A H.%%%%$/)" 4!9%%%%2 V!.%%%%;+ S#%%%%:+" '%%%%P5)J 4B%%%%( L9)%%%%0=& )%%%%5e(pivoting) 'P5)J2 _5,D A1:= N#+" _F0= A5,:/+" O+> N9R/)+!9A/5.(

<PP: "#>A T.5*JY '5)JP+" L)J5F+" J)G(strictly diagonally dominant).

niaan

ijj

ijii ,,2,1;1

&-

!"

!!!!"#$ %&A '!!!!(')*+#, -!!!!,.%/(positive definite) -!!!!0(12 3!!!!(,2* 4!!!!$/( 5!!!!"67 8 9#:"+; <=> ?%#.+ @A)+;Ax = y %',-7%B!C/+; @%B!C+ D('!,* E& F!/> 4 80*!G/ -!(=/H+; IAJ 4%$*% K1(stable) #!,(10*+; L#!2MN, 1ON!** P E& 5!"67 Q+#!*+#,% 8

F(=)* 4$/(A 4((O=O/+; #R(=/#> <+SL , U.

#7%B!C/+; 4!/ T#!M U%" -(120+; -(ODO #7%BC/+;; F/*!V* 5!(7 -7%B!C/+1#B!!!CW; 4!!!/ 1!!!(O$ <!!!=> -!!!,%=2/+; -(,#!!!G)+; #!!!(=/H+; ''!!!> F!!!(=0* 4!!!$/( Q+#!!!*+#,% 8

#R(=/#> <+S -7%BC/+; F(=)*+.

QJ -(120+; -(ODO -7%BC/=+ -/#H+; K1%C+;:

#########

$

%

&&&&&&&&&

'

(

"

)

)))))

nnnn

nnnnnn

aa

aaa

aaa

aaa

aa

A

1,

,11,12,1

343332

232221

1211

00

0

00

00

00

!

!

!!

Q!!!7 2!!07 '!!!.%* 4& 4!!$/( -7%B!!C/+; IA!!!J Q!!7 -(1B!!!C+; 1!!(X 1!!C#"H+; 4& E&+; 1!!20+;K1!!V#,/ I1#!!G( <!!=> %& K1!!V#,/ 5!!"(/( <!!=> %& Q!!G(Y1 . K1%!!C+; 2(!!G,* 4!!$/(%

Q+#*+; F$V+#, #R*,#*$, -7%BC/+; IAR+ -/#H+;:

©�Ëd� WI¹dÞ® WO¦K¦*« UNK�«uŽ v�≈ W¹dDI�« WOŁöŁ �U�uHB*« qOK%

(Crout) Factorization of Tridiagonal Matrices


155

(*)

111

333

222

11

0

0

########

$

%

&&&&&&&&

'

(

"

)))

nn

nnn

A

*+,*+

,*+,*+

,*

F(=)* 4$/( 5"& Z1B"A -(+#*+; K1%C+#,) %1$ -0(12:( A = L.U

[():

#########

$

%

&&&&&&&&&

'

(

"

nn

ii

d

d

d

d

d

L

1

1

1

1

0

0

33

22

1

############

$

%

&&&&&&&&&&&&

'

(

"

)

)

1

1

1

1

1

1

1

1

1

3

2

1

0

0

n

i

i

u

u

u

u

u

u

U

2H( ;A!!!!!!J% E%#!!!!!!G( P'#!!!!!!H/+; 4!!!!!!/ \;''!!!!!!> #!!!!!!"(3n-2 4!!!!!!/ 5!!!!!!GB" ''!!!!!!H+; Q!!!!!!7F(J#./+;- .iii ud ,,1 .QJ P'#H/+; IAJ%:

11 d"* (1)

niii ,,3,2;1 ""+ (2)

nidu iiii ,,3,2;1 1 "/" )* (3)

1,,2,1; )"" niud iii , (4)


156

#!H/+; IA!J 9;'M*!GP \P#!O/ ]1" 4& F, % 4(*7%B!C/+; 1!C#"> 4(!(H* Q!7 P'L,U E& U%!"+; ;A!J 4!/ D/#H/+; -7%BC/ 4%$* 4() Q2M+; 9#:"+; F) <+S 1(V"

-(120+; -(ODO -7%BC/(tridiagonal).

4& Z1B"A (=)* 4$/( 5"&% -(120+; -(ODO -7%BC/ 4((O=O/+; #R(=/#> <+S #R=L , U %J 5=) _%=2/+; Q2M+; 9#:"+; 4&%:

Ax = y

0 LUx = y1

4& Z1B": Ux = z (1)

0 Lz = y1 (2)

9#!!:"+; \P%& F!!)"7(2) 9#!!:"+; 9!!O(1) _%!!=2/+; <!!=> F!!C)"+x 4!!/ F!!^& ''!!H, ?%#.+ @A)+; -0(12, F)+; ;AJ <=> F%C)=+ -,%=2/+; #(=/H+;) 4%', E& F!(=)*

-7%BC/+;A #R(=/#> <+S.(

!"#$%& '()%& *"!#%& +,- ./ (0%& ./121 .3-45$%& +/6,7 ./$8 &-)

-!(120+; -!(ODO -7%B!C/+; F!(=)*+A 4(!(O=O/+; #!R(=/#> <!+S \#0,#!G -!71H/+;L , U

\#0,#G 4(71H/+; Q2M+; 9#:"+; F) 9O 8Ax = y -(+#*+; ;%2M+; a,*":

9:-; :%& +/6,7.3-45$ A:

FH.; 11 *"d

_G); 111 / du ,"

9(0=+i = 2, 3, ... , n-1 _G);:

iii

iiii

ii

du

ud

/

1

1

1

,

*

+

"

)"

"

)

FH.; 11 ))"

"

nnnn

nn

ud

l

*

+

9"/#"1 :'()%& *"!#%& +,:

) &( 9#:"+; F) Lz = y

W¹dDI�« wŁöŁ wD)« ÂUEM�« qŠ

Solution of Tridiagonal Linear System


157

FH.; 111 /dyz "

9(0=+i = 2, 3, ... , n _G);: - .11 1 ))" iiiidi zyz

) _( 9#:"+; F) Ux = z

FH.; nn zx "

9(0=+i = n-1, n-2, ... , 1 _G);:

- .- .

11

32

23/1/

23

21121

33

222

1222

)""

)")""

")))")"

+

,

*

du

ud

1 !" iiii xuzx

!"#(7-3): ! !"#:

!

"

####

$

%

&

&&

&&

&

'

2100

1210

0121

0012

A

$%&"'()* +,-A ./01,(.2 3)4L, U ) $/51!67&!/8(9 :./;#)* +/- </=&>&Ax = y

?1-:

!

"

####

$

%

'

1

0

0

1

y

!"#: :./;#)* +/-& $/1!65)* $/1@A@ $%&"/'()* +/1,-B $/1(C!*&D 7*&6D E.FBGF

H,1 .( <=# $1!65)* H@A@ H6D)*:

$%&' : ()&*+,"# -.!/A:

11

2/1/

2

22

111

11

&''

&''

''

(

)

*

du

d

+ ,


158

! ! !

! !45

43121

11

43

34/1/

34

32121

3444

44

333

2333

"###"#"

#""

#"#""

"###"#"

ud

du

ud

$

%

&

$

A = LU

'''''

(

)

*****

+

,

#

#

#

'''''

(

)

*****

+

,

#

#

#"

''''

(

)

****

+

,

#

##

##

#

1000

100

010

001

100

010

001

0002

2100

1210

0121

0012

4

3

3

2

2

1

4

5

3

4

2

3

!"#!$ :%&'() *!+#() ,-:

! ! ! ! ! ! ! ! ! ! ! ! ! ! !

1

14111

4/5111

41

3110

3/4111

31

2110

2/3111

21/

44

3444

4

2333

3

1222

2

111

""

"##"#"

"##"#"

"##"#"

""

zx

zyd

z

zyd

z

zyd

z

dyz

! ! ! 1

21

21

132

31

143

41

2111

3222

4333

"##"#"

"##"#"

"##"#"

xuzx

xuzx

xuzx

- ! "#$%&' (): 14321 """" xxxx

.+-/0:

*+++,-&' "#+++$%&' (+++) .+++/0,1&' .+++/232 .4 5+++67&' (+++/8)9& :0 ;<+++7&' .+++/7=0' -&'>#; #78; #?1/@,9 >;7/ (@#17&':

nidi ,,2,1,0 "./

>//&#9&' >/,0A&' >7 BC D1)9 '<E F#)/)6 > ;/ '<! :

) C( G/G)9&#@ .@H 7 "#$%&' *4 I37#J7&' .4 567.

) K( F#/J,L ./0,1&' :0,/M&' ,0A D1)9 "#$%&' *4 I37#J7&' .4 567.


159

N<++++! D++++/@,9 .++++/%#;7E O++++11)9 >7++++P/ F#/4#++++PE F#,0++++A *++++,J9 .++++/&#9&' .++++/0$%&' ./7=0' -&'.

."1+#:

>C Q05%A 2 .4 567 ./0,1&' ./23] *4 F#1@#M .4R0J7&#;(*) .[>C Q05%

1,,3,2,0. #"./ niii &%

>#; '<E:

nn

iii ni

%$

&%$

&$

0

#".12

0

,1,,3,2

,11

>S4A "/+L T:G0+5%7 0+/U .4 5+67 > +;9id > +;9 .1@#+M&' .+/7=0' -&' #+?/,J9 *+9&' "/L V/7H& ./056 0/Ui ) "/18& BCi = 1, 2, ... , n.(

.+/,-&' "$%+&' Q+J@ *+4Ax = y I3-G+7&' *+4 .+5/5, I'0+/W9 BGX+9) 0+6#%Y I37#++J7&' .4 5++67A O++H97&' 0++6#%Y Cy #++7?/8; C ( O++H97 *++4 :0++/@; I'0++/W9 Z++&E

(+++)&'x .% #+++?%E "$%+++&' N<+++! (+++27& (#+++1/,0+++A&' .+++8/8Y "+++$ . I'0+++/W9&' N<+++! IGC '<E #+++7%/@,0A&' >M) "#$%&' >E (#1/4 ()&' OH97 *4 .5/5, I'0/W9 Z&E I3-G7&' *4 .5/5,&'.

0 ,!$(8-3):

*,-&' "#$%&'Ax = y >/9&G#J7&#@ [R0\J]7&':

0001.320001.1

32

21

21

"1

"1

xx

xx

>C BC:

'(

)*+

,"'

(

)*+

,"

0001.3

3,

20001.1

21yA

G/) &' ()&' O&

'(

)*+

,"

1

1x

◊dA�« WKOKF�« rEM�«Ë ◊dA�« WM�(« rEM�«

Well-Conditioned and Ill-Conditioned Systems


160

*8/ #7; "#$%&' ^@6/ _/)@ F#5/5, F'0/W9 I37#J7&' .4 567 I0/W9 '<E #7%/@:

0001.329999.0

32

21

21

"1

"1

xx

xx

^@6/ _/)@ F'0/@; F'0/W9 ()&' 0/W9/

'(

)*+

,#"

2

1x

O+++H97&' 0+++/W9 '<E `&<+++; y *8+++6a' "#+++$%&' *++++4Ax=y ^@++++69 _+++/)@ F#+++5/5, F'0++++/W9'<;! >#9&G#J7&':

9999.220001.1

32

21

21

"1

"1

xx

xx

()&' Z8Y (6)% #%%S4'(

)*+

,#"

2

1x *86a' ()&' >Y F#7#79 [89-7&''

(

)*+

,"

1

1x.

(++++)&' >C :0!#++++$&' N<++++?& *++++MG%?&' (++++/8J9&'x >/7/19++++M7&' V,#++++19 .++++,1% (++++27/"#+++++$%&' *9&G#+++++J7@ >/40+++++J7&' .Z+++++,J7&' "#+++++$%&' *+++++4 >/7/19+++++M7&' >C _+++++/) yAx "

F#+@/019 >#+/=' 97) .+,1%&' *+4 >#+J,#19/ (1 , 1) (;+A&#@ >/+@7 +! #+7; ( 0+/W9 BC >S+4 Z&E BGX/ GL >/7/19M7&' >7 BC .&G#J7 *4 [/5,V,#+19&' .,1% *4 0/@; 0/W9 BC T

#?%Y :G/J@ .7/G1&' .,1%&' >Y F#7#79 .589-7 :G/GH&' V,#19&' .,1% > ;9.

(;++++++A&' >++++++7 b0++++++% `&<++++++;) >/7/19++++++M7&' *9&G#++++++J7 >++++++7 C21 1,1 ( .++++++,1% >C

(++27 !0,33ax >/7/19++M7&' G++)C Z++8Y V++19) ++! 11 (0++-c' >++7 .++@/0L !21 Z++%J7 T


161

!!!"#$ %&!!!'– ()*+,- !!!.– )/0, 1),-2 !!!+3-% 405,!!!6 7 !!!. 7 !!!).)/0, 78!!!5 1*9,!!!6 !!!"#$ :$ ;<= !!!0,-% >!!!=0# 1!!!? @2!!!)+. >!!!0)05-% A!!!B A!!!' !!!3#). ; 1),-2 !!!+3C- . 1$ D/!!!# 1$ !!!##93)*

>=0#-%(3, 0) E F#C- 7 ).)/0, 785 1*9,6Ax = y G 65. !"#$%& '(#$%&(residual

vector) )A.6%/-% HI,3-%.(

!

"#$

%&

!

"#$

%'

!

"#$

% !

"#$

%&

'&

0002.0

0

0001.3

3

0

3

20001.1

21

yAxr a

/ )+3 1JB K-&.*(norm) :* 6) HI,3-% %&'

0002.0max21

&&(() iirr

])*+,$:

L/+M) '(#$ -. /012$(vector norm) z >N8+- .

inizz

(()&

1max

O!!)5n H!!I,3-% /!!P #? 22!!?z/ !!)+3-% %&!!' Q3!!6)* ; :/ !!)+3-%)1 /)) !!+3 K !!#'* ;)CN 2+. R% 1&J. D/#6 39 "S)/+, 193) D/T$U.[

1$ F!!!!58# %&!!!!9'*@/)V!!!!P A!!!!0.,3-% H!!!!I,3-% / !!!!)+3 >!!!!3)N G!!!!)/0,-% 1J!!!!B K!!!!-& E!!!!W/* ;

!

"#$

%&

0

3ax X)5!P-% U!5-% 1? 2)+. H!I,3-% / !)+3 1J!B >!0)05-% A!B* ;ae xx ' ) *!'*

=*.Y3-% U5-% 1). 4/ S-%ex A.)/0,-% U5-%*ax ( :* 6)

21

2

0

3

1

1&

'&

!

"#$

%'

!

"#$

%&'

))

)ae xx

@/).9 H,3)N 1$ :$.

1$ <!!N*,# !!##JB 7 !!.)/0, 1))Z%*!!,3 /!!)W 1 3)0,!!63-% !!")B 1*!!9) A!!,-% >!!- 5-% A!!B* 7%/)VP 7 )0.,3 7 "I,3(small residual vector) 7 0)N2 7 .)/0, A#+).

? >!!!-[2C- E !!!F#-% >!!!62#' Q!!!C? 2 !!!3,?[% !!!##93) [ H!!!#JB 7 !!!3*3?* >!!!)# 93\ Q!!!C]89!!!^3-% _&!!!' U!!!`3 O*2!!!5 . 4!!!)/= 1!!!? ] !!!3*C+3-% _&!!!' Q!!!-\ U!!!P*,# 1$ 1!!!93)*

=/!!!^-% 22!!!+. Q3!!!6) !!!3 >!!!3)N >!!!B/+3(condition number) >!!!3)N Q!!!C? 2!!!3,+) :&!!!-%


162

]83 !!+3-% >B*S!!P3 / !!)+3A "!!6*9+3 / !!)+3* . =/!!^-% 22!!? 1!!3 aU!!9 L!!)/+, U!!.N*>)- ,-% ] S)/+,C- b ,5# >B*SP3-% / )+3*:

#31/2:

L/!!c+M) '444(#$%& /044412$(vector norm) ] !!!"I,3-% >!!!?*3I3 Q!!!C?nR ]%*& 2+M.-%n >)0)05-% ] .9/3-% ]%*&*(real components) >-%2 H#d.. 1!3nR Q!-\R

>)- ,-% e%*T-% 405,:

n

n

n

Ry

RR

R

*+,(,

**+&

&-&

*+.

,x,yxyx

x&,xx

0x0x

x,0x

///

. 13*% 1) "S)/+, 1933-% >SC,T3-% /)) +3->)- ,-% /)) +3-% /9&# ;:

/ )+3-%11: 0&

&n

iix

11x

/ )+3-%21: 1 2 2/1

1

2

2x 0

&&

n

iix

/ )+3-%)1: .

1maxx i

nix

(()&

/ )+3-% Q36) @2 ?*21 : :2)CNf% / )+3-%(Euclidean norm) HI,3C-x O)5 H!!!I,3-% 1 !!!9 %&\ !!!3 >!!!- 5 A!!!B U!!!Pg% >!!!=0# 1!!!3 >B !!!63-% U!!!`3) H!!!#$x A!!!BR *$2R

*$3R.

HI,3-% 78`3B3 4tx 2,1,1 ''& 1%/ )+3-% H-:

3 4 3 4 3 41 2 22,1,1maxx

6211x222

2

&''&

&',',&

)

31/2#:

L/c+M) )56748$%& /0412$(matrix norm) >!)0)05-% ] B*S!P3-% >!?*3I3 Q!C? >+./3-%nn5 >!)0)05 >!-%2 H#d.. 1),+./!3 1),B*S!P3 :g >!)- ,-% =*/!^-% 4!05,

A , B A0)05 22? :$*/:


163

BAAB

BABA

AA

AA

A

&

,&,

&

&-&

.

//

00

0

1 9 %& h%*. QC? HI,3 / )+3 *'nR 1JB

xmax1

AAx &

&

>!!!+./3-% ] B*S!!!P3-% >!!!?*3I3 Q!!!C? >B*S!!!P3 / !!!)+3 L/!!!c+M)nn5 Qc3!!!6M)* ; 9&/044412$ 90121": (natural norm).

/)) +3-% 1). 13*)>)+).=-% (E%2T,6% /`9) ] B*SP3C-:

3-% / )+)1: )&))

& xmax1AA

x

/ )+3-%21: 212

2xmax AA

x &&

]# 9 %&\ H#$ ] .`\ 193)*3 4ijaA & >+./3 >B*SP3nn5 1JB:

0&(()

&n

jij

niaA

11max

La/i+M# U`3- .*:

0&((

&n

iij

njaA

111max

>3)N 2 I)f 78`3)

A >B*SP3C-

!

"

###

$

%

'

'

'

&

115

130

121

A

41213

11 &',,&0

&jja

41303

12 &',,&0

&jja

71153

13 &,',&0

&jja

1 2 77,4,4max &&6)

A


164

)*+,$:

>N8+- . Lc/+M) :&-%* >B*SP3- A+).=-% / )+3-%

xmax1AA

x &&

>)- ,-% >jB 93-% >N8+- . 7 Y)$ Lc/+M) 1$ 193):

x

x

0max

A

xA

7&

7 Y)$ / )+3-% %&' Q63)*;<= >+?$%& /012$%& (induced norm).

)1/*@:

]# 9 %&\ax E !F#C- 7 !).)/0, 78!5Ax=y ]!# 9*ex %&!"- =*.!Y3-% U!5-% A!' ]# 9* ;E F#-%A 1 ,N8+-% 405,, A+).= / )+3 :g H#JB ; @2/S#3 /)W >B*SP3:

1ea .rxx '(' A (i)

yr

..xxx 1

e

ea '('

AA (ii)

) 1$ =/^.0y,0xe 77.(

O)5r A0.,3-% HI,3-% *'(residual vector) A!.)/0,-% U5C-ax >.!6#- . E !F#C-Ax

= y.

A0B/"%&: >)- ,-% >)2)"3,-% ] .`f 7[*$ b ,5#:

)1=1C$#:

H!!!!I,3 :gx :* !!!!6) 2!!!!+M. :&n >B*S!!!!P3 :$*A >!!!!+./3nn5 ) / !!!!)+3 ]%& A+).=A:(

x.x AA (

)1=1C$#%& A0B/":

xx..x

xx

.xx0x

A

AA

&

&67

) 1gx.x // &.(

1$ #Y/B %&JB xxu &


165

1JB:

1xxu &&

1JB K-&.*:

A

A

A

.x

1u

umax.x

u.xAx

&

&(

&

)A+).=-% / )+3-% L)/+, 13.(

>- 5-% AB >5Y%* >)2)"3,-% 1,3 >5P*x = 0.

31/2#: 1 3I!6#3 !3"#\ >B*S!P3 / !)+3* H!I,3 / )+3- U 0)(compatible) !005 %&\ >N8+-% x.x AA (

] "I,3-% <)3I- K-&*x ABnR >+./3-% ] B*SP3-% <)3I-*nn5 0)05-%)>.

)1/*@%& A0B/" D"0#:

)$( >#) .,3-% ] .`\(i):

3 4rxx

xxxx

xr

1ea

eaea

a

'&'6

'&'&

'&

A

AAA

yA

) 1gA @2/S#3 /)W >B*SP3(

r.

rxx

1

1ea

'

'

(

&'

A

A

)>)2)"3,-% 4).=,.(

)G( >#) .,3-% ] .`\(ii):

ee x.xy AA (&

yx1

e

A(6

>#) .,3-% <3 >#) .,3-% _&' E%2T,6 .*(i) 1$ 2I#:

y..rxxx 1

e

eaA

A'('


166

1)!!!!!!,)39-% 1$ 1 !!!!!!)#+, >!!!!!!)/F#-% _&!!!!!!' !!!!!!,#) .,31'A * 1. 'AA 1$ 1!!!!!!93)A!!.)/0,-% U!!5-% >!!N2* A!!0.,3-% H!!I,3-% 1)!!. >C!!P-% Q!!C? >!!-[2C- 32T,!!6M) . 1J!!B 7 !!3*3?*

A.6#-% d=T-%eea x/xx ' 2*2!53 d!=T-% %&!'* ;_/!)W 13 /`9$ #3") :&-% *'–

>!!!)# `-% >!!!!#) .,3-% K!!!-& Q!!!!C? U2!!!, !!!39– /%2!!!03-% G/!!!!Y U!!!P 5.1'AA >!!!!)0.-%* >).6#-%y/r G!)/0,-% %&!"- G!6 #3 / !)+3 :$ E%2T,!6% 1!93)* ;A.)/0,-% U5-% %&"-

>06,3 @/*P. 7 3j%2 H3%2T,6% UF) 1$ QC?.

=/!!^-% 22!!? Lc/!!+M)K(A) @2/!!S#3-% /!!)W >B*S!!P3C-A / !!)+3C- >.!!6#- . .

>N8+- .:

3 4 1A.AAK '&

>)- ,-% @/*P-% AB >0. 6-% >)/F#-% A,#) .,3 >. ,9 193) Z3/-% %&' E%2T,6 .*:

3 4A

rea xx .AK(' (i)

3 4y

r

x

xx

e

ea .AK('

(ii)

>B*SP3 :g H#g 7%/F#*A >N8+-% 405,, @2/S#3 /)W:

3 4AKAAAAI &(&& '' 11 ..1

<!N*,M) H!#JB– 1 !,/)Tg% 1 !,#) .,3-% K!-& Q!C? #-2!, !39– 22!? 1 !9 !3C9 H!#$ =/!!!^-%K(A) 1!!!3 7 !!!.)/N1 K*C!!!6 1J!!!B(behavior) >B*S!!!P3-%A *$ 7 #!!!65 1*!!!9)

7 )*6(well)!P 7 !)0.,3 7 !"I,3 1$ Q!#+3. ; 7 !0)N2 7 !).)/0, 78!5 A!#+) 7%/)V 1 !9 %&\ !3#). ;. /!!.9$ =/!!^-% 22!!? 1!!3 /!!)`91 K*C!!6-% >)*!!6 1*!!9, [ >B*S!!P3-% 1J!!B ;(not well-

behaved).

(Condition Number) ◊dA�« œbŽ n¹dFð


167

31/2#:

>B*SP3- U 0)A !"#\ @2/S#3 /)W:/4E%& )@4F+ )56748$ (well-conditioned

matrix) =/!!^-% 22!!? 1 !!9 %&\K(A) 1!!3 !!.)/N >B*S!!P3-% _&!!"-1 . !!"#\ U !!0)* )44G1GH

:/E%& (ill-conditioned) 9$ =/^-% 22? 1 9 %&\ 13 /)`9. /.1.

E !F#-% 1\ U !0)*Ax = y :/4E%& A4F+ I04*@ (well-conditioned system) %&\ >B*S!!!P3-% ]!!!# 9A H!!!#\ U !!!0)* ; =/!!!^-% >#!!!65:/444E%& J4441GH >B*S!!!P3-% ]!!!# 9 %&\A

=/^-% >C)C?.

$ J0K(9-3):

E F#-% 1$ ].`%Ax = y U `3 AB Q=+3-%k-l =/^-% U)C? E F#.

J+%&:

!

"#$

%&

20001.1

21A

0&(()

&2

121max

jij

iaA

0001.320001.1

321

2

12

2

11

&,&

&,&

0&

0&

jj

jj

a

a

1 2 0001.30001.3,3max &&8)

A

!

"#$

%

'

'&'

50005.5000

10000100001A

1 2 200005.10000,20000max1 &&)

'A

Aj "#8-% / )+3C- >.6#- . H#$ :$:

3 41002,60000,200001.3

. 1

99&5&

&)

')AAAK

13 /)`9. /.9$ =/^-% EN/ EI5 1$ :$1 =/^-% U)C? E F#- B EI5 /i.m9* ; EN/-% %&' %&!"- A!0.,3-% H!I,3-% Q!C? 7%n !#. A!.)/0,-% U!5-% >!N2 QC? >?/6. E95-% 2+.,6# #C+I)

U5-%.

WOD)« �ôœUF*« rE½ ∫Y�U¦�« qBH�«3 WOD)« �ôœUF*« rE½ ∫Y�U¦�« qBH�«

168

)!"0F%& )1/*@%& )(1#@:

3 43 4

yr

.xxx

yr

.1e

ea AKK A

('

(

A0B/"%&:

3 4

ea

ea

eaa

xx.

xx

xxyxr

'(

'&

'&':

A

A

AAA

ea

axx

yxr'(

':

A

A

A (1)

y.1y1ex

1e

e

yx

xy

'('&

'&6

&

AA

A

A

(2)

>!!!#) .,3-% A!!!B/= >3!!!60.(1) Q!!!C?y.1'A =/!!!^-% 22!!!? L!!!)/+, E%2T,!!!6 .* ; >#) .,3-%*(2)1$ 2I# ;:

3 4 e

ea

1

ea

xxx

y.

xxyr

.1 '(

'(

'AAK

>#) .,3-% E%2T,6 .*)( ;ii G,9# 1$ ##93):

3 4 3 4y

r.

x

xx

y

r.1

e

eaAK

AK(

'(

=/!!^-% 22!!? 1 !!9 !!3C9 H!!#$ 1),#) !!.,3-% 1), !!' 1!!3 F!!58#K(A) 1!!3 7 !!.)/N1; /%203-% 1JBy/r A.!6#-% d=T-% >3)N 13 H,3)N G/,0,eea x/xx ' H!#$ :$ ;

A.!!!!!6#-% d!!!!!=T-% %&!!!!!' >!!!!!3)N 7 !!!!!.)/0, o!!!!!9+) . 7%2!!!!!I 7%/!!!!!).9 =/!!!!!^-% 22!!!!!? 1 !!!!!9 %&\ !!!!!3#).3 4 199AK /%203 1 9 1 h%* Q,5 ;7%/).9 1*9) 2N A.6#-% d=T-% 1JBy/r 7%/)VP

7%2I.

>B*S!P3- =/!^-% 22!? G !65 7 !3*3? U"!6-% 13 o)- H#$ /9&- . /)2I-% 13* >!6*9+3-% >B*S!P3-% /!P #? G !65 G!C=,) K-& 1$ O)51'A b !,5) _/*2!. %&!'* ;

H!,%&. >-d!63 2!+M)B @2!)2? ] . 65 Q-\ .. >B*S!P3-% ]!# 9 %&\ K-&!9A /!^-% >!C)C? =


169

!""#$ !"#1 A - $%""&'( ))""* +',""-',./– 0""$12'( 34""5 6 78 9:,""1;<) /""=; 7 >/""? :(%";.= $%"&'( ))"* @"AB ,"= (48 ,"C 0"#%DC :,"C/C* ,"2CE; F"2G H%;.= 0;C56 I(4 I?;'

7 @6 :()"""A .."""$'( J"""D. )"""A/-/ /) $%"""&'( ))"""D' 0""";.;%1- 0"""C;< K,"""?B' 0""";))D'( L%0?/=DC'( 0#/MNC'( K,?B.

@,""O2'( P""B )""2* $%""&'( ))""D' 0"";.;%1- 0""C;< K,""?B' @)Q-""?- +""-'( R;""N'( ""C/ +""$Q'(Ax = y ))""D' I,.,""?B'( S(%A!""./ >4""BT' U/,""A 0""1;%$ P,CD-""?,.t ""C @,<%G((t-digit arithmetic) 0;',-'( 0V;N'( 9:

! " ax/z.10tAK #

W;B:

ax: +$Q'( @,O2T' +.;%1-'( PB'( Ax = y

z: +$Q'( @,O2T' +.;%1-'( PB'( Az = r

W;B:

r: +1.-C'( FA-C'( axyr A $

6 ,2""""X%# (48ax F""""A-C'( @()Q-""""?( ,""""22=C;# +""""$Q @,""""O2' +"""".;%1- P""""B /""""5 +1.-C'(r 9 @,"O2T' $/."XC'( P"B'( /B2 K%,1-- 0D.,--C 0;.;%1- P/TB YT* P/NBT'

0'/.1C 0<)' PN2 ;B 0;%;%=-'( 0;TCD'( 345 ></2/.

/""5 +""$Q'( @,""O2'( 6 J%""M2Ax = y$/.""XC'( P"""B'( 6/ 9 /""5ex 6/ 9+.;%1-'( PB'(/ $/.XC'( PB'( ;. L%,M'(ae xx x $

/5 +.;%1-'( PB'( (4E' +1.-C'( FA-C'(:

! "! "

r x

x

xyx

y xx

yxr

e

e

a

$

$

$

$

$

A

A

AA

A

A

©Íd¹dJ²�« 5�×²�« Ë√® Iterative Refinement W¹d¹dJ²�« WOHB²�«

(or Iterative Improvement)


170

F""""A-C'( 6 Z6 x +""""$Q'( @,""""O2'( P""""B /""""5rx $A . @,""""O2'( (4""""5 P""""B./'(/YT* P/NB x ,22=C;) :,;%O2 ( $/.XC'( PB'( YT* P/NB'(ex W;B:

xaxex %$

='/– 0";TCD'( 0";B,2'( "C– K";%1-'( S,"$Q6 [,"25 !"#(rounding errors)

\"$QT' 0";.;%1- 0"C;< YT* PNB2 ,22!# +',-',./ x +',"-',./ 90$/."XC'( F"-C;< U;"'/$/."XC'( P"BT' 0";.;%1- 0C;< YT* PNB2 ,22!#ex L)6 0"C;1'( 34"5 /"=- 6 P"C\2/ 9

,E-1.,? Cax . %;%=-'( ,22=C;/(iteration) 0";TCD'( 345 %(%=-.– K,"?B 0";TC* x

0""C;< K,""?B/ P""BT' 0"";.;%1-ex - 0""'/.1C 0""<) I(4 0""C;< Y""'8 @;""1'( K%,""1-- 6 Y""'8 .

$%"&'( 0"T;TD'( @O2"'( ]"C 0";%;%=-'( 0;M"N-'( YC"?- +-'(/ 01;%$'( 345 @)Q-?- H),*/ .

P"""";TB-'( @()Q-""""?( """"=C;/LU 0#/M""""NCT'A ) P/""""NB'( )""""2* F"""";T* P""""NB2 Z4""""'(/YT*ax $.>4"B'( 01;% ( @,"O2'( P"B'rx $A . ,2C)Q-"?( (48 H),"*/k :,;%"&* :,"C<%

+TNG( @,O2'( PB K,?B')),A;8ax ( :,M*,XC :())* @)Q-?2 ,22!#(2k) @,<%G( C +<(/""".'( K,"""?B' 0;%"""&D'((residuals) S,"""$QG( /6 x .)"""A'( """C/ 6 0"""OB_C',. %;

+""#/ 9$%""&'( 0""T;TD'( @O2""'( ]"";CA ]""C H%/%""X',. `T""N- 7 0"";%;%=-'( 0;M""N-'( 0""1;%$+T""NG( @,""O2'( P""B I,.,""?B +""# %"".=6 0""<) @)Q-""?2# ,E-;B_""N @)""* 0""',B . P,""aC'(/

01;%$'( 345 I(/$Q b( 4!. `X/; +',-'(.

!"#$(10-3): +$Q'( @,O2'(

&&&

'

(

)))

*

+

$&&&

'

(

)))

*

+

62

51

78

x

172025

141720

202533

( F' $/.XC'( PB'! "t111x $.

) 6( ;;%""&* ;C<%""' 0""1;<) I,.,""?B'( S(%""A8 ]""C 0$;""?.'( >4""B'( 0""1;%$ @)Q-""?(0;.;%1- 0C;< ),A;c ['4/ $1# ;;/2DCax PBT'.

) K( Z%"";%=-'( ;""?B-'( 0""1;%$ ""C ;%"";%=- L"".$– 0;/2"""DC @,""<%6 0""D.%6 :,C)Q-""?C"""A-C'( K,"""?B +"""#F -C'( +"""1.– P"""BT' P"""X#6 ;.;%"""1- Y"""T* P/"""NBT' %,"""</ 9

$/.XC'( PB',. %;QG( K(/A'(.


171

!%& :

) 6( YT* PNB2 9 ;;%&* ;C<%. O,M-B7( ]C U/,A' >4B'( 01;%$ L;.$-.:

&&&

'

(

)))

*

+

80.0:79.000

4.3:8.18.10

78:202533

YT* PNB2 345 C/

&&&

'

(

)))

*

+

$

79.000

8.18.10

202533

U

&&&

'

(

)))

*

+

$

80.0

4.3

781yL

&&&

'

(

)))

*

+

$

156.076.0

0161.0

001

L

)A2/ /5 K/?BC'( +.;%1-'( PB'( 6: ! "t0.189.01.1ax $

) K( +1.-C'( FA-C'(– 0;/2DC @,<%6 0D.%G I,.,?B'( S(%A8 ]C– /5

! "t3000.01300.05500.0r $

),A;c/ x:

! "tLU

LU

A

002920.02055.05500.0

r x

r x

r x

1

$

$

$

$

YT* PNB2 @,O2'( (45 PB./

! "t003696.01105.01026.0 x $

( /=; ['4./P/G( %;%=-'( )D. +.;%1-'( PB':

! "t004.1000.19974.0 xax1x $%$

,C2;. F26 OB_2/: 11.0xx ae $ ,


172

!#: 004.0xx 1e $ ,

+.;%1-'( PB'( 0C;< +# %;.= ?B- W)B Z6.

YT* PNB2 01;%$T' %Qd %;%=- L;.$-./:

! "

0002.0xx

xxx

004034.00001538.0002737.0 x

2e

112

1

- .

%$.

$

,

t

'(%)$:

%;%=-'( ;?B-'( 01;%$ +.;%1- PB Z6 YT* ,E1;.$- =C; Zax ) )< ,2= S(/?""NB,""5%;e @6 >4""B'( 0""1;%$. F"";T* ,2T( )""D. Z%"";%=-'( ;""?B-'( 0""1;%$ L"".$- H),""*/ 9

>4B'( 01;%$. +$Q @,O2 PB YT* P/NB'(.

H%"""&,.C'( L%"""$'( +"""#– ,"""? H%/=4"""C',= :,1.– :())"""* Z%"""A2 0""";$Q'( @O2"""'( P"""B'K/T$C'( PB'( Y'8 P/N/T' I,;TCD'( C :())BC . 6)".2 ,"22!# 0";%;%=-'( L%"$'( +# ,C6

PB Y'8 PN2 6 Y'8 0D.,--C I(%C %(%C-?,. F2;?B- P/,B2 @a 9 PBT' Y'/6 K;%1-.0;#,= 0<) Z4 .H%";.='( 0";$Q'( @O2"'( P"B )"2* 0";%;%=-'( L%"$'( ].-- H),*/ I(/4 Z6

7),DC'( C %;.='( ))D'( I)P";5,AC'(/( 0"',B'( 34"5 +"# Z)f"- H%"&,.C'( L%"$'( G 9,""5f(%A8 @-""; +""-'( I,"";TCD'( H%""a= K.""?. K"";%1-'( S,""$Q6 %,""&-2( Y""'8 . g;"".2 +""T; ,""C;#/

0;%;%=-'( L%$'( C ;-1;%$ 4;M2- 0;M;= b( /D..

/5 FTB K/T$C'( +$Q'( @,O2'( 6 J%M2: yAx $/5 :_;NM- %a=6 H%/N./:

nnnnnn

nn

nn

yxaxaxa

yxaxaxa

yxaxaxa

$%%%

$%%%

$%%%

2211

22222121

11212111

6 J%M2/: niaii ,,2,1;0 $/

) ,= (4 h(/0$iia @;< C 02;DC 0C;1'i 0.?,2C H%/N. I7),DC'( P;).- =C;#.(

(Iterative Methods) W¹d¹dJ²�« ‚dD�« ∫UO½UŁ


173

0";',-'( I,"<_D'( K"-=2 6 ,"22=C; 01.,"?'( I7),DC'( 0*/CAC C +"$D- +"-'( I(%;V-C'( @;<nxxx ,,, 21 ,E?M2 @;1'( 345 0'7).:

! "0 1! "0 1

! "0 1 22

3

22

4

5

%%% $

%%% $

%%% $

nnnnnnnnn

nn

nn

axaxaxayx

axaxaxayx

axaxaxayx

/

/

/

11,2211

22232312122

11131321211

(1)

H%N-QC H%/N. /6:

niaxayx ii

n

ij

jjijii ,,2,1;

1

$

&&&

'

(

)))

*

+

6 $

/$

(2)

I(%"";V-C'(/ I7),""DC'( K"";-%- )"";D2 ,""22!# H),""*/)P"";5,AC'( ( %)""< L""1B2 W"";B. 0"";%$1'( H%$;""?'( $%""& 0*,$-""?7((diagonal dominance) 0""C;1'( 6 +""2D; Z4""'(/ 9

Z%$< %N2* ZG 01T$C'(iia C %.=6 /=-– Z/,?- /6– 0"1T$C'( @;"1'( i/"CAC FMN +# j%QG( %N,2DT') @<% >N'(i(02;,.-C',. ['4 * %.D2/ 9:

niaan

ij

jijii ,,2,1;

1

$67

/$

)* ( +,-."/ '0123 [Jacobi's (J) Method]

)& '0123 -*'4,"55$& 6",1205( (Successive Approximations Method)

P"";5,ACT' 0"";'/6 @;""< ;""CQ-. 6)"".2 0""1;%$'( 34""5 +""#nxxx ,,, 21 9 ,""226 J%""M2/k/C%',. @;1'( 345 Y'8 %;&2:

)0()0(2

)0(1

)0( ,,,x nxxx 8

I,""<_D'( 0""*/CAC +""# Y""2C;'( >(%""$G( +""# @;""1'( 34""5 J/""D2 @""a(1)""C P""NB2' 9 A I,".;%1- Y"T* j%"?;'( >(%$G(P";5,ACT' H)";) @;"1'( /6 I,".;%1-'( 34"5 Y"'8 %;"&2/ 9

k/C%',. H);)A'(: )1()1(2

)1(1

)1( ,,,x nxxx 8

@;1' H);)A'( I,.;%1-'( 345 Jg/D2nxxx ,,, 21 0"*/CAC +"# Y"2C;'( >(%"$G( +"# I,<_D'((1)6 I,.;%1- YT* j%?;'( >(%$G( C PNB2' 9))A:

)2()2(2

)2(1

)2( ,,,x nxxx 8


174

P""X#6 I,"".;%1- Y""T* P""NB2' 0"";TCD'( 34""5 %%""=2 @""a)3(x 34""5 L"";.$- +""# %C-""?2/ 9@;< +# %=4; %;V- )A/; 7 F26 j%2 6 Y'8 0;TCD'(nxxx ,,, 21 .

I,.;%1-'( C 0D.,--C K,?B. @/12 ,226 +# L.? ,C l;QT- ,22=C;/

,x,x,x )3()2()1(

0<_D'( @()Q-?,.:

! " )3(,,2,1;

1

)(

1 nixiia

n

ij

j

kjxijaiy

ki $$

6

/$

%

0./T$C'( 0<)'( Y'8 PN2 6 Y'8 @;< `.N- ,. 9x 0<)'( 345 )/)B +# 0-.,a :,.;%1-.

0;m()-.7( I,.;%1-T' 0.?2',./)0(x 0;%MN :,C;< 4Qf- H),* ,E2!#

0x )0( $

' ['4 ='/ :,;%/%X U;.

0<_D'( +#%$ C P= 0;,E2 4Q\. F26 OB_;/(3):

,= (48 xxlim )( $,.

k

k

!#x 0;TNG( 0'),DCT' :_B /=;Ax = y.

0"'n)oDC'( @;"1'( "C :,";6 @)Q-"?2 7 +./=,"A 01;%$ +# ,226 OB7 ['4=) H)";)A'( 02"""" n?BC'( /6improved """"X#6 :,"""".;%1- 0"""". n%1C'( /6P ( :_C,""""= :(%"""";%=- +""""E22 6 )""""D. 78

(complete iteration) I,"""".=%C P""""= K""""?B2 /6 )""""A/2 Y""""-B Z6 9(components)

FA-C'(x ,5))*/ 9n.

!"#$(11-3):

I7),"""DC'( 0"""*/CAC' P"""B ),"""A;c +./=,"""A 0"""1;%$ """C I(%""";%=- U"""CQ L""".$0;',-'( 0;$Q'(:

32

1642

1124

21

321

321

$%

$%%

$%%

xx

xxx

xxx

0"";),;-*7( %/""?='( @)Q-""?( K"";%1-'( S,""$Q6 K""2A-' I,.,""?B'( S(%""A8 )""2*) )""< +""-'(0;%&D'( %/?='( @()Q-?( C p-2-.(


175

!%& :

G 0""a',a'(/ 0"";2,a'( ;-'),""DC'( Pn)"".033 $a K"";-%-',. I7),""DC'( `.""N- ['4""./+',-'(:

1642

32

1124

321

21

321

$%%

$%

$%%

xxx

xx

xxx

0;',-'( I,<_D'( YT* PNB2 I7),DC'( 345 C/:

213

12

321

41

214

21

23

41

21

411

xxx

xx

xxx

$

%$

$

%,-Q2– :,;%,;-Q(– +m()-.7( FA-C'((starting vector) 0;'/G( I,.;%1-T' +',-'(:

! "

! "

! "

! "

&&&&&&&&

'

(

))))))))

*

+

$

&&&&&&&

'

(

)))))))

*

+

8

1

1

1

x

03

02

01

0

x

x

x

P/G( %;%=-'( +# 0;',-'( @;1'( YT* PNB2 ['4./: ! "

! "

! "

! " ! "tx

x

x

x

41322

413

41

214

221

23

241

21

411

1

13

12

11

$.

$ $

$%$

$ $

,2;$D; +2,a'( %;%=-'(/: ! "

! "

! "252

412

214

252

21

23

1615

413

412

21

411

23

22

21

$9 9 $

$9%$

$9 9 $

x

x

x


176

P""""aC',./UC,""""Q'( %"""";%=-'( Y""""-B %(%C-""""?7( ,""""22=C; . 34""""5 pm,""""-2 +""""T; ,""""C;#/I(%;%=-'( W;B 9k %;%=-'( @<% Y'8 kC%;.

k 1x 2x 3x

0 1 1 1

1 2 2 413

2 1615

25

25

3 87

3263

3293

4 128133

1631

128393

5 512519

256517

256767

2 P"""""B'( """""* >"""""T-Q- UC,"""""Q'( %""""";%=-'( +"""""# 0./"""""?BC'( I,""""".;%1-'( 6 O"""""B_ $/."""XC'(! "3,2,1 321 $$$ xxx )/)"""B +"""# :,""".;%1-q %P"""B'( (4"""5 """C . """=C;/

%a=6 I(%;%=- PCD. 0<)'( H),;k.

O""B7– j%""Q6 H%""C– P""C,= )"";)A F""A-C K,""?B K""A; 0""1;%$'( 34""5 +""# F""26 PBT'(whole new solution vector) .<+',-'( %;%=-'( +# @)Q-?; 6 P.

)7 ( !"# $%&'(– )*&+ [Gauss-Seidel (G-S) Method]

!!"#$% &'!!()% *+ ,-!!'-.)% /0!!1'2(3)% + 4!!5 063(10!!" !!7 8!!('29)% :;!!< =!!>3?3(improved values) – @'#!!!A)% B!!!#>) C2!!!D$%– 8!!!'2'2E3)% F'!!!A)01 81G!!!"#H)%G

IG0. 8('29 45 0J< &-?3"3– 10"# -2.H1 B-'K06 L!5 41GE0!. 8!('29 4!5 0!HJ'1 M 0<--7G /%2'N3H)% &'D O'H. C"#3 + -P1 QR &-?3"3 Q ,-'-.)% &'()%n.

IG0!. 8!('29 4!5 8!'2'2E3)% F'A)% BSH3 43)% /0DTP)% 8103E 0JJEH' U);1G–

8')03)% ,2GA)01 B-'K:

! !

! ! !22

32

11212

12

112

11

1

axaxayx

axayx

n

j

k

jjkk

n

j

k

jijk

"#

$%&

'())*

"#

$%&

'()*

*

++

*

+

) ,-'-.)% 8H'()% 0J< 0JH-?3"% 0JJ+ V#Q !11

+kx 8(10")% 8H'()% H WQ-1 !kx1.(


177

! ! !

334

3

2

1

1

331

3 axaxayxn

j

k

jjj

k

jjk

"#

$%&

'()()***

++

!"#$%& '$!( )*+ ,-$.$/.ix 01!12%&– 3( ! 4#%& 5+k+1 – '(67%-8 9:;#:

! ! ! !

ni

axaxayx ii

n

ij

k

jij

i

j

k

jijiki

,,2,1

4;1

1

1

11

!*

"#

$%&

'()()*

+*

)

*

++

<-==#;> -==>>? -==>@ A==;B– '==A;% C? 5==+– )!D==E#% CF G==H+ 1==;&. 9==! H#ix I ==!"#$ CF G==H+ 01==;&. '==$!( )!D==E#% C?ix J==%K. I-==L!M/ ->M==N; '==$!( O1==;? 5==@.

GH+ 01!12%& 3!H%& ,-$P&1 31E#:> ->>F.

!"#(12-3):

Q-R$ '%S:$ Q; 1/?T-T U.-2 'H! G )$ V& ! 4# 'R6R ,-$1E#:$-Q1!D.

$%&:

U.-===2 '===H! G% '===! ! 4#%& V-===(67%& 5===+ )===$!F& W ===G%& 5===+ ,-===$P&1 31E#===:>–

V& !"#$M% ->!1% 0 +.#$ 3!( O1;? Q1!Dix.

'!P&1#8B& 3!H%& KES>: ->>? X Y>: !

"""

#

$

%%%

&

'

*

1

1

1

x 0

->!G7! Q.F& ! 4#%&: !

!

!8

1925

412

214

252

21

23

241

21

411

13

12

11

*,),)*

*,+*

*))*

x

x

x

->!G7! 5>-R%& ! 4#%&.: !

!

!256783

64125

41

3229

214

64125

3229

21

23

3229

819

41

25

21

411

23

22

21

*,),)*

*,+*

*,),)*

x

x

x


178

O%-R%& ! 4#%& Z& 2[ ->>4$! QR$%-8. .5M! -$!+ \P-#>%& ]EM>.:

k 1x 2x 3x

0 1 1 1

1 2 25 8

19

2 3229

64125

256783

3 10241033

20484095

819224541

U.-==2 '==H! G 9 -==H# )? ^8-==:%& Q-==R$%&. Q-==R$%& &K==@ )==$ A==;6>-? Q1==!D _ ==:58.4-==2 '===H! G 9 -==H# )==$ ==!R48 ,-==$P&1 U!===% )==4%. 01-==/ `!;===N &K==@. I) ).===4# 1==(

a1;[ -H# _ :? )!#H! G%&a EF& )$ ,-8a EF& 1/-8##. -$@&1;[ 9 -H## 1(. I.(

V-==Y! 7#%& X==78 '==72& $% ,B.? <-==#;> '==! ! 4#%& ==G%& 9 -==H# V-==! A> '==:& 1%5GE%& 82%& 5+ '$.M7$%& \P-#>%&..

#%&' ()*&+%& (#),%& -)./*012)##%& (%3!/#%&' 4*&+%& 56*

(Eigen Value, Eigen Vector, and Characteristic Equation)

)? X ==Y>A '==78 $ '+.Y==N$nn, '==!H!H; 1&1==/? )==$ . )? X ==Y>.- 11==/) 5=====H!H;real C1=====H/ .?complex ( 3-=====A>%& )? G =====b%& ^=====H;!xx -*A 3-=====A>%& .?

! 0x *) IA !"#$ !##%& '##( )##*+, )#-x . ..##/-0 12 '3##4% !!"#$% !!&"' (eigen

value) 567"$,8-A 93#:;8- 5#%+0< 5#,%= 7> ?xx !A . '#( )#*+, @ '3#4% A-<#B7x );2 5%+0<-0 5,%4-0 C<D- 'E34, !"$ !%&(#$% )*#& (eigen vector) ) 7> 567"$,8-

93:;8- ('E34, 5%+0<-0 5,%48- .

5###-.3/,-0" # 0x !$ IA F3###G;3*+, F3###%HI F3###,3:; '###J,+(homogeneous) 1###,KL.3###/,-0 . )###63+ !###%& '###( 93###:;-0 0<###D- .###*7%7(nontrivial solution) – !###%& >

!"###$– 13###B 0<2 H###460!$ IA MN###%,,-0 5###-.3/,-0 O,###G+ 5###%!E* 5###-.3/, C<###P7(characteristic equation) 567"#$,8-A 5#*!.-0 1#, 5#-.3/, Q#P7 ?n Q#6 1#B,%7 ?5%-3+-0 M!7$-0 Q6 3D+E3+B:

Convergence of Iterative Methods W¹d¹dJ²�« ‚dD�« »—UIð


179

0

321

2232221

1131211

!

%%%%

&

'

((((

)

*

$

$

$

nnnnn

n

n

aaaa

aaaa

aaaa

!

!

> 0!+!n

5%!E*-0 5-.3/,-0 C "$,8- 5%+0<-0 9%4-05%-3+-0 56758E34,-0 5%+0<-0 K3D*+,-0 .*7>7 ?:

%&

'()

*

$!

14

12A

'(-0:

2,3

060

14

12

2

$!!,

!$$,!$

%&

'()

*

$$

$!$

IA

IA

)i ( 3,.;R3! :

21

21

21

2

1

044

0

0

0

314

132

xxxx

xx

x

x

!,-./

!$

!+$

%&

'()

*!%

&

'()

*%&

'()

*

$$

$

!E+/; 1> 3;;B,%6 %&

'()

*!%

&

'()

*

1

1

2

1

x

x

)ii ( 3,.;R2$! : " #

%&

'()

*!%

&

'()

*%&

'()

* +

!$

0

0

14

122

0x

2

1

x

x

IA

12

21

214

04

04xx

xx

xx$!,

-./

!+

!+

<IS; 1> 1B,% %&

'()

*

$!%

&

'()

*

4

1

2

1

x

x


180

.,#/01&:

T - 5###%+0<-0 5###,%4-0eigen value MN###U%,,-0 5###,%4-0 O,###G+ F3###;3%(>characteristic value ? Q+0<-0 )*+,-0 O,G% A-<B7eigen vector NU%,,-0 )*+,-3Echaracteristic vector.

V- 13B 0<2x < F3D*+, F3%+0 1W6 ?0x ) X%(0 ..R > ( F3Y%> Q+0< )*+,.

"2/3:

567"$,-0 K;3B 0<2A 5#%!H4-0 M!H%G-0 5%/H=(strictly diagonally dominant) >1> niaa

n

ij

jijii ,,2,1,

1

!!123

4!

567"###$,-0 1W###6A M.!###";, !###%& 17###B+(nonsingular). 567"###$,-0 K###;3B 0<2 A-<###B7 M!7.,-0tA 4-0 M!H%G-0 5%/H=5%!H1> > ?:

njaan

jii

ijjj ,,2,1,1

!!123

4!

567"$,-0 1W6A M.!";, !%& F3Y%> 17B+.

"2/3:

567"#$,-0 K#;3B 0<2A 5#/E!,-0" #nn5 .#%.(+-3E 5#E*7,(positive definite) ?M.!";, !%& 17B+ 3D;W6.

4"25# : (6"78$ 2798$ 4:3 (A) ;<6:&8A

(Spectral Radius of a Matrix A)

%= !EB> 7P567"$,8- 5%+0<-0 9%48- 548H, 5,1> > ?:

" # " #AA ini 6

77!

1max

X%(i 567"$,8- 5%+0< 5,%= QPA.

"2/3:

1> Z!";A 5/E!, 567"$,nn5 :" #nRLA8 ] X%(" #nRL [3#Y"-0 7P 5##%4%4(-0 K367"#$,8- Q#HI-0nn5 .[1> Z!#";7)(A6 567"##$,-0 !#H= \##$; 7#P

Q"%H-0 . ] _ ]6!3%/,. K367"$,8-5=a/-0 b4(++ ?:

" # AA 76


181

"2/3:

1> Z!####";" #nRLA8 ? 1> >A 5####/E!, 5####%4%4( 567"####$,nn5 . Z!####";71>

2. X].## c(d,-0 567"##$,-0 !3##%/, 7##P(induced matrix norm) 12 . !3##%/,-0 0<##P

5=a/-0 F3,e0. b4(%:

)(2

AAA t6!

"2/3:

1> Z!";A 58J3,+,7 5%4%4( 567"$, . 7"#$,8- 5#%+0<-0 9%4-0 56A 3#D8B 17#B+1%+;%3E+,-0 b4(+7 5%4%4( F0.0.R>:

nttt RA 8177 xxxxxxx maxmin

X%(min 7 5%+0< 5,%= !f$> QPmax 567"$,8- 5%+0< 5,%= !EB> QPA.

"2/3:

] 58J3,+,-0 5%4%4(-0 567"$,-0A .%.(+-3E 5E*7, [9:

] 567"$,8- 5%+0<-0 9%4-0 'BA F3/H= 5E*7, 5%4%4( .0.R>.[

matrix A: real, sym., pos. def. ii 139 0

/01&:

567"####$, >A 5#####/E!,nn5 Xa#####J g7#####,*, M!7#####$ Q#####6 h#####Y7+ 1> 1#####B,% 3P0.(2 K367"$,"D" 5%!H= i!I>7 ?"L" H4-0 3P!#$3;R 5%8"#G 5#%J8J, ?5%!"#$ 5#%!

5J-3J-07"U" 8J,5%!"$ 5%!H4-0 3P!$3;R 5%78R 5%J0<BP ?:

%%%

&

'

(((

)

*

;

nnn

n

aa

aa

A

!

!

1

111

A = L + D + U

%%%%%%

&

'

((((((

)

*

!

$ 0

0

0

0

1,21

3231

21 0

nnnn aaa

aa

a

L

!

"


182

%%%%%%

&

'

((((((

)

*

!

nna

a

a

a

D

"0

0

33

22

11

%%%%%%

&

'

((((((

)

*

!

$

0

0

0

0

,1

223

1312

0 nn

n

nn

a

aa

aaa

U

"

!

!

1%+67"#####$,-0 1> :#####(LL,U '#####%8(+ 1#####, 1%+*+3#####;-0 1%+67"#####$,-0 3#####,P 3+#####G%- 567"##$,-0A 3R O###-23###D%8, !3H###j;0 1##, 13###+*+3; 3###,P 3##,; k07 ?(splitting) 567"###$,-0A

[0N*> M.R O-2'J, i!I> M!7$E !3Hj;L0 0 1B,%7 ?: )()( ULDA $$$$!

7> )( UILDA <++<!

X%(

%%%%%

&

'

(((((

)

*

!

1

1

1

0

0

"I

%%%%%%

&

'

((((((

)

*

!<

$ 0

0

0

0

1,1

33

32

33

31

22

21 0

nna

nna

nnana

a

a

a

a

a

a

L

"

%%%%%%%%

&

'

((((((((

)

*

!<

$

0

0

0

,1

22

2

22

23

11

1

11

13

11

12

0nna

nna

ana

a

a

ana

a

a

a

a

U

"

"


183

####P .####/E 1l07 b!####H-0 m!3####4+ n!.####; oe3####+;-07 K3####"%!/+-0 Z/E####- 5####/*0!,-0 C< 5%!%!B+-0 .. 93#:;-0 '#%7(+ 5%8,R O8R ',+j+ b!H-0 C<P 9:/,7Ax = y 93#:; O#-2

M!7$-0 Q6 p63B,:

cxx +! T

X%(T 5/E!, 567"$,nn5 !#%!B+-0 567"#$, O,G+7(iteration matrix) ?7c , )##*+ .. Qe0.##+EL0 )##*+,-0 !3##%+I0 .##/E7" #0x vector)(initial Q##E%!4+-0 '##(-0 >

b%EH+E A-<7 ? 5/E3++,-0 5%E%!4+-0 '78(-0 K3D*+, 1, 58G8G .%-7+E 974; ? Qe0.+EL05=a/-0:

" # " # (*)cxx 1 +!+ kk T

9%48-: k = 0, 1, 2, ...

)4;-0 5###4%!H7 5###4%!H-0 C<###P 1%###E )E3###j+-0 :###(L Q###6 3P3;###G!. Q###+-0 5###+E3J-0 5###HbE3G-0 '$"-0.(

5###=a/-0 C<###P b###%EH+- )###*3+(; <###-0 K###=7-0 1> 5###%8,/-0 5###%(3;-0 1###, :###(a%7 M!###j3E,-0 b!###H-0 )###*3+(+ <###-0 K###=7-0 b7###"% 5###%63B 5###=. O###-2 '7###$7-0 O###+( F3###%!0!B+M!%f##$-0 .3##/E@0 K07< > M!%f##$-0 5##%HI-0 9:;##8- 5E##G;-3E n73##*- \<##(-0 5##4%!HB .

3##,> C<##P 1W##6 ? 5%!"##$-0 !##$3;/-0 1##, 5##%-3R 5E##G; 3##D%6 Q##+-07 M!##%EB-0 9:;##8- 5E##G;-3EK#=7-07 1%NI+-0 K3E8H+,- 5EG;-3E 5%-3R M[3"B K07< 17B+ 5%!%!B+-0 b!H-0 . 5#G0!.-7

54E3G-0 5=a/-0 m!34+ 5G0!.- q3+(; b!H-0 C FL7> K##EJ; 5##=a/-0 n73##* 5##4%!H7 QE7B3##* 5##4%!H 1##,– h##Y7 1##B,% '.##%N

M!7$-0 Q6 3D;7;3= (*)5,3/-0 M!7$-0 C > ?.

=><?: !"#$% &'()$% *+ *,(-". /0'1 2(3"4 56(:(*)

0$"7$% &'()$% *+ /0'1$% 89: 2(3"4 56( 2-!0:


184

! !

,2,1

;,,2,1

;1

1

"

"

# $%"

&"

$

k

ni

xxn

ij

jiiaiyk

jiia

ijaki

,2,1;,,2,1 "" kni

!"#$% &'()$% 5,70 ;3< =<(*)>0? @

!UL

T

nna

nna

nnana

nnana

ana

a

a

a

a

ana

a

a

a

a

'$'%"

((((((((

)

*

++++++++

,

-

%"

%0

0

0

1,21

22

2

22

23

22

21

11

1

11

13

11

12

*: *,(-". /0'1$ '0'-7$% +(A)! 2< =<:

)( ULT j '$'%"

(((((

)

*

+++++

,

-

((((((

)

*

++++++

,

-

"

(((((

)

*

+++++

,

-

"

n

nna

a

a

nnany

a

y

a

y

y

y

y

c!!

2

1

1

22

1

11

1

22

2

11

1

0

0

! ! !

! !yc

yxx

y

1

1

11

1

%

%

%$

%

"

$%"'$'%".

$'$'%".

"

D

ULDULT

DUL

D

j

kk

*B0 "!- .073$% 89: C$D E()($% "60< "33-!0(:

yx)( x

yx)(x

yx)(

yx

11 %% $$%"

$$%"

"$$

"

DULD

ULD

UDL

A

: 2F G737( /0'1$ 0$"7$% 0+(A)!$% H0)$% 4I#$% 89J 0'0'-7$%:


185

! !

yc

)()(

yx)(x

1

1

111

%

%

%%$

"

'$'%"$%".

$$%"

D

ULULDT

DULD

j

kk

!"#!$: J(". /0'1 2(3"4 56(– !"#$% &'()$% *+ EK0L(*):

0$"7$% &'()$% *+ 2(3"/$% %9: 56( 2-!0:

! ! !

,2,1

;,,2,1

)1...(;1

1 1

11

"

"

# # $%%"%

" $"

$$

k

ni

xxxi

j

n

ijiiaiyk

jiia

ijakjiia

ijaki

2< =<: ! ! !

! ! !

! ! ! ! yxx

yxx

yxxx

1111

11

111

%%%$

%$

%$$

'$$'$%".

$'%"'$.

$'%'%"

DLIULI

DULI

DUL

kk

kk

kkk

$% 5,70 2(3"/$% 2< =< !"#$% &'()(*) >0?:

! ! yc 11

1

%%

%%%

'$"

//'$%"

DLI

TTULIT SeidelGaussSG

2(3"MMMM4 5MMMM6( "MMMM33-!0 "MMMM!-G-S MMMM!"#$% &'(MMMM)$% *MMMM+(*) *MMMMB0 "MMMM!- N'MMMMO< H0MMMM), :

/,"P$% 4I#$% *+'1 Q'6,(1) 2(3"/$G-S *+iia R,)7:

! ! !

,2,1;,,2,1

1

1

1

11

""

# $%#%"$"

%

"

$$

kni

yxaxaxan

iji

kjij

i

j

kjij

kiii

2< =<: ! ! !

! ! !

! ! ! ! !

! yc

yxx

yxx

yxxx

1

1

111

1

11

%

%%

%%$

$

$$

$"

$%".

$$$%"

$%"$

$%%"

LD

ULDT

LDULD

ULD

ULD

SG

kk

kk

kkk

"/7 J'K3 2S%( !"#$% &'()$% "T$ *7$% 0'0'-7$% U'1$% Q'(*).


186

&'()$% *+ 0'0'-7 H0) Q'"/7$ *+"-$%( ='('6$% 1'V$%: ! !

0c;,2,1

;cx.x 1

&"

$"$

k

T kk

K0?($% E?$% C$D: x = Tx + c

*W%K7,% Q0'/7 =X, 20WK7,! ! nR00x (: @

! 11T2

%!&'()* +',-: ! !

! ! !

! ! cccx.

cc.x

ccx.

cx.x

22

12

1

1

$$$"

$$"

$$"

$"

%

%

%

$

TTT

TT

TT

T

k

k

k

kk

! ! !

! !

! !cTTTIT

TTIT

T.TT

TTT

kk

k

k

k

$$$$$"

"

$$$"

$$$"

$$$"

$

%

%

%

201

223

223

22

x

..........

cx

cc.cx

cccx.

1'V$% QB170 &K"F Q'"/7$%(:

0lim "3.

k

kT (1)

2(-0 *- *+"-$%( ='('6$% 1'V$% Y"60< (: %9: 2-$(:

! ! 12lim%

3.%"$$$$ TITTTI k

k

*MM+ '(-9MM!$% 1'MMV$% UMM/?7 %9ZMM+ %9MM-:((1) MM0"T3$% 2ZMM+ @ !kk

xlim3.

"MMT$ 2(MM-0

K(.() =("P7(x YI[!( 4I#$% CBF E)?3( @:

»—UI²K� WOÝUÝ_« W¹dEM�«


187

! !

cxx

cx.x

cx

c0x1

$"

"%

"%.

%$" %

T

T

TI

TI

;.7!$% 2< =<x YI? E[!0.. M4I#$% CBF K!7#0 E? C$D Q'"/7$% 2Z+ *$"7$",((1) @(: 4I#$% 89: U/?7$ *+"-$%( ='('6$% 1'V$% 2< \(B#!(:

ii 41 ;15

*$"7$",(: (T) < 1

./012:

]3"- "!B- 'P< /0'1$% Q'"/7 2"- @'H)<.

3!$2(14-3):

*1O$% \"_3$% E?$ *,(-". /0'1 \KO7P3P "33< 'A3

0;yx1

4&"(

)

*+,

- %6

66

;MM//?7 2< QMM.0 =9MM$% 1'MMV$% KMM.(< /,"MMP$% MM0'_3$% \%KO7MMP",6 89MM: Q'"MM/77 CMM7? @ *W%K7,% ;.7! =a /0'1$% !0x.

30)*: (: /0'1$% Q'"/7 1'V:

!

(()

*

++,

-

%"

1

0

0

1

1

4

6

6

2

j

j

T

T

*: &L0!!$% $K"#!$%:

!6

66

2

55

2

22

42 0

"

7"."$

iT

i

Q'"/7$% 1'V 2(-0 b$9,(

.8.8.

1

21

1

2

2

66

6

2%16 (<286


188

% 'MM1/$% cMM)3 Q"MMP? Y"MM!(!F ETMMP$% 2MM! J0MM$ *MMA01$2 ]"+(AMM)!B$ ,MMP3$", &'MM0,-$% !39n MM0+"- Y"1('MMV 2(MM-7 >MM0?, "MM:'",7O% ETMMP0 1('MMV ]MM[ d?e, KMM/+ %9MMT$( @

(sufficient conditions) MMM0'0'-7$% U'MMM1$% 89MMM: Q'"MMM/7 "MMMT3! =< 2!MMM60 . 89MMM: 2MMM! ('V$% +(A)!$% '"0#! Q"P?, UB#70 "! 13!( @!B$ 0'1/$% &'10P$% 1'V "T +(A) @

70 "! "T3!(K0K?7$", ,.(! +(A)!$% 2(- QB1 0$"7$% ]"0'_3$% b$9 R6(7 "!- @.

."'/#: 0'0'-7$% H0)$%:

! ! ,2,1;cx.x 1 "$"$ kT kk

K0?($% E?$% C$D Q'"/77 x = T.x + c

*W%K7,% E? =X, 20WK7,!nR0)0(x1'V$% U/?7 %9D @:

11T

0? >T '0'-7$% +(A)!$ '"0#! =< (:T.

%!&'()*:

Y%'MMMM_3 ;MMMM3a– /,"MMMMP MMMM0'_3 2MMMM!– MMMM4I#$% UMMMM/?77 ]"+(AMMMM)!B$ '"MMMM0#! =a 0$"7$%: ! TT :2

1'V$% U/?7 %9Z+ 11T

2Z+ ! 11T2

'"/77 b$9,(Q'"/7B$ 0P"Pa% 0'_3$% CBF Yf"3, 0'0'-7$% /0'1$% Q.

E"[! *+ YI[!g-hi:

64"

3jT

1'V$% U/?7 %9D: 14 16

=<: 14

86

=<: 486 (<4%16

77 *,(-". /0'1 2Z+*W%K7,% *,0'/7 E? =X, 20WK7,! Q'"/.

X1OB$ K(K? f"1FD 5! /,"P$% 0'_3$% E!V7 0$"7$% 0'_3$%(error bounds).


189

."'/#: +(A)! '"0#! =a *$"7$% 1'V$% U/?7 %9D 11/ T;

]"T.7!$% #,"77! 2Z+ ! ,2,1,0;cxx 1 "$"$ kT kk

Q'"MM/77– *W%KMM7,% ;MM.7! =a ! nR00x - ;MM.7!$% CMM$DnR0x UMM/?77 ;MM3< "MM!- @ 0$"7$% X1O$% K(K?:

! !

! ! !

! ! !kkk

kk

kk

xxxx

xx.xx

;xx.xx

1

1

1

01

1

0

%:%

%:%

%:%

$%

$

%

;;

;;

;

MM/0'1B$ MMB["!! MM/0'1( /,"MMP$% MM0'_3$% \%KO7MMP", MM0'_3$% 89MM: MM3:', 2MM-!0 MM/0'1, MM$K"#! '9MM. Q"MMP? *MM+ XMM1O$% '0KMM/7 *MM+ U,"MMP$% EMM)A$% *MM+ ]MM#,7% *MM7$%

3$% MM7,"[$% MM1/) MM/B1!$% MM!0/$", '"MM0#!$% E%K,7MMP% 5MM! .( 20MM, 'MM0,-$% ;,"MMV7$% _MM?I0( /,"MP$% &'M0Oa% M30",7!$% 20,( 7,"[$% 1/3$% /0'1 *+ X1O$% 30",7! . M30",7!$% 89M:(

1'V$% U/?70 20? '0'-7$% c"/0D 2! "33-!7 &'0Oa%. ! ! <:%$ kk xx 1

>0?< !$% R!"P7$% E[!0;, j(!P . c"M/0k *$"M7$% 1'MV$% \%KO7MP% "3e3d- l!e0 "!-'0'-7$%:

! !

! <:$

%$

:: 1

1

1max

kix

kix

kix

ni

) 1'V, !0x

1ki &$.(

./012)1(:

+(A)!$% K".0D ETP$% 2!T *,(-". /0'1$ ,P3$",)K".0D =<jT U,P "!-"MMMTA0'#7(!$% K"MMM.0D Y"MMM!(!F Q#MMM)0 2MMM-$( @ +(AMMM)SGT % +(AMMM)!$% *MMM:(T MMM/0'1$ J(".– EK0L . M$mK, 207/0'M1$% 2M! EM- Q'"M/7$ "M0+"- Y"1'V *1#7 0$"7$% 0'_3$%(

0B)a% +(A)!$%A ) \"_3$% E?$yx "A ( '0'-7$% +(A)! $mK, J0$(T.


190

R00x.

.444/012: 2(MMM-7 MMM0'1/$% &'10MMMP$% 1'MMMV UMMM/?7 +(AMMM)! =< 2< ]"MMM,[D 2MMM-!0K0K?7$", ,.(!.

.4"'/#: +(AM)!$% ]M3"- %9DA *M1O$% \"M_3$% *M+Ax = y !KM0K?7$", M,.( @ /0'1 2Z+G-S *W%K7,% ;.7! =< '"07O% 5! Q'"/77 0'0'-7$%.

3!$2(15-3):

) <( /0'1 2! YI- 2< 2n0,J /0'1(G-S *1O$% \"_3$% E? K3F Q'"/77

6 4

8310

5 2 8

321

321

321

"$%

%"$$%

"%$

xxx

xxx

xxx

K0?($% E?$% C$Dx *W%K7,% *,0'/7 E? =a ! nR00x.

) Q( 1 U,13MMMMP "MMMM33< 'MMMMA3 MMMM/0'J \"MMMM_3$% %9MMMM: EMMMM?$ *MMMM,0'/7$% EMMMM?$", 20WKMMMM7,! @ ! !t000x 0 " . 'MMMM0'-7$% ]%'MMMM! KKMMMMF 'nKMMMM4(k) CMMMMBF E(MMMM)?B$ MMMM!LI$%

2% =< @ 0'VF \"4'< 7P$ Y"?0?) E?$%: ! 6105.0xx %=:% k

30)*:

) <( ]I!"#!$% +(A)!: (((

)

*

+++

,

-

%

%

%

"

411

3101

128

A

MM0'1/$% &'10MMP$% MM0#14 +(AMM)! @ 2MM! YIMM- 2ZMM+ /,"MMP$% MM0'_3$% CMMBF f"MM3,( /0'1J /0'1(G-S *W%K7,% *,0'/7 E? =< '"07O% 5! Q'"/77.

) Q( /0'1, E?B$J 0$"7$% &'()$% *+ ]mK"#!$% 563:

!"#$)2:(

!"!#$ % !&'#( )*+, -#,-*'. / -#+ !!#0 -,!#1' 2#3 45165#( 7-"*8#+. 9#:&.(k)

;##<,=+ ;##>! ?##.@ A"##B"C. ;##6"C:+. +. )##+ D##E', 5##+* F A##,C> !##=6 G-*8<##( H8##. A5##I) A5I+15-3.(

%%%&'!(: ;3"J###B+. /###11$ 8@A 5K-"!###+ "LtA ;###,=:1. ;###,-:1. 7-:,###(. :-###M

(strictly diagonally dominant) N###3 ;###1,-: )###+ 4O###* )J ;###1,-:"G-S %5###P<. A###$.yx A !,$". A$. ?.@ Q-51''x 2R !'6 26,-1' A$ HS! " n


191

! ! ! 4/6

10/38

5/25

213

312

321

xxx

xxx

xxx

"#$

#"#$

"#$

!"#$!%&: ! !t000x 0 $

'(): ! ! ! ! ! ! ! 5.1xx

46

108

85xx

46

108

85x

01

01

1

$#

#$#

#$

%

t

t

'* +, -.&: ! ! !01

1xx.xx #&# #'

' kk

/0"1 !.2%3 ,4()J 556 /" " 7#8,k 9%1" :#; <, =8, '=

! ! (*)105.0xx 601

1

## (&#'

' k

'> ?#."): ! 6105.0xx #(&# k

@AA%" 2#8, BAA;8, '&AA7" C> !kx /" AAD6 E!AAF > /#AAG8 +!;";AA1 HAA"8I !.JAA1& C4AA8,.

/."!%#=8, '78&(*) – !"K=8, 4$L. M"; @N!O.P8,– @.K#:

! )(*105.05.1. 6

1)(& #

#'' k

M"; %$ jT'

****

+

,

----

.

/

#

#

#

$

041

41

1030

101

81

820

jT

! !

2351653.22

105.05.15.01

5.0)(*

21

21,

104,

83max

6

$010

(&(#0)

$$$

#

%

kk

T

k

j'


192

!

xx

xx

! "# $!%& '()* +,%-.% /,0,()1* 2*3450 6721, xx *.89(,06 ! "#$%1,91* :3;1*:

" !"#$ %&'( )*+,-.

%&'( )*+,-– ./0,&(.

12&3'( )*+,-)4 )+,56$/ 7+*$'2 '8092: ;/,+,39.(

%&'( )*+,-– <0+=)4 *$'2 '8092: ;/,+,39+)+,56$/ 7.(

( )*+,-2 >$!3 0(&? %&'– .: @3 ./0,&(: )(det,1 AA

;A0'B:$/ )C&:(: <" 0(&? )+$'9$/ D: %&'($ !"$/ )*+,- 7/0E9F'2 G'-E? H? IJ(9$ >$!& K )*+L0$/ )+0'+9CA/ )+,F3$/ )M+6$'2 ;'2'F"$/ )N'3 G/,(O

I+,*9$/ )(+9J.

051

41

31

041

31

21

131

21

321

321

321

!""

!""

!""

xxx

xxx

xxx

-

,52 )*2'F$/ )$PF:$/ <" 0C? 1$Q IF'" 'R9'2'F" G/,(S2 7&*+F H!$/ .? TU2'F$/ <"$/ D: .,'L& K;L& H? 1N -*N .++&JB: .+:L,2 V'59"A/ D+-9F+ . W/,VJ&

(1-3) s¹d9

(2-3) s¹d9


(3-3) s¹d9

;'N&56: X,&6 1N )+$'9$/ )+JY/ ;A0'B:$/ )C&:(: I93/yx !A:

12 4

53

21

21

!

Y�U¦�« qBH�« �UM¹d9


193

Z![ .: 0&6*:$'N K 7'L,\/ .: 0&0": 00B2 V'59"A/ D+-9F+ IF'" H? .\ ] &9 .? )$PF:$/– ': 0" _$O– ;'2'F"$/ Z![ <`: 1N a0"+ ':.

-!

1#+ ':+N.+9+JQ .+9+-E .+9$0'B: .: )J&3: )C&:(::

031

21

121

21

21

!"

!"

xx

xx

)?( )i( )+N&56:$/ X,&6$/ 1N )C&:(:$/ DÂx = y.

)ii( )*+,-2 )C&:(:$/ <"G-J ;'2'F"$/ )N'3 G/,(O D: K !"#$I+,*9$/ G'-E? IJ(9$ )-&2^:$/ )+,F3$/ )M+6$'2.

)iii( .+:L,2 V'59"A/ D: )C&:(:$/ <" 0C? 00C H? 1N -*N .++&JB:<"$/ ;/&-E )N'3 <@E. U2'F$/ <"$/ D: .,'L P-E$/ IF"/& K

.: <3 1N 12FJ$/21, xx.

)I( )i( ;@:'B:$/ )N&56:$ )+9/!$/ 7+*$/ 0(&?A.

)ii( )*+,- H0b9 .? .&:^:$/ .: <[G-S )C&:(:$/ <" _$O )+,+,39$/ c)*2'F$/)$O I,'*99F <[ H?c)+8/092/ 7+L HP2 .+8092: <"$/ _.(

)iii( .+9:+*$'2 W'8092:: # $ # $5,5 0

20

1 ! ! xx

-2/,+,39 )B2,? U; )*+,- .:G-S d,9 /!': ,VJ/& K:

cA 7? -&2^:$/ <"$/ &"J )B2'99:$/ 7+*$/ I,'*99 <[

)2'(e/ ;J'3 . f/& : .+9+8'RJ$/ .+9:+*$/ .: <3 1N P-E$/ IF"'N K7BJ .+,+M9:#$21, xxg8'9J .: h+#C <6"9 ': _#C U#C& K.

-"

;A0'B:$/ )C&:(: <"$ 70E9F9 19$/ %&'($ !"$/ )*+,- .? 7&#B:$/ .: .&39 .? <"$/ ;/&-E G'J`? 1C&, /!O U0? g8'9J 7&:B$/ _#C 1-B9 )+JY/ )+-E$/

(4-3) s¹d9

(5-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

194

,9 U+,- .C >$!& K.3:+ ': ,23? H=39,:$/ ,6JB#$ )*#-:$/ ):+*$/ ;A0'B:$/ I+9;/,+M9:$/& . _$O <&6&#$ ;A0'B:$/ I+9,9 )+:[? .'+2 )$PF:$/ Z![ .: 0&6*:$/

g8'9J$/ 1N )L0$/ Z![.

)+-E$/ ;A0'B:$/ .: )+$'9$/ )C&:(:$/ <" I&#-:$/ .? T,5J: 0.50 x + 1.1 y + 3.1 z = 6.0

2.0 x + 4.5 y + 0.36 z = 0.020

5.0 x + 0.96 y + 6.5 z = 0.96

.+:L, i7[\ X'-B: ;@:'B:$/ a+" . g8'9J .? T,5J I+,*9$/ G'-E? <+#*9$&7'L,? )`@` 7[\ I939F ;@:'B: H?& )+$'9$/ ;'2'F"$/.

)?( ,++M9 H? .&0 X'-B:$/ ;A0'B:$/ )C&:(: <" !"$/ )*+,- 7/0E9F'2 0(&?'R2+9,9 1N.

)I( (: <" !"$/ U+,- 7/0E9F'2 d,E? X,: 0(&? D: .3$& ;A0'B:$/ )C&: .3:+ ': ,23? H=39,:$/ ,6JB#$ )*#-:$/ ):+*$/ .&39$ 'R2+9,9 ,++M9) W'8092:

)`$'`$/& _$&\/ .+9$0'B:$/ <+0292.(

)j( .+*2'F$/ .+#"$/ .+2 .,'L.

-#

)+JY/ )+-E$/ ;A0'B:$/ .: )+$'9$/ )C&:(:$/ <" I&#-:$/ .? T,5J:

1

22

2 10 13

!""

! "

!""

zyx

zyx

zyx

& &*+ A IF'"$/ .? .: ,`3? .+=E92 7kl )+,mC 7'L,? <:'B: .? _JB:2 Kx

W/,56 Z,29B+F _$&\/ )$0'B:$/ 1N.

) ?( <nnnB( 0nnnB2 X'nnn-B:$/ ;A0'nnnB:$/ )nnnC&:(:$ ]+"nnn6$/ <nnn"$/ )nnn*+,- HPnnn2 0nnn(&? <:'B:x W/,56 _$&\/ )$0'B:$/ 1N.

) I( 3 X'nn-B:$/ ;A0'nnB:$/ )nnC&:(: <nn" %&'nn($ !nn"$/ )nn*+,- 7/0E9nnF'2 0nn(&? 'nn: In+,*9$/ D: .3$& K <"$/ G'J`? ;A0'B:#$ <+029 H? .&0& ,++M9 H? .&0 1[

(6-3) s¹d9


195

)rounding( D:($'nnnnn3 )+2'nnnnnF"$/ ;'nnnnn+#:B$/ G/,nnnnn(O 0nnnnnJC <nnnnn"$/ ;/&nnnnn-E G'nnnnnJ`? W/0( X,+23$/ 7'L,\/ IJ'(2 X,+M6$/ 7'L,\/ <':[S2 >$!& o,-$/&.

) j( :$/ ;A0'B:$/ )C&:(: <" !"$/ )*+,- 7/0E9F'2 d,E? X,: 0(&? 'n:3 X'-B )nn*#-:$/ )nn:+*$/ .&nn39$ ;A0'nnB:$/ Inn+9,9 <+0nn29 Dnn: .nn3$& ,nn++M9 H? .&0 1nn[

.nn3:+ 'nn: ,nn23? H=nn39,:$/ ,nn6JB#$)nn292 W'80nn92:)nn+J'`$/& _nn$&\/ .+9$0'nnB:$/ <+0( KI+,*9$/ D: >$!3&.

) 0( .: <3 1N <"$/ .,'L)I ( K)j ( 1N <"$/ D:)?.(

-$

%&'( )*+,- .: ;/&-E a@` U2-– "$ <0+=.+9+JY/ .+9$0'B:$/ <:

1925

64

21

21

!

!"

xx

xx

.+,+M9:$/ .: <3 ):+L 1N P-E$/ IF"/ 7`21, xx . .+9+8/092A/ .+9:+*$/ !E0,0 21 !! xx.

-%

)+JY/ ;A0'B:$/ .: )C&:(: 1#+ ':+N:

13 10

20 10

952

321

321

321

!""

! "

!""

xxx

xxx

xxx

)*+,- 7/0E9F'2G-S Z![ <" IF"/ )M+6$/ U+2-92 >$!& )C&:(:$/ _$O <69 )L0 0&0" 1N )+#:B$/ I,'*99 .? _$O W'+,/,390.1 . ):+*$/ .? T,N/

.: <3$ )+8/092A/32 , xx W/,56 H&'F9.

(7-3) s¹d9

(8-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

196

-&:

)*+,- 7/0E9F'2 )+$'9$/ )+JY/ )+-E$/ ;A0'B:$/ )C&:(: <" I&#-:$/ %&'(– <0+=.

0 2

0 2

02

1 2

432

321

43

21

!"

!"

!

!"

xxx

xxx

xx

xx

I2F$/ .'+2 D: ;A0'B:$/ I+9,9 WA&? 0CPN X,&,^ >'J[ ;J'3 /!O . U2- 7`)+2+,*9$/ 7+*$'2 W'8092: X,&3!:$/ )*+,-$/ .: .+,+,390!ix 7+L D+:($i.

-'(:

.? T,5J:

!A

a+"#

# $ AA :det.

X0:C? &? &56 H? <+0292 7*9 A . )+,F3$/ )M+6$'2 ;'2'F"$/ ,(? )+0'+9CA/)mB$/ A)+, (I+,*9$/ )(+9J G'-E? H? IJ(99$ <6"9$ 1$'9$'2& K

)-&2^:$/ <&#"$/ _#C.

(9-3) s¹d9

(10-3) s¹d9

!

"

###

$

%

&

!

"

###

$

%

&

0

0

1

y,

1

5

1

4

1

3

1

4

1

3

1

2

1

3

1

2

1

A

!

"

###

$

%

&

!

"

###

$

%

&

!

"

###

$

%

'

'

&

0

1

0

v,

1

0

0

y,

10.0

12.0

03.0

z

) ( !" #$%& '()&* +,-./*#0 1 2– 3$%,4 5*3/#%:

-1 +,!0&* 673$8-&* +9#-%- :)yx

(txxx 321x &

-2 +;#<8-&* +=#>?-&*1'A.

-3 +=#>?-&* @33)-


197

) ( !"#$%&'( !)#*+&'( ,(-./"011 A !2'0/'( !23.'( 405#&6&'( 7#89 -6#::

.vx,ux,zx !!! AAA

-!!:

!2;<( 4=-0%&'( >& >2/2'0/'( >2/5#&6&'( >& 7$':

) :( ?) !5#&6&'( /$( !2)#*+&'( !@2+'(yx !A.

) ( A2B ,: C-A*;& !5#&6&'( 4;0$ (DE 0& -F-9 G#06' HD9'( !I2A3 ,(-./"01C-A*;&.

) J( G#06' HD9'( !I2A31 -6#: C-A*;& A2B !5#&6&'( 4;0$ (DE : 79'(x K 4L&0%&'( !)#*+& C--9&#A M:det (A).

)1( 6 4 2

3 2

0

31

321

321

!

! "

!""

xx

xxx

xxx

)2( 42

4422

20224

321

321

321

!"

! "

!

xxx

xxx

xxx

-!":

!23.'( A2B !2;<( 4=-0%&'( >& !5#&6& ?82 0&2):

4.21.040

0.11.03.0

1.005.01.0

2

2

22

!""

! "

!"

xyy.z

xzxy

zyx

) :( !5#&6& 79' 4(A& NLO !%10//&'( 4012AI/8' !2A2A$/'( !I2A3'( !@2+ P13!2Q(-/1=( !212AI/'( ,2I'01 R0Q-/1& !I10"'( 4=-0%&'(:

x = 0, y = 0, z = 0

$&2 7S ,$;– 0T285 ,/8+9 ?/'( UQ0/;'( V85 (RW0;1– 3#1X&'( 79'( >2&./(exact solution) Y4=-0%&'( >& !5#&6&'( ZDT'

(11-3) s¹d9

(12-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

198

) ( G#06 !I2A3 ,2&%/ >$&&'( >&– !5#&6&'( 79 ?) ,-./"/' !2A2A$/'( 7-2[!23.'( A2B 4=-0%&'( >& !I10"'(.

2AI/'( ,2I'01 R0Q-/1& !I2A3'( ZDS >& >2A2A$/ P13!I10"'( !2Q(-/1=( !21 0T"*;.

3#1X&'( 79'( >2&./ ,$;$&2 7S \A.: CA&.

?) !I10"'( !I2A3'(# !I2A3'( ZDS >21 >A0]):.(

-! :

) :( !Î2A3' !^2A2A$/'( !@2+'( >& 4(A2A$/ !OLO P13G-S !^ _&&_%a&'(] W[^6'( A^b;() (!I10^"'( !'c^"&'( >^& [!^2'0/'( !^5#&6&'( 7^9' <( 4=-0^%&'( >^& A^2B !^2;

!23.'(!2Q(-/1=( 4012AI/'01 R0Q-/1& K :

x = 0, y = 0, z = 0

) ( 4^;0$ (D d(# Y0T285 ,/8+9 ?/'( UQ0/;'( >& 3#1X&'( 79'( >2&./ ,$;$&2 7S!^106e( : Y?;0Ô'( A^2A$/'( UQ0^/; ?^) !^]-'( !^6A- ?^S 0^&) K ,^%;) --^5 ,^$ M:

,2^] ?^) !929^+'( !2A^f%'( ,0^]Ag( x, y, z A^2A$/'( -^%1 0^T285 ,/8^+9 ?^/'(Y?;0O'(.(

- !:

!"#$%&' !"&(!)*&': !

"#$

%&'

!

"#$

%

0

01.0x

5.01

01.101.2

$+,-*&' !*./0( )tx 21 &'.

)1 ( 2#+3)* 4!561 2787& 9!,!:0&' ;'6<= >* ?@0&' 2A#6$ 9A,$ '@= B31 9,7'C&= DE" F%G" #"&(!)*&' HI J $AI:

!

"#$

% &'

!

"#$

%

& 00498.0

01.0x

00298.00

01.101.2

+K L+:0*&' M0&' HI N&!"&!,+: ( )tx 67.1834.0 &'

)L ( L+:0*&' N,#6A"&' M0&' '@K #:0"& 2#6#6O"&' 2#PQ"&' 2A#6$ 4(%":'.

(13-3) s¹d9

(14-3) s¹d9


199

" - #:

) 1( 2#&!"&' 9!I+PQ*&' * MO& 2/,!A*&' 2#"'@&' 9!.<"*&'+ 2#"'@&' 4#A&' L:0':

!

"#$

%'

!

"#$

%

&

&'

0

0

21

12

2

1

2

1

BA

!

"

###

$

%

'

!

"

###

$

%&

'

300

021

012

700

430

021

DC

) L( 2A,!:&' 9!I+PQ*&' * MO& NP#$&' 6$A&' ?Q3 2*#5 (<+1.

) R( S2#,6!A" 2A,!:&' 9!I+PQ*&' T1)6#6O" 9!I+PQ* 6,")G" #0 T1.(

" - $:

#:0"&'+ U+!<& ?@0&' 2A#6$ 4'(%":!, N&!"&' N$%&' 4!V3&' M0 (<+1T6#6O"&' .$6W&' ((X 2*#5 6Y(Z5 [&@O+.

8.3778.321550.5

12.6809.58003.1

21

21

'*

'*

xx

xx

) L!:0&'4!561 2),6\.(

" - %:

N&!"&' N$%&' 4!V3&')Ax = y(: !

"#$

%'

!

"#$

%'

!

"#$

%

0001.3

3

20001.1

2 1

2

1

x

x

$+,-*&' B/0( )t11x ' .2I+PQ*&' 1 ]6P3A ^#0, _!P#P$ _'6#`" 96#`" (5

90,Q1 !

"#$

%

29999.0

2 1

(#(<&' N$%&' 4!V3&' M0 (<+1 !

"#$

%'

!

"#$

% !

"#$

%

0001.3

3

29999.0

2 1

2

1

x

x

,!:0&' ;'6<H,4!561 2:*%& 9! . 2I+PQ*&' MKA S$6W&' 2/#/X

(15-3) s¹d9

(16-3) s¹d9

(17-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

200

" - &:

7/7*&' !.#/*!X C&= #"#&!"&' #"I+PQ*&' * 8O M/0# #L, U +O# ^#0,ilii +'1 ) 6W!,*&' M#/0"&'– 2A#6$M"#&+(.(

)1(

!

"

###

$

% &

533

933

112

)L (

!

"

###

$

%

&

&

&

55.44

201

35.12

" - ':

, 2I+PQ*&' M/0 NO:&+W" 2A#6$ 4'(%":!A L6- MQ!0 C&= 2#&!"&' a6+Q&' NI #"I+PQ*tLLA .' ^#0L 2#/P: 2#7/7* 2I+PQ*.

!

"

###

$

%

&

&&

&

'

210

121

012

A

" -():

,! 2#*E6'+% b!,")9+6O ( * MO M0 (<+1 2#6$A&' 2#787&' 2I+PQ*&' M#/0"& ##&!"&' ##$%&' #*!V3&':

)1(

5.12

1242

0

32

321

21

'*&

&'&*&

'&

xx

xxx

xx

)L(

1 2

3 2

3 2

32

321

21

'*&

&'&*&

'&

xx

xxx

xx

(18-3) s¹d9

(19-3) s¹d9

(20-3) s¹d9


201

" -( :

,! 2#*E6'+% b!,")9+6O ( M0 (<+1 2#6$A&' N787&' N$%&' 4!V3&' M0& N$%&' 4!V3&'Ax = y ^#0:

:A 2#6$A&' 2#787 2I+PQ*10 ×10 2#&!"&' 9!58)&!, ?c6)G":

1

9,,3,2;1

2

9,102,1

1,1,

&''

''&'

+'

&*

aa

iaa

ia

iiii

ii

:y B<"* 110, Z)G*2#&!"&' 9!58)&!, ?c6:

9,,3,2;0

1101

''

''

iy

yy

i

" -((:

6#6O"&' 2I+PQ* 1 ]6P3T 2#6#6O"&' 2`#Q&' NI

( ) ( ) ( )*cx.x 1 *'* kk T

2I+PQ*&' NK !

"#$

%'

aa

aT

60

6#6O"&' L6!A"& NI!O&'+ T6+6-&' $6W&' 1 9,7'(*) M0 C&= x = T.x + c

N,#6A" M0 T\ Nd'(",'( ) nR-0x +K: -1/3 < a < 1/3

-(":

2#6#6O"&' 2`#Q&' L6!A"& 2#:!:\' 2#6V3&' e,$ ( ) ( ) cxx 1 ! kk T

!"#$ %#&!' ()#*+G-S ,- #"#.'+) /0123 4"5:" # ULDT SG

1$$ $!

,$6+) 7&89+) :5+ !"#$+) ;<- =>$' ?"5 @+<0Ax = y 4"5:

0;1

4%&

'

()*

+ $! ,

,,

A

(21-3) s¹d9

(22-3) s¹d9

(23-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

202

-!":

!"#$ ?3 A. ?B C>D)J !"#$0G-S ,E)*'>) ,>"#!' :5 FG %#&!'' *9H I

,$6+) 7&89+) :5: &&&

'

(

)))

*

+

$

!&&&

'

(

)))

*

+

&&&

'

(

)))

*

+

$

$$

$

5.0

5.20

5

311

141

118

3

2

1

x

x

x

-!#:

!"#$ ?3 :. %#&!' J#*)J !"#$0G-S ,$6+) 7&89+) :5+ &3K!">$' *9HAx = y 4"5:

)B( &'

()*

+

$

$!

12

21

a

aA

)%( 0;1

4%&

'

()*

+

$

$! a

a

aA

)L( 0,0;2

1--&

'

()*

+

$

$! ba

b

aA

-!$:

!"#$ &9!>$ )<MG-S ,$6+) 7&89+) :5+

22

124

21

21

!

!$

xx

xx

,>"#!' ,E)*'>) :5 FG !"#$+) %#&!'' :K/" #0x 7N9> 7'>OB ? P)0 Q)<&3+0 QR 7B 7./ I #"#.'+) C)#3 **H(k) +) Q "#SH 7&T#B N>#G U&5"52 :5+) VWH :025W+ 3XA

9"&>'3+) ="!5'+ 3XA+) FB: " # 4105.0xx $./$ k

-!%:

"$6+) CR*&N3+) 7&89Ax = y Y"/: &&&

'

(

)))

*

+

$

!&&&

'

(

)))

*

+

!

2

2

2

y;

10

1

01

a

aa

a

A

(24-3) s¹d9

(25-3) s¹d9

(26-3) s¹d9

(27-3) s¹d9


203

)B ( 7"!+ Z*3 FGa /0123+) [>2'A Q*"*5'+&> >O03

)% ( \#]]]]19a=-0.25 . ]]]]!"#$ ?]]]]3 ?"#]]]]"#.' =]]]]>$G-S >3 ,E)*]]]]'>R) :]]]]5+&> U&E*]]]]'F#12+) .Q)<&3+0 QR 7B !"#$+) ;<- %#&!'' :-

-!&:

DAD+ C&>&(5+) %"#!' ^3 ,.(+0S' !"#$> ,+&'+) ,$6+) 7&89+) :5 *O0B7&T#B: Ax = y

4"5:

&&&&

'

(

))))

*

+

$!

&&&&

'

(

))))

*

+

$$

$

$

!

6

4

8

7

;

4200

2511

0131

0114

yA

-!':

,$6+) 7&89+) &9"*+ ?B \#19Ax = y 4"5:

&&&&

'

(

))))

*

+

$!

&&&&

'

(

))))

*

+

$$

$

$

!

12

5

9

8

;

4200

2511

0131

0114

yA

!"#$ %#&!'' :-J ?3 C)#"#.' DAD =>$ Q)<&3+0 Q7&89+) )<- :5+ &-&9!>$ )<M F#12+) ,E)*'>R) :5+&> U&E*'>3 !"#$+) ;<-" # 0x 0 ! . >0W$3+) C)#"#.'+) **H 7.

Q7&T#B '( V+M :2' T* VWH :025W+

- (:

,$6+) 7&89+) ?B \#19Ax = y Y"/:

&'

()*

+!&

'

()*

+!

13

17;

67

89yA

:"W5'+) 7)*6'(&> 7&89+) )<- :5 *O0BLU *N> H0$!3 C&>&(5+) _)#OM ^37&T#B DAD.

(28-3) s¹d9

(29-3) s¹d9

(30-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

204

$#S+) **H 3"T #*T0 I "#"#.'+) "12'+) !"#$ ?3 a*5)0 a0$6 =>$K(A). #&"N3+) 7*6'()11.

- ):

?B \#19:

&'

()*

+!&

'

()*

+!

6

7;

115

136yA

C&>&(5+) _)#Ob> 7&T#B DAD V+M >#!3:

)B ( /0123+) :W5A a#02+) ,/A = LU ,$6+) 7&89+) :5 *O0B0 IAx = y.

)% ( .'+) ?"(5'+) !"#$ ?3 U)*5)0 U)#"#.' =>$ CW25 F<+) :5+) ?"(5'+ F#"#Y"WH $#S+) **H 3"T #*T0 IK(A) . 3"!+) ^3 3"!+) ;<- ?#&T 7D

$0>c3+) .#&"N3+) 7*6'()01.

- !:

?B \#19A, B, C N>#3 C&/0123nn. ?3 A. ?B0 IA, C #"d /0123 ?B0 I a*#193y ?3 ?0.3 YO'3n #29H.

YO'3+) #2&9H a_&1.> %(5' e". f#S)x4"5 I:

yx 11 $$! BCA

H :025W+ !"#$+) ;<- ,/ >0W$3+) ">&(5+) C&"W3N+) **H 7.0 VWxQ

- :

,$6+) 7&89+)Ax = y Y"/:

&'

()*

+!&

'

()*

+

$

$!

5.1

0y;

25.11

12A

(31-3) s¹d9

(32-3) s¹d9

(33-3) s¹d9


205

!"#$ ?3 C)0$6 4AD <19G-S :5+&> U&E*'>3 I,$6+) 7&89+) )<- :5+ ,E)*'>R) ,>"#!'+)" # 0x 0 !. #"#.'+) /0123 *O0BSGT $ #"#.'+) C)#3 **H #*T0 I

VWH :025W+ >0W$3+)x 7&T#B "9&3D+ U&5"52 .#&"N3+) 7*6'()01.

- ":

?B \#19A: /0123nn. a*#193 #"d.

x : YO'31.n.

#)*!3+) %&(5 ?.3" e".

" # xx1$

! tt AAQ

>&(5+) C&"W3N+) **H 7. Q "+&H a_&1.> %&(5+ &KO&'5' ,'+) "QQ

- #:

?B \#19:

&&&

'

(

)))

*

+

!&&&

'

(

)))

*

+

!

4

6

8

,1

14

16

9

8

5

2

1

8

5

4

5

21

yA

,$6+) 7&89+) :5+ ,.(+0S' !"#$ 7*6'()Ax = y 3"T %(5) 7D ?30 I#)*!3+): yz 1$At

4"5: &&&

'

(

)))

*

+

$

!

1

3

2

z

- $:

&O ?09&T =">$'+ >(&93 a#02 ,/ ,+&'+) ,$6+) 7&89+) %'.),>0.:

24

6 4

2 4

32

321

21

! $

!$ $

!$

xx

xxx

xx

(34-3) s¹d9

(35-3) s¹d9

(36-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

206

?09&!+) )<- ?3 C)#"#.' DAD =>$ F#]12+) :5+&> U&E*'>3 F#"#.'+) #*T0 I7&T#B (36 V+M :2' T* VWH :025W+ >0W$3+) C)#"#.'+) **H.

- %:

:"W5'+) *O0BLU e012W+ CA"*>' FB :3H ?0*> "+&'+) /0123W+:

&&&

'

(

)))

*

+

!

13248

81216

1894

A

7D ?30 ,$6+) 7&89+) :5 *O0BAx = y ?3 :.+:

)B( &&&

'

(

)))

*

+

$

$!

3

12

11

y )%( &&&

'

(

)))

*

+

$

!

7

4

11

y

- &:

=>&(+) :)g(+) :5 *HBh-hi /0123W+ >(9+&>:

&&&&

'

(

))))

*

+

$$

$

$

$$

!

1110

4312

1101

1211

A

?"KO'3+)0:

)B( &&&&

'

(

))))

*

+

$

$

!

2

3

3

2

y )%( &&&&

'

(

))))

*

+

$

!

1

2

1

1

y

- ':

O0B#"#.'+) /0123 *jT ,$6+) 7&89+) :5+ =>$' ?"5 ,>0.&O !"#$+:

(37-3) s¹d9

(38-3) s¹d9

(39-3) s¹d9


207

55

10 5

5 5

321

321

321

!"

! "

!

xxx

xxx

xxx

!"# $%&' ()#

jT*+',-.' /0-! 12 $345-- *.6740 5"38 9: 9;"<. () 9!6 = .

>.8? 13@AB' *+',-.C' /0-!B4. D4+,-.! E'3"37-.

-!":

F3@G #HIB4. J<8IK- E4L0-!B' 9! M%M% $4%& $6M8!B' 9::

kiyCBAx ii ,,2,1;11 !!

N"&A, B, C: I.3! E4O6@A!nn$ = !6MI!

9! P76A, B: =Q,3@G! 3"R O6@A!

6kiyi ,,2,1;: ! E4L0-! M%M%

! AE

! BF

! *B'6-B' JMS

$%&'ii Gyx !

! *B'6-B' JMS

;@G*M" 4! T:

):( *8UB' (4VGB' P&: iiBB yzUL !

)$( $%&': ii Czd !

(40-3) s¹d9

1 n !"#$!.

%"&'()*!+, -#. /"01#+ 2'*3456 7#4 '!4i

x:

!"#$ %&'()!$:

-1 891,1!

.

-3 891,G = ECF.

-4 :43#+ki ,,2,1

.

%'*+,!$ %&'()!$:

-1 /#1A 24#!'. -+; 244<#<!AA

UL.

-2 /#1B 244<#<! 24#!'. -+;BB

UL.

-3 :43#+ki ,,2,1

.

-2 891,1!

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

208

)W ( !"#$ %&'(#$ )* iiAA dxUL

+,-./-0!#$ ,1 )2 3 4567!1#$ 4-5&8*#$ 9&-71:#$ ;;< =&5-0/. >6&8- %2+)?3@ &1A-@6

-!":

,@ B0C(A 45.0 ,1 D;0C(1 0-E 4:501 436CF1n . 4-1G0$6"#$ ,@ 95H$ I62:1#$ J8*. 4-#&.#$1!A:

! )7*A D06F#$ 3LU.

! !"#$ %&'(#$ )*:

nieLUx ii ,,2,1;

K-*ie %L0 ;61:#$ 6Mi N.&(#$ )*#$ OP.1 ,62-3 QD;*6#$ 436CF1 3ix

%L0 ;61:#$ 6Mi 436CF1#$ 31!A.

4-5&8*#$ 9&-71:#$ ;;< %2)418/#$6 J0?#$ ( J&8*# 4567!1#$1!A RSA5!#$+4/-0

436CF1#$ I62:1 J&8*# 4/-0!#$ RSM T5!:

""""

#

$

%%%%

&

'

11932

4844

4421

9621

A

-!#:

,@ B0C(:

""""

#

$

%%%%

&

'

""""

#

$

%%%%

&

'

!

!!

!!

!

0

0

0

1

y,

2100

1210

0121

0011

A

) @( )-7*.#$ ;P6@LU 436CF17#A.

) J( !"#$ %&'(#$ )*Ax = y )-7*.#$ =&1;".81LU 3)@.(

(41-3) s¹d9

(42-3) s¹d9


209

) U( I62:1#$ ;P6@1!A !0V#$ ;;< J8*$ %H QK(A) 36CF17# 4A =&1;".81 0&-:1#$(1.

-! :

!"#$ %&'(#$ )* ;P6@Ax = y K-*:

""""

#

$

%%%%

&

'

""""

#

$

%%%%

&

'

!

!!

!!

!

3

2

2

3

,

4100

1410

0141

0014

yA

) @( 28#6V. 4/-0! %$;".8&5.

) J( 4/-0! ,1 )2 %$;".8&5J 4/-0!6G-S OP.1#&5 @;5$ ,-./-0!#$ ,1 )2 36 Q>0CF#$ 0x )0( 4W-F#$ T-5!. 3 01.8$6 Q X#Y )F. ,@ X#Y 4-0-02.#$

!0V#$ T/*.- ,Z5 Q4-0V< %&L0@ 4H[H \75. 4L;: ) * ) *

) *3

1x

x1x

105.0 !

(+

(!+

,-k

kk

(43-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3


‰ULJ²Ýô« »U�Š

Interpolation

WO�ULJ²Ýô« !«dłô W¹œËbŠ º

WO�U�_« ‚ËdHK� sðuO½ WGO� º

WOHK)« ‚ËdHK� sðuO½ WGO� º �dH¹≈ WGO� º

v�Ë_« ”ËUł WGO� º

WO½U¦�« ”ËUł WGO� º

QD)« —UA²½« º w�JF�« ‰ULJ²Ýô« º

�U¹œËb(UÐ w¾¹e−²�« V¹dI²�« º

WO³OFJ²�« W×¹dA�UÐ ‰ULJ²Ýô« º W²]³¦*« W×¹dA�«Ë WOFO³D�« W×¹dA�« º

WOFO³D�« WO³OFJ²�« W×¹dA�« WO�“—«uš º

W²]³¦*« WO³OFJ²�« W×¹dA�« WO�“—«uš º

lÐ«d�« qBH�« �UM¹d9 º

213


‰ULJ²Ýô« »U�ŠInterpolation

!"# !"# $% &'()*+ &%), &-!# ./012 &-34% 5'-! !67 89 "):2: . #;<"2" !"#;;-! .;;/012- $)2/;;4/(0 $)2#;;()9 =);;%"%, . &;;-!# >#;;1?% &;;-!#+ #+2;;4: $/;;@ );;::A );;%7B"A

C;D)'%-)E );F/-G H);2@: 8;2-! I);/C%?-! J?+ K!0<G LF4/ M-6 $N9 *4+A&;C%)E%-!" & .

&/4/(0-! >#()'-!" "!#<-! $% )%E24B! 87 &/:)O-!&/-)2-! &-34%-! )F+ 8:?:" P.

!"#$:

)% &-!# Q/R ):/*,SA)(xfy *)1:-! J?+ #:,:

nixi ,,2,1,0; !

Q/1-! $A TA:

" # nixfy ii ,,2,1,0; !

):/#- &%"C?% I@+5A$//?2 ):% .CS* QO P &-!#-! &%/Ry &;*1: #;:,x >#";<"% 0;/U>)*?%-! &%"C?%-! Q/1-! "#< 89.

I)/#"#;;;;;;;;@-! "A #"#;;;;;;;;@-! I!0;;;;;;;;/OE 8;;;;;;;;7 .;;;;;;;;/012C- =)%!#V2;;;;;;;;4! !"#;;;;;;;;-! 0;;;;;;;;OEA"(polynomials) !"#;;;;;;-!" P &;;;;;;/OCO%-! (trigonometric functions) &/;;;;;;4W! !"#;;;;;;-!"(exponentials) !"#;;-!"&/+;;4:-! (rational functions) . X6;;7 0;;OEA 8;;7 I)/#"#;;@-!"

I)+;;;4)@-! Y;;;C, Z[;;;*\! Y;;;C, =)%!#V2;;;4! !"#;;;-! $;;;% I);;;,"%<%-! .. ;;;5'-! 8;;;9" )%E24B! 89 I)/#"#@-! I)1/+*2" ]!"V 8-)?2 ^! $"?+ _0#: 8-)@-!.

&/-)2-! &1/0*-! =)%"%, `+2: I)/#"#@-! Q!#V24)+ =)1+)4 >0"E6%-! &-34%-! @-":

!"#$!% &' ()*+:

&-!#-! .L01S:f(x) &/#"#@+p(x) &<0# $%n &;/#"#@-! Q/R Z+*:2 a/@+p `;% &-!#-! Q/Rf *)1:-! #:, nixi ,,2,1,0; !

$A TA:

" # " # nixfxp ii ,,2,1,0; !!

214

&;;;;%/1-! $/;;;;/?2- &;;;;/#"#@-! X6;;;;7 Q#V2;;;;4: Q;;;;O)(xp;;;;%/R $A 0;;;;+2?:" P&;;;;-!#-! & f

&;;*1:-! #;;:, &;;+"C*%-!x $A TA P&;;%/1-! X6;;F- &/");;4%)()( xpxf $ . X6;;7 Y%;;42" &;;;;<0#-! "6 )%E2;;;;4B! &;;;;1/0*-!n &;;;;<0#-! $;;;;% &;;;;/#"#@ Q#V2;;;;4: );;;;::W Pn =);;;;:)/@A" P

b%0-! )F- Q#V24:np. : )%E24B! Q"F'% c/D"2-"8-)2-! */4+-! )O%-! 0E6+ A#+:

&,-$ :(1-4) &-!#-!)(xfy 8-)2-! "#<-! 89 &:/+%-! *)1:-! #:, )F%/R Q"C?%: x 1 2 4 5

y = f(x) 0 2 12 21

&%/R .4@!f(3) Q!#V24)+:

d- 8?/+02-! )%E24B!(quadratic interpolation) *)1:-! Q!#V24)+1, 2, 4.

e- E24B! *)1:-! Q!#V24)+ 8?/+02-! )%2, 4, 5.

f- 8+/?E2-! )%E24B!(cubic interpolation).

&'!%:

) A( $A J0'::

210

2210

00)1(

)(

aaap

xaxaaxp

%%!&!

%%!

(1)

210 4222)2( aaap %%!&! (2)

210 1641212)4( aaap %%!&! (3)

)FC@ &/*V IB#)?% &,"%<% X67": 1,1,0 210 !'!! aaa

$A TA: 693)3()3(

)( 2

!%'!$(

%'!

pf

xxxp

*./ &': $A J0'::

642)1)(2(2)1)(2()3()3(

)2)(1(2)4)(1()(

2)2)(3(1212)4(

1)2)(1(22 )2(

0)3)(1(00 )1(

)2)(1(

)4)(1()4)(2()(

22

11

00

2

10

!%!%''!$&

''%'''!(

!&!&!

'!&'!&!

!&''!&!

''%

''%''!

pf

xxxxxp

ccp

ccp

ccp

xxc

xxcxxcxp


215

‰ULJ²Ýô« »U�Š ∫lÐ«d�« qBH�«

) .( *);1:-! Q!#V2;4)+2, 4, 5 gTA" P &;/#"#@-! $;, 0;/+?2C- $/21+);4-! $/21/0;*-! $;%)()( 2 xpxp YC, 5@: P:

667.5235)3()3( 2 !!$ pf

82</2: Q!#V24)+ )::E%/")A ( P). (0L#1: $A

17.083.5)3(

83.52667.56)3(

)!

!%*

f

f

) H( + *)1: `+0W! Q!#V24)(1, 2, 4, 5) YC, 5@: $A )::E%/:

833.5635)3()3( !!$ pf

$A J0;;;;;': $3;;;;;+ Z+);;;;;4-! );;;;;O%-! 8;;;;;9 );;;;;7):0E6 8;;;;;2-! &;;;;;1/0*-! Q/;;;;;%?2 );;;;;::E%/)()( xpxp n $A TAp &;;;;;;;;;<0# $;;;;;;;;;% &;;;;;;;;;/#"#@n &;;;;;;;;;%"C?%-! Q/;;;;;;;;;1-! J/";;;;;;;;;?2+" P

nyyy ,,, 10 202-! YC, &C+)1%-! *)1:C- ./nxxx ,,, 10 8*V Q)h: YC, 5@: $;;%n+1 )7##;;," &;;/#"#@-! I[%);;?% 8;;9 &;;/:i &;;-#)?%n+1 !6;;7 $A I);;+OG );;::E%/" P

#/@" @ j- Q)h:-!.

&;'C2V%-! *)1:-! ):/#- $A J0':nxxx ,,, 10 - )7##;,"n+1 &;*1:– Y;C, $A" P 8;1/1@-! 0";@%-!f(x) );% >0;29 Y;C, &;90?% Q/;1-! &;/1/1@ &;-!# I = [a, b] %2;k2

*);;1:-! X6;;7 Y;;C, . &;;/#"#@ $/";;E2 .";;C*%-!"p(x) T");;42 "A $;;% ;;RA &;;<0# $;;%n P %E242 a/@+ f ) `% Z+*:2 TAf(x) ($A TA P >0"E6%-! *)1:-! #:,:

" # " # nixfxp ii ,,1,0; !!

&;/#"#@ $;% 0;OEA #;<"/ B $;E-" P &/#"#@-! X67 O% &+)2E- Z0* >#, M):7"&/-)2-! &/0h:-! M-6 $/+2 )%E *0k-! !67 Z1@2 >#@!":

)*01:

&;;;<0# $;;;% >#;;;@!" &;;;/#"#@ 0;;;OEW! Y;;;C, #;;;<"/+n %E2;;;42f(x) *);;;1:-! #;;;:,&'C2V%-!nxxx ,,, 10 , 82-! )7##n+1 &*1:.

Lagrange interpolating polynomial WO�ULJ²Ýô« !«dłô W¹œËbŠ

‰ULJ²Ýô« »U�Š ∫lÐ«d�« qBH�«4

216

2,3*4!%:

$/2/#"#;@ M);:7 $A J0;':p(x), q(x) &;<0# $;%+n );%F:% ;E %E2;42 Pf(x) >0"E6%-! *)1:-! #:,.

$A J0':: h(x) = p(x) – q(x)

h(x) &<0# $% &/#"#@+nM-6E" P

" # " # " # " # " # nixfxfxqxpxh iiiii ,,1,0;0 !!'!'!

$A TAh(x) ! Y;;;C, );;;F- ;;;RWn+1 l;;;C2V% 06;;;<) l;;;C2V% 0';;;5distinct

zeros .( &;;;<0# $;;;% >#;;;/@"-! &;;;/#"#@-! $A a;;;/@"+n );;;F- 8;;;2-!"n+1 0';;;5 8;;;7 &/0'5-! &/#"#@-!(zero polynomial) $A 8:?/ !6F9h(x) 0 8-)2-)+" P :p(x) q(x).

2;;;42 &;;;/#"#@ I#;;;<S" !6G j;;;:A I;;;+O2 &1+);;;4-! &;;;/0h:-! &;;;-!#-! %Ef *);;;1:-! #;;;:,>);;;*?%-! >#;;;/@" $";;;E2 &;;;/#"#@-! X6;;;7 $N;;;9 P(unique) . #";;;<" I;;;+O2 &;;;/-)2-! &;;;/0h:-!"

(existence) &/#"#@-! X67 O%.

)*01:

&;;;<0# $;;;% >#;;;@!" &;;;/#"#@ ;;;RW! Y;;;C, #;;;<"/+n %E2;;;42f(x) *);;;1:-! #;;;:, &'C2V%-!nxxx ,,, 10 82-! )7##,n+1 &*1:.

2,3*4!%:

) A( 9 $"E2 82-! &5)V-! &-)@-! 89 =B"A &/0h:-! $70+:/ &-!#-! )Ff 0';5C- &/")4% &;;;-!#-! Z;;;1@2 $3;;;E P >#;;;@!" K):O2;;;4)+ )%E2;;;4B! *);;;1: `;;;/%< #;;;:,f I);;;R[?-!

=[O% &/-)2-!:

" #ki

kixf i ,

!!

0

1

&;<0# $;% &;/#"#@ $;, a;@+: &;-)@-! X67 89+n *);1:-! `;/%< #;:, Y;k[22

ix K):O24)+ kx .*0k-! !67 Z1@2 &/-)2-! &/#"#@-!:

" #

" #" # " # " # " #nkk

n

kii

ik

xxxxxxxxxx

xxxg

'''''!

- '!

%'

,!

1110

0

)(


217


$A cD!"" :

" # " # 00

,- '!

,!

n

kii

ikkk xxxg

*)1:-! `/%< $Wix =)D09 &'C2V%.

$N9 8-)2-)+":

- ''!!

,!

n

kii

ik

i

kk

kk xx

xxxgxg

xL0)(

)()( (1)

$A TA P &5)V-! &-)@-! X67 89 )%E24B! &-34% @ "7:

niki

kixL ik ,,1,0

0

1)( !

./0

,

!!

) .( $"E2 )%#:, &%)?-! &-)@-! 89f(x) $N9 P &/0)/2V! &-!# TA:

)()()(0

xLxfxp kk

n

k

1!!

(2)

&;;;;;;;;<0# $;;;;;;;;% &;;;;;;;;/#"#@ 8;;;;;;;;7+n %E2;;;;;;;;42" Pf(x) *);;;;;;;;1:-! #;;;;;;;;:, &;;;;;;;;'C2V%-!

nxxx ,,, 10 .

;;/-)2-! &;;/0h:-! 8;;9 $/21+);;4-! $/2/0;;h:-! `;;%< );;::E%/ $m!"& Y;;C, ]:;;2 8;;2-!#"<" (existence) &/:!#@""(uniqueness) &/-)%E24B! &/#"#@-!.

)*01:

Q/;;;;1-! &;;;;/1/1@ &;;;;-!# );;;;:/*,SA !6G(real-valued function) f(x) *);;;;1:-! #;;;;:, &;;'C2V%-!nxxx ,,, 10 )7##;;, 8;;2-!n+1 &;;*1: >#;;@!" &;;/#"#@ *+;;D-)+ #;;<"2 j;;:N9 P &<0# $%+n %E242f(x) >0"E6%-! *)1:-! #:,.

5)*67:

&R[?-!(2) Y%4/ )% 8*?2 )!,$87#9% ):;:'<! =1%*>9 ?)@4. I)/#"#;@-!")(,),(),( 10 xLxLxL n ;;;R[?-! );;;F9 L0?S2 8;;;2-!" &(1) Y%;;;42=1%*AAA>9 B,):;:AAA' *);;;1:C-

nxxx ,,, 10 ./202-! YC,.

&<0#-! $% &/-)%E24B! n:!0<B &/#"#@ $A TA+n 87:

)(,.)( (x)pn

0kn kkkk xfyyxL 1!

!


218

002010

21 .)()()(

)()()( y

xxxxxxxxxxxx

n

n

''''''!

11312101

)320 .)()()()(

)(()()( y

xxxxxxxxxxxxxxxx

n

n

''''''''%

nnnnn

n yxxxxxx

xxxxxx.

)()()()()()(

......

110

110

'

'

''''''%

%

knkkkkkkk

nkk yxxxxxxxxxx

xxxxxxxxxx.

)())(())((

)())(())((

n

0k 1110

11101'''''

'''''!

! %'

%'

$ 0'C:

&;;+)2E+ &/-)%E2;;4B! n:!0;;<B &;;/#"#@- &o/;;5-! X6;;7 Y;;-G ";;5"-! =);;D/A );;::E%/ &<0# $% &%), &/#"#@n );O% ;@ 8;9 );7):?+2! 8;2-! M;C2- &;CO)%% &o/;5 89p-d

] Kb<-!)A (– )0Vi @ [(I;+!"O &-B#+nccc ,,1,0 &;%"C?%-! *);1:-! J/";?2 Q;O P2):-! 8*V-! Q)h:-! @"8C/ )%E P I+!"O-! X67 $//?2- n:

1 '%'''''!!

n

0k

n )()1)(1()1)(0( (x)p nxxkxxkxxxxxxkc

I;;;+)O TA &;;;%/R Y;;;C, ";;;5@C-"kc $E/;;;-"Kc a;;;/@K Q/;;;R q#;;;@G 8;;;7 k

J"?:: kkn y )(xp !

YC, 5@:9: )())(())((

1110 nKKKKKKK

K

xxxxxxxxxx

yKc

'''''!

%'

n:!0;;<B &o/;;5 Y;;C, */;;4+ Z;;/+*2E"$A J0;;': P n=1 . );;:/*,SA );;::A TAf(x)

$/2'C2V% $/2*1:"1,0 xx .$"E/ 8-)2-)+":

01

0

10

1 )(1,0 xx

xxxL

xx

xxL

''

!''

!

" # " #

" #0.)()(

)0(

)()(

)1()0(

)1(.)(1)0(.)(0

1.)(10.)(0)(2

01

01

10

1001

01

0

10

1

xxxx

xfxfxf

xx

xfxxxfxx

xfxx

xxxf

xx

xx

xfxLxfxL

yxLyxLxp

'''

%!

''''

!

''

%''

!

%!

%!


219


8*V-! )%E24B! r/5 q#@G 87 X67"(linear interpolation).

&,-$)2-4:(

&;;;-!#C- 8-);;;2-! "#;;;<-! $;;;%f(x) &;;;%/R 0L#;;;Rf(0.6) Q!#V2;;;4)+ n:!0;;;<B &;;;/#"#@&/-)%E24B!:

x 0.4 0.5 0.7 0.8

y f(x#2 -0.916291 -0.693147 -0.356675 -0.233144

&'!%:

32

)3.0)(2.0)(1.0(

)2.0)(1.0)(2.0()6.0(

61

)8.04.0)(7.04.0)(5.04.0(

)8.06.0)(7.06.0)(5.06.0()6.0(

).6.0()6.0(

1

0

0

!''''

!

'!''''''

!

1!!

L

L

kykLfn

k

" #" # " #" #

" #" # " #" #

509975.0

233144.061356675.0

32

693147.032916291.0

61)6.0(

61)6.0(

32)6.0(

3

2

'!

''%'%

%'%''!

'!

!

f

L

L

0'C$:

&;-!#-! Q/;R 8;7 Z+);4-! "#;<-! Q/;Rf(x) = ln x &*"+;D%-! &;%/1-)9 8-);2-)+" P87:

000851.0

510826.06.0ln)6.0(

!&

'!!

Error

exactf

*)1:-! $% &,"%<% ):/*,SA !6G:

niixfiyix ,,1,0;)(, !

(Inverse Interpolation) w�JF�« ‰ULJ²Ýô«


220

&;;;%/R #);;;</G );;;:% .;;;CS*"x &;;;%"C?% &;;;%/R ;;;+)12 8;;;2-!y ) *);;;1:-! "#;;;< 8;;;9 I;;;4/->)*?%-! ( )%E24B)+ Y%42 &-34%-! X67 $N9 P &%/R #)</G "7 ."C*%-! $W 84E?-! 0/o2%C-x _/-"y.

!#;+G `;% n:!0;)*?% &/+/012 &%/R #<"A Px &-!#-! 0'5-f(x) P &-#)?%-! 06<- TAf(x) = 0.

ix 0.2 0.3 0.5

)( ixf 0.4 -0.1 -0.45

&'!%:

&/-)2-! >0"5-! 89 Z+)4-! "#<-! I):)/+ `D" $E%/: 0y 1y 2y

iy 0.4 -0.1 -0.45

)( iygx 0.2 0.3 0.5

&%/R #)</G "7 ."C*%-! c+5/"g(y) $A QC, !6Gy = 0. " # " #" # " #

" # " #" # " #

" # " #" # " #

.x . x

. . - .-. .- .

. .-

. .- . . .. .

xyyyy

yyyy

xyyyy

yyyyx

yyyy

yyyyyg

!

333%3

33%3

33!

''''

%

%''''

%''''

!

26252090

50350850104030

350504504020

8505045010

.

1.)(

21202

10

2101

200

2010

21

0'C$:

06< #)</G &-34% 0)+2,! $E%/x &-#)?%C-f(x) = 0 $2"/: &1/0*+– $"49!084E?-! )%E24B! YC, =)1/+*2.

!"#(3-4) :


221


!"#$%&'() *+),-( .!/ 01 Accuracy of Lagrange Interpolation

!"#$% &!'( )*+!,- .!/00$% .!0 1*!$npf 2!$%+$% &*!3 "!,4 5!6078 %9:f +!!68 ;<%=0$%nxxxx ,,, 10 ! 2!$%+$% .!8 >)!#? @50=7A0 24)A0 B'* CD Ef . .5!*D$=

5F++8 G-$% 2*$5-$% "5,6$% 28=0'0 56*+$ .? H)-I6 J$9n+1:

" # " # " #)(,,,)(,,)(, 1100 nn xfxxfxxfx

K*(nxxx ,,, 10 2I7-#0 5LA*0'.

2!*+=+($% M5!050- ++!(- "5!,6$% N9Fnp .- =? O!*)A-$= 2!$%+$% +!*+(f &!P) 5!66/0* 5F++!!!!!8 G!!!!!-$% "5!!!!!,6$% N9!!!!!LD )!!!!!0* Q!!!!!6(60 R?n+1 . S)5!!!!!I$% C!!!!!A' 5!!!!!66/0*4 G$5!!!!!-$5D=

)]()([ xpxf n M%)*D/– 5F)5-#6 20*3 R? Q$: T-0*3 CU- K*(D– ;<=0 R? +68x ;<%=0$% CV#nxxxx ,,, 10 ! S)5I$% K*()]()([ xpxf n R=5P* M%)IU.

!!!"#$% )*+!!!,- 5!!!66/0* T!!!6? Q!!!78npf 2,-!!!W0 2!!!$X+Df 2!!!D-)$% @%9(n+1) %9:2*$5-$% 2*)Y6$% J$9 .*D- 50/ E @+'Z=:

2222!,3+: 6!!!!I .? H)[a,b] "5!!!!!,6$% C!!!!/ Q!!!!!78 R=!!!!-(- [)!!!!-4 R? G!!!!!Fnxxx ,,, 10

5F++8=n+1 2",6 . C%=+!$% .? H)!I6=)(,,,, nffff $$$ G!4 27!U-0= +=!'= 5!L$ [)-I$%[a,b] . 2,-W0$% .? H)I6=)1( %nf [)-I$% G4 +='= 5L$a < x < b . 5!6*"8Z? %9\4

2",6 R?],[ bax& ++8 +'=* T6\4' ) Q78 +0-A*x ([)-I$% G4 (a,b) .? K*(D:

)!1(

)(.)()()()()(

)1(

0 % ! (

%

n

fxxxxxpxfxE

n

nnn

'

4#5,6"):

)% N9F .? Y(V* 50L60 C/ .5F)D +0-A*= E )=7*5- 2*+=+( 2*)Y6 TDW- 2*)Y6$2"P=-0$% 20*,7$ C=) 2*)Y6 Q78.( .*-$%+$% .0 MV/ .] M%)Y6:

,)()()(, 10 nn xxxxxxpf

"5!!,6$% +!!68 )5I!!U? 5!!L$ :nxxx ,,, 10 +!!68 )5I!!U? 5!!L$ 2!!*$5-$% 2!!$%+$% M5!!<*? J$9!!/4"5,6$% 5LPI6:

),()()()()( 0 nn xxxxxpxfxg % ! )

K*() @D5^ R?.


222

)5-#6P ._%=) .? .0<6 K*(Dg C!3]% Q!78 5L$n+2 )I!U . .? 5!6+)? %9: 2!!",6$% +!!68 !!"#$% )+!!,6x = * K!!*( E],[ ba&* )5!!-#6 5!!66\4 E) C!!A'- K!!*(D

0)( !*g .? Q$: R+ * %9F= E:

)()()()(0 0 nn xxpf % ! **)**

))()((.)()()()(

)()()(

)()()()(

0

0

0

****

****

)

nn

nn

n

n

pfxxxxxx

xpxfxg

xxpf

!+

!+

2!!$%+$% .? a!!<%=g .!!8 C!!,* X "5!!,6$% .!!0 ++!!8 +!!68 M%)I!!U R=5!!P-n+2 N9!!F= C0!!!!W- "5!!!!,6$%nxxxx ,,, 10 ! J$9!!!!/=*!x "5!!!!,6$% N9!!!!F )=!!!!U-6 .? .!!!!/0*=

B*-)-$% %9LD 2D-)0 . 2!$%+$% Q78 C=) 2*)Y6 S*D"-D=g 2!*bc'$% @%)!-I$% .!0 C!/ +!68(subintervals) ) 5F++!!8=n+1 ( .? d-6-!!P6 E [)=5!!'-0$% )5I!!U]% e%=c? .*!!Dg$ 5!!L$

C3]% Q78n+1 )I!U .2!$%+$% Q!78 C=) 2!*)Y6 S!*D"-D &!^ g$ @%)!-I$% .!0 C!/ +!68 2!!*bc'$%) 5F++!!8=n ( .? d-6-!!P6 E [)=5!!'-0$% 5F)5I!!U? e%=c? .*!!Dg $$ C!!3]% Q!!78 5!!L$

n )I!!U . .? d-6-!!P6 2!!*I*/$% N9!!LD 2!!*)Y6$% S!!*D"- G!!4 )%)0-!!PX5D=)1( %ng Q!!78 5!!L$)I!!U C!!3]% [)!!-I$% G!!4 +!!(%= (a, b). .? H)!!I6' )5I!!U]% N9!!F +!!(? =!!F. 27!!<5I0D

2$%+7$ [)*#]% 23VA$%g(x) ++8n+1 Q78 CU(6 @%)0$% .0:

))()((.)()(

)!1()()(

0

)1()1( **** n

n

nn pfxx

nxfxg

%

! %%

++8 27<5I0$5D T6] J$9=n+1 G!I-#- E @%)!0$% .!0np +!($%= E1%nx d-5!6$% B)!!!!!!!!!<$% C!!!!!!!!!U5( .!!!!!!!!!0)()( 0 nxxxx Q!!!!!!!!!$: Cc!!!!!!!!!-#Z*(n+1)! . H*=!!!!!!!!!A-D=

0)()1( !% 'ng Q78 CU(6 2,D5P$% 2$+5A0$% G4:

))()((.)()(

)!1()(0

0

)1( ****

' nn

n pfxx

nf

%

! %

;!!!<== 2!!!$+5A0$% N9!!!F +=+!!!( B!!!*-)- [+5!!!8: +!!!AD M%)!!!*#?=x .!!!0 MX+!!!D* C!!!U(6Q78:

.!)1()(

.)()()()()1(

0 % !

%

nf

xxxxxpxfn

nn

'


223


378$:

)*+!,-$ e5!-(6 2!*)Y6$% N9!F G!4 [5!"A0$% "#$% )*+,- 2f*U &%+#-PX)1( %nf .

2!!!0*3 O)!!!A6 X [+5!!!8=' !!!'*: Q!!!$: !!!"#$% )*+!!!,- +!!!68 !!!'764 J$9!!!$= Q!!!78]% +!!!($% +5 2,-W07$)1( %nf ^0$% .0 .*D-* 50/5G$5!-$% C . N9!F 2!0*3 )*+!,-$ 2!,*)" +!'=- T!6? Q!78

5L6*D 50*4 @545P0$% 2*=5P-0 "5,6$% .=/- 50+68 2,-W0$%.

9#:$)4-4(:

2!!$%+7$ .*!!-0*3 G!!"A* G$5!!-$% C=+!!'$%f(x) .*!!-",6 +!!68 . C=+!!'$% %9!!F &+#-!!P% C50/-PX%= 20*3 +5'*g G"#$%f(3.16) . !"#$% )h+3=) !"#$% 2f*!U &%+#-!P5D ( G!4

.? &78 %9: 5L*78 CU(- G-$% 2'*-6$% xxf ln)( (.

x 3.1 3.2

f(x) 1.1314 1.1632

97"): G"#$% C50/-PX5D:

15048.11.0

0318.006.01314.1

)()().()()16.3()16.3(

01

01001

!,%!

%!- xxxfxf

xxxfpf

1505.1- 2*)W8 &53)? 2AD)] ( )

2'*-6$% G4 "#$% )*+,-$=f(3.16) "#$% 2f*U &+#-P6 5L*78 567U( G-$%:

!2)(

.)()()()()( 1011

'fxxxxxpxfxE

$$ ! !

,1)(ln)( xxfxxf !$+(

21)(x

xf !$$+

21 21)2.316.3(.)1.316.3()('

, !+ xE

.? K!*(=' .*!D ;!,-3.1, 3.2 .*!D ;!,* !"#$54 J$9!$ E0.00011, 0.00013 E.? R?: 00013.0)(1 .xE

2*)!!W8 &5!!3)? 2!!^V^$ a*(!!U T!!*78 567!!U( R9!!$% B%=!!'$% .? R?)C!!3]% Q!!78 ( %9!!F= E C=%+!!!'$% .!!!0 5!!!L*78 C!!!U(6 G!!!-$% 2!!!0*,$% .? K!!!*( ;!!!3%=$% ;!!!0 S!!!I-*– &5!!!3)? 2!!!AD)]

2*)W8– GF: ln 3.16 = 1.1506


224

[+!!(%= [+!!(=D 2(*(!!U$% 2!!0*,$% ;!!0 O!!7-#- 5!!L*78 567!!U( G!!-$% 2!!0*,$% .? R?4 ",4;D%)$% R)WA$% ;<=0$% G.

5!!0 2!!",6 +!!68 d6%)!!'X 2f*!!U G!!4 2*$50/-!!PX% 2!!*+=+($% 2!!0*3 B5!!P( B!!7"-* .0 M%++8.8 C,* X 2*D5P($% @5*70A$%:

3n + 1 B)< @5*708 /20P3

=2n + 1 ;0' @5*708 /i)".

2f*!!U$% N9!!F G!!4 d6%)!!'X @5*+=+!!( @5!!05,0 B5!!P( +!!AD J!!$9= . B!!7"-* 5!!06*D 2!!!!!!')+ .!!!!!!0 2!!!!!!*+=+( 2!!!!!!0*3 B5!!!!!!P(n C#%+!!!!!!-0$% B)!!!!!!<$% &%+#-!!!!!!P5D(nested

multiplication) )6)=F 2,*)" R?:

n B)< 2*708

=n ;0' @5*708.

d6%)!!!'X 2f*!!!U S!!!*D"-$ 2!!!D=7"0$% 2*D5!!!P($% @5!!!*70A$% [)!!!^/ Q!!!$: 245!!!<g5D=2*+=+( @0j7Z8 %9: T6? =F= &F? )#k B*8 J56L4)(xpk &!^ "5,6$% .0 28=0'0D )0-

2!!*+=+($% e5-6-!!P% 5!!66/0* V!!4 2!!8=0'07$ >)!!#? 2!!",6 @I*!!<?)(1 xpk% !!*+=+($% .!!0 2 2!!0=7A0$%)(xpk .(56$% .!!0 T!!6] 27/!!W0$% N9!!F )!!LY- 5!!0 M5!!D$5l= O)!!A* X 2!!*70A$% 2!!*

Mm%+-D% m)0$% – [+58– &%+#-PV$ e5-(* C50/-PX% "5,6 .0 &/ . @!65/ %9\4 G$5-$5D=)(xp j 2!!!!!!!!!*+=+($% G!!!!!!!!!F) 2!!!!!!!!!')+ .!!!!!!!!!0/j (0/-!!!!!!!!!P- G!!!!!!!!!-$% Cf(x) "5!!!!!!!!!,6$% +!!!!!!!!!68

jxxx ,,, 10 B5!P( G4 9# * +(%=$% .\4 E ),(),(),( 210 xpxpxp G!4 C!Y* . !D "5,6 ++8 [+5*c C50/-PX%– 2*$50/-PX% 2*+=+($% 2')+ G$5-$5D=– CU* .? Q$:–

J!!$9 MV!!0k– C=!!D,0 B!!*),- Q!!$:)(xpk 2!!$%+7$f(x). $% N9!!F C!!^0 G!!4= 2!!*70A– 2!!*708

©œUFÐ_« W¹ËU�²*« ◊UIM�« bMŽ ‰ULJ²Ýô«® ‚ËdH�« Â«b�²ÝUÐ ‰ULJ²Ýô« mO�

Interpolation Formulas by Use of Differences (Interpolation at

Equally Spaced Points)


225


2!!!!AD5--0 @5*+=+!!!!( B5!!!!P(– 2A*!!!!<0 .=/*!!!!P d6%)!!!!'X 2f*!!!!U &%+#-!!!!P% .? )!!!!LY* B5P( +68 T6] E@3=7$)(xpk !"# $%&%' (%) *"*+ ,! -&$./012 $%%3!" 11 kp.

2&4/502 657." (5%89 :;54) <$=0)> ,"==0?2 ,"@A?>B2&4/50$= 6$!3/5012 C"5D B

E>;.?23/012 F$*% ,>3/ G"+ H$A%"= $!"9 I$9$0!?2 ">$0/! 6$! J;K/0% LM" L9> H .M/4!?2 E>;.M? I$.";K/?2 JK= LFK% J2;K/012 2@N 6=#> H C"D?2 O@N JK=.

!"#$%&'( !)*+&'( !%,%-' .($)&' /!$01

Forward, backward, and central differences

$!3/012 ,P2 Q/+ $%0;& 5?2&? =50%?$= 6f &&5' &5%' $5A!"# B>5MK!1!n ,5! 555.M/4!?2 555";$"/412 F$555*%?2 . 555=/;! F$555*%?2 555'>!R! ,) J;/.%5550 ,P2 ,555! ST2&555/=2>

U>$0/> /=$V ,"/"?$//! ,"/F*% U) ,"= 9$0!?2 ,) G"+=>h ,) U) H:

nihixx

hxhxx

hxx

i ,,2,1,0;.

.....

2

0

012

01

"!"

!"!"

!"

W;55KX" 22!%,%-' .($22)&' 322%,0% (forward difference operator) - ;VY55" U@55?2> ?2&?2 QM'f – "?$/?2 #ZK?$=: #f(x) = f(x + h) – f(x)

[5555!;?2 $%!&4/555502 2@855559if Q5555M' 6&5555"?)( ixf I$5555#ZK?2 <5555/3% ,) $5555%%3!"9 H "?$/?2:

$ % $ %121

01000

fff

ffxfhxff

"#

" !"#

\2 E>;.?2 </3%0> 5"%$V?2 "!$!(second differences) -;>5D?2 L59)(2 xf#

#ZK?$= W];K/>:

$ %$ %

$ % $ %

12312

2

2

2

)()(2)2(

)()()()2(

)()()(

)()(

ffff

xfhxfhxf

xfhxfhxfhxf

xfhxfxf

xfxf

! "#

!! !"

! ! !"

!#"#

##"#


226

."3?$=> " $A0.% QM'\2 E>;.?2 W";K/ ,3!"(higher differences) SZV!9 H:

$ %

)()(3)2(3)3(

)()(

)()(

22

23

xfhxfhxfhxf

xfhxf

xfxf

!!! !"

# !#"

##"#

S$!>!'>:

$ %)( )(

,2,1,)()(

0

1

xfxf

nxfxf nn

&#

"##"# !

$%K7> 2@2>: )()( 0 ihxfxffy iii !&&&

I$#ZK?2 =$/3 $%%3!"9:

)()(

2

1

122

1

iii

iiii

iii

yyy

yyyy

yyy

!#"##"

! "#

"#

!

!!

!

F$*%?2 ,! '>!R! $%"F'X) 2@2>:

niyx ii ,,1,0,),( "

.M/4!?2 E>;.?2 <$0+ $%%3!"9:

,2,1, "# kk

) G5"+#&#1 ( 5?2&M?fy & 55.M/4!?2 F$5*%?2 &5%' L559 E>;5.?2 _5M/ `57>> H" 6>&55R550 E>;55.?2 6>&55R Q!(difference table) E>;55.?2 F55F4! >)(difference

scheme) L?$/?2 6>&R?2 ,"=" $!3 H– E>;.?2 Q/+4# - B"*?y -$FK!?2. y # 2# 3# 4#

0

1

1 14

15 36

16 50 24

65 60

81 110 24

175 84

256 194 24

369 108

625 302

671

1296


227


"?$/?2 -;>D?2 "!$!\2 E>;.?2 6>&R @4a" S$!>!'>:

0y

0y#

1y 0

2 y#

1y#

03 y#

2y

12 y#

0

4 y#

2y#

13y#

05 y#

3y 2

2 y# 1

4 y#

3y#

23y#

4y 3

2 y#

4y#

5y

W];5555bKX" 65555V!?$=> 2222!)*+&' .($2222)&' 32222%,0% (backward difference operator)

#ZK?$=: )()()( hxfxfxf "'

"!$!\2 E>;.?2 ?$+? A=$c! *";F= QM'\2 E>;.?2 W];K/> SZV!9 H:

)2()(2)()(2 hxfhxfxfxf ! "'

% 55'>!R!? 55".M4?2 E>;55.?2 6>&55R $55% ]>b3 2@d (55%) e55+Z"> ,>3/5509 -$55FK! F$55* $A0.% LN 6>&R?2 2@N ;D$%'– "&&K?2 !"*?2 G"+ ,!– E>;5.?2 6>&5R ;D$%'

F$55*%?2 55'>!R!? 55"!$!\2$A550.% H 55.M/4! ,>553/ $NT$!550) >) ;55D$%K?2 [>55!; ,) 1d H ;555D%K?2 SZV555!91f' ;555D%K?2 >555N0f# (5550.% 555!"*?2 $555!A? U).%$A5550 L?$555/?2 63555c?2> H

,5!> HF$*% `=;) ,! -$FK! '>!R!? ".M4?2 E>;.?2> "!$!\2 E>;.?2 L?>&R LFK" 63c?2,"=/% ">$0/!?2 M=$*/!?2 ;D$%K?2.

0x 0f 0x 0f

0f# 1f'

1x 1f 0

2 f# 1x 1f 2

2 f'

1f# 0

3 f# 2f' 3

3 f'

2x 2f 1

2 f# 2x 2f 3

2 f'

2f# 3f'

3x 3f 3x 3f


228

Q!50" E>;5.M? G5?$V 65!$K! &5R>" 5MV$!/! -;>5D= E>;5.?2 ,' ;"=K/M?> 32%,0%

!"#$%&' .($)&' (central difference operator) #ZK?$= W];KX">:

$ % $ %hxfhxfxf21

21)( !"(

6D+% $A%!>QM':

$ % $ %$ %)()(2)(

22))(()(2

hxfxfhxf

hxfhxfxfxf

! !"

!"" ((((

L5/?2 I$5#ZK?2 J5K= L5M" $5!"9 ;3@5% 5.M/4!?2 E>;5.?2 IZ!$5K! W5";K/ &K=> I$555.";K/ B2&4/5550$= I$555#ZK?2 O@555N I$555=Vd f2 ,@8555= 6A5550"> HIZ!$555K!?2 O@555N $555A**+/

IZ!$K!?2.

.($)&' 45%,0% ,67781 91&' 4,:50&' ;0<:

$#ZK?2 "!$!\2 E>;.?2 6!$K! E*+" "?$/?2 I:

))(()(

)()())()((

))(())((

1 xfxf

xgxfxgxf

xfxf

kk ##"#

#!#"!#

#"# ))

,Z!$K!?2 E*+">(,# *=$0?2 GZV?2 I$#ZKM? MV$!!?2 I$#ZK?2 _?@3.

I5555%$3 2@2>)(xpk 5555R;& ,5555! 5555"&>&+k,$55553> Hka U@5555?2 &5555+?2 65555!$K! >5555NQM' 6!/c"kx H,89:

0)(

! )(

1 "#

"#! xp

hakxp

kk

kkk

k

S$!>!'>:

,)()1( )(k

0i

ihxfi

kxf ikk !**

+

,--.

/ 0"#

"

G"+:

!)(!!iki

ki

k

&**+

,--.

/

,"M!$K!?2 ,"= S$!>!' "?$/?2 #ZK?2 E*+//>'#,:

)()( khxfxf kk !'"#


229


8'"=' 3%,0% /!$01 (Shifting operator) E:

#ZK?$= 6!$K!?2 2@N W;K": )()(

)()(

1 hxfxfE

hxfxEf

"

!"

"?$/?2 I$#ZK?2 g/%/0% W";K/?2 2@N ,!>:

' "

"#"'"'

#"#

#!"

""

1

1

1

1

1

11

E

EEE

EE

E

EEEE

hDeE

EEE

"

"! "# 222 )1(12

G5555"+D 655557$./?2 65555!$K! >5555N (differentiation operator) W];5555bKX" U@5555?2> #ZK?$=:

dxxdf

xDf)(

)( "

55";K/ B2&4/5550$= *=$5550?2 I$555#ZK?2 h$/%/55502 $55%%3!"> 555+2[i2 6555!$K! W I$555=Vi SZV555!9 H #ZK?2: #!"1E

,) :;%:

)()(

)()()(

)()()()1(

xEfhxf

xfhxfxf

xfxfxf

"!"

!!"

#!"#!

:;4) -;>D= >):

kk

kkk

kkk

Eyy

yyy

yyy

""

!"

#!"#!

!

!

1

1

)1(

#ZK?2 I$=Vi>: EE #"#

,) :;%:

12

1

12

1

1

)()()(

)()()(

!!

!

!!

!

!

"

#"#"#

"

"

"#"#

kk

kkk

kk

kk

kkkk

yy

yEyyE

yy

EyEy

yyEyEyE


230

#ZK?2 I$=Vi>:

' " 11E

,) :;%:

kk

kkk

kkk

yEy

yyy

yyy

11

1

)(

)1(

""

"

' "'

#ZK?2 I$=Vi>: 1222 ! "# EE

,) :;%:

kkk

kkk

kkkk

kk

kkkk

yEyyE

yyy

yyyy

yy

yyyy

! "

! "

"

# #"

#"##"#

!!

!!!

!

!

2

2

)()(

)()(

2

12

112

1

12

:;4) *";F= >):

12

1.11..

)1)(1(.

2

2

! "

! "

"##"#

EE

EEEE

EE

#ZK?2 I$=Vi S2;"4)>: hDeE "

,) BMK%:

!111!11!1!"

!"

)(!3

h)(!2

h(x)fhf(x)

h)f(xEf(x)

32xfxf

);>M"$/ _>3.! B2&4/0$= _?@>.(

$ %)(

)(!3

h!2

hhD1

)(!3

h)(!2

hhDf(x)f(x)

3322

3322

xfe

xfDD

xfDxfD

hD"

!!!!"

!!!!"

I$55=Vd L559 $55N$%/=V) L55/?2 -;55"4\2 55#ZK?2 O@55N B2&4/5502 $55%%3!"> 55"?$/?2 55#ZK?2,"M!$K!?2 ,"=D,( :

hDUU &" ,2

sinh2(


231


LM" $!3 _?@>:

$ %2/12/1

2/12/1

2/12/1

2/12/1

!

"2

"

"

"

EE

yEE

yEyE

yyy

k

kk

kkk

(

(

,3?>:

2sinh22/2/ Uee

eeE

UU

UhD

" "2

"" (

:;4) -;>D= >):

2sinh2 1 ( "U

55555"F4?2 IZ!$55555K!?2 J55555K= W55555";K/ M555550M0 L555559>(linear operators) U) IZ!$K!?2P #ZK?2 E*+/ L/?2:

PgPfgfP 3434 !"! )(

,! 63 G"+34 , jU) L5*"*+ I5=$V(real constant) ,5! 653> Hf, g 5?2& U) LF4?2 k2;.?2 Q?d L!/%/(linear space) ] k2;.?2 >N>F F;c?2 E*+" U@?2:

gfFgbfaFgf ,.., 56!26

,! 63 G"+a, b U;$"/42 I=$V.[

F550>/!?2 55!"*?2 655!$K! S2;55"4) Wl;55K%(mean value operator) 7 55#ZK?$= "?$/?2:

89:

;<= !!" **

+

,--.

/**+

,--.

/

2221)( hxfhxfxf7

L559 W55";K/?2 2@55N B2&4/5502 $55%%3!">?2 I$55=VdIZ!$55K!?2 ,"55= ,"/"?$55/?2 ,"55/#ZK

E,,(7:

S1>): $ %1

21 " EE7( ,\ _?@>:

$ %

$ % $ %

$ % $ %

$ % kEE

kyEkEykk

kkkk

kkk

y

yy

yyyy

yyy

1

111

11

2/12/1

21

21

21

21

21

" !

! !!

!

"

"

"

" 77(


232

$"%$V: $ % 14

112 "!

(7

,\ _?@>:

$ % $ %

$ %

2sinh4

2sinh2

2cosh

2cosh

21

21

21

22

2/2/

2/12/12/12/1

UU

U

yhD

kyee

yEEyyy

k

hDhD

kkkk

"2"

"2

"

!"

!"!"

!

((

7

7

$ % $ % 12

2sinh12

cosh4

12/12/12 "!"!2

UU(7

.M/4!?2 IZ!$K!?2 ,"= I$#ZK?2 m4M" L?$/?2 6>&R?2>:


:;5% 5".M4?2 E>;5.?2> 5"!$!\2 E>;5.?2 L?>&5R ,"5= 5#ZK?2 65=# ,! $%"); $!3>!> H 5"[3;!?2 E>;5.?2> 5"!$!\2 E>;5.?2 L?>&R ,"= #ZK?2 ,P2 f2 ,@85= n57// $5A%

E>;.?2 ,! VZV?2 o2>%p? VZV?2 6>2&R?2 ,"= #ZK?2.


233


0f#

5.0f(

1x 1f

02f#

1x 1f

12f(

1f#

03f#

5.1f(

5.13f(

2x 2f

12f#

2x 2f

22f(

2f#

5.2f(

3x 3f 3x 3f

[!;M? =0%?$= (%) e+Z"# ) 5"!$!\2 E>;5.?2 ( I2;5cY!?2 ,>53/(subscripts)

/=$V='6V! H 6.0\ ;&+%! 6q$! F4 ;:

.....,,,, 03

02

00 ffff ###

[555!;M? =5550%?$=>' ) 555".M4?2 E>;555.?2 ( 555/=$V I2;555cY!?2 ,>5553/555=' 6555q$! F5554 ;6V! H QM'\ &'$D:

.....,,,, 33

32

33 ffff '''

[55!;M? =550%?$=>( ) 55"[3;!?2 E>;55.?2 ( 55/=$V I2;55cY!?2 ,>553/55=' H L55*9) F554 ;6V!:

.....,,,, 26

24

22

2 ffff (((

;D%K?2 ,) e+Z"> (0.% H[>5!; 5VZV &5+a= (5"?d ;$5c" ,) ,53!" 6>&R?2 L9 r;55F?2 55"M!' ,55! g/$55%?2 ;55D%K?2 SZV55!901 yy (55"?d ;$55c" ,) ,553!" [>55!;?2 &55+a=

VZV?2: 5.010 ,, yyy ('#

! "!: 5.010

yyy !"!#

!"#$%&: 3

5.24

3

1

3 !"!# yy

... !"#$.

!""#$%& '""()*+,-. /!""012-.3 '""%4+-. /.567"",-. 38 /1,!""2,-. '""9.5: :""2& ;<.3=35(-. >.:+*9!& ?!,@*9A. B-!2* C. ;32& D5:$.


234


!"#$% &%'()*+, -+./)*0%:

>%E-. >)2$ !$$8 F5($:

! ....,2,1,0;)(, ""#$ ixfyx iii

>%0 ;83x '%3!9*, G3"4+-. ;83 !#$%& !,%H /!H!9,-.(step) ) ;%"& '"*&!6-. 'H!"9,-. I8;%*%-!**, ;%*,%0 ( I3!9*h;8 I8 :

......

,...2,,,,2,

21012

020100102

0

yyyyy

hxxhxxxhxxhxx

ihxxi

%%

%% &#&#%#%#

&#

h px

3%x 2%x ' 1%x ' 0x ' 1x ' 2x ''

',%0 :!J%K L3)4,-.y ',%E-. :$M:

11;0 ((%&#$ pphxxx p

;8 F5($py ',%0 NO y !,:$M '&3)4,-.pxx #

0

0

00

)1(

)()()(

y

yE

xfEphxfxfy

p

p

ppp

)&#

#

#&#$*

032

032

3211

321)2)(1(

21)1(

1

yppp

yppppp

p

+,

-./

0&)11

2

3445

6&)11

2

3445

6&)11

2

3445

6&#

1234

56 &)77

%%&)7%&)&#

&)112

3445

6&)11

2

3445

6&)11

2

3445

6&# 0

30

200

321y

py

py

pyyp

'"""012-. PQ"""O B,"""9*312223.+.4% !"222#5$ 6)!22237 1832229 (Newton forward

difference formula) '%-!*-. '012-. B)M ?3RS-. !$$@,% ?6,-!&3 :

!

02

001

0

21)1(

1

)1(

ypp

p

yyEyEy pppp

1234

56 &87

%&8&#

8%### %%%


235


&8112

3445

6&811

2

3445

6&811

2

3445

6&# 0

30

200

321y

py

py

pyyp

'""012-. PQ""O B,""9*31223#5($% !"22#5$ 6)!2237 183229 (Newton backward

difference formula).

1:;<.:

'"",%E-. ;3""@* !,:""$M '""%,!,T. =35""()- ;*3""%$ 'U%""R >:+*""9V*px '""&%50 38 :""$M ;3""@* !,:""$M '""%()+-. =35""(-. 'U%""R >:+*""9* !"",$%& G!""42,-. >%""E-. ?3:""J '""%.:& ;"",

',%0px ?3:J-. '%!#$ L50.

* N"*-. W)* NO3 >O8 X5+8 Y%R W!$O3 !"#$, 5@Q"$3 '"%Z@5,-. =35"(-. >:+*"9D3!J N*U%R3 /5(%K 'U%R.

="#3> 1839 (Everett formula):

&112

3445

6 &&11

2

3445

6 &&&

&112

3445

6 &&11

2

3445

6 &&#

04

02

0

14

12

1

5

2

3

1

5

1

3

1

yq

yq

qy

yp

yp

pyyp

99

99

[%S : pq %#1

'&!*@& 'U%R-. PQO B)M ?3RS-. ;@,%3:

000 )( yeyeyEy pUpUpp ###

'""""012-. >:+*""""9$ >""""62/sinh2 U#9 $3 X3""""0 N""""H ')""""9)9*, G53""""R N""""H W""""(?,!2,)- '%J3Z 9. ;%*%-!*-. ;%*U%R-. B)M ?3RS-. ?6,-!& !$$@,%3:

?$!4% @!+A 1839 (Gauss 1):

&112

3445

6 &&

&112

3445

6 &&11

2

3445

6 &&11

2

3445

6&# %%

04

2/13

02

2/10

4

2

3

1

2

1

1

yp

yp

yp

yp

yyp

9

999


236


137+B$% @!+A 1839 (Gauss 2):

&112

3445

6 &&

&112

3445

6 &&11

2

3445

6&11

2

3445

6&#

04

5.03

02

5.00

4

1

3

1

21

yp

yp

yp

yp

yyp

9

999

!& ?!,@*""91- !""O!$5@Q N""*-. D"",+-. Y%""R-. '""&%@5* ?""6,% N-!""*-. ?@""\-. >.:+*""94!E$& '()*+,-. >%E-. ]%H ?6,V* [%S =35(-. 44+, ;%&% 3O3 =35(-. . 5"9%T. :3,2-!H

=35"(-. ?"6,% N-!"*-. :3",2-.3 B"-3T. =35"(-. ?"6,% ]"%)% IQ"-. :3",2-.3 '"-.:-. >%0 ?6,%.Q""@O3 '""%$!6-. . 38 '"""%.:& :""$M ?!,@*"""91- ;!,:+*""9* ;*3""%$ N*U%"""R ;_""H !""$5@Q !""",@3

:"""J-. '"""%!#$?3 .'"""%5`$-. '"""%S!$-. ;""", ?!,@*"""9A. '"""9.5: N"""H ;.:"""%(* D3!"""J !*U%"""R3 .

?3:J-. aR*$, L50 'R!+ ?!,@*91- Y%R-. :%H8 ;, :2V* /5(%K 'U%R3.

=35(-. >.:+*9!& ?!,@*91- '()*+,-. Y%R-.

!"#$ %"&'Gauss 1

!"#$%& '(#)Gauss 2

*+,!-Everett

!,./%& 01(!" Newton backward

!2#23& 01(!" Newton forward


237


!"#(4-5) :

!"#!" $%&%' $() *+,%!" -./ 0*.1 0!+2!" 34#5!".

x 4 6 8 10

y f(x)! 1 3 8 20

'16 #547 34#5!" "89 :':

) 7( f(4.5) ';" :241% <1= >"#?2@+/ 1'+.

) A( f(9) 1B(?!" :241% <1= >"#?2@+/.

) C( f(6.4) DEB1F <1= >"#?2@+/.

$%&:

G4EB!" 34#5 HI47 :J4K%:

ix iy "! 2" 3"

4 1

2

6 3 3

5 4

8 8 7

12

10 20

) 7( phxxp #$ 0

E+2?%40 $x ) L+*.'!" '1,(! 34#5!" 0M '16 AE67 0944.5.(

41

245.40$$$ %

%

h

xxpp

1'+';" :241% <1= 0M -14.2!+/4$() 3=&% N:

1615

321

2141

14.47

43

41

3.43

41

21)5.4(

$#

##&#$

&&

''

(

)

**

+

,%

''

(

)

**

+

,%

&

''

(

)

**

+

,%

f

) A( E+2?% 100 $x 21

2109 %$$ %p


238


83154

32125

23

21

721

23

21

122120)9(

!!!

"""#

#!!

""#!"

$$%

&''(

)$$%

&''(

)$$%

&''(

)

$$%

&''(

)$$%

&''(

)

f

) ( !"#$% 60 x

7321

)8.0(2.02.18.02.0)4.6(

8.02.011

2.02

64.60

!!!"!!

#! *

" "

" "

f

pq

h

xxp

p

632.3

3321

)2.0(8.08.138.0

!!!"!!

#!#

!"#$: &'(!) "%*+$#,- -./%&'() *'+,&*('0- 123 456% 7:

76.3321

2.02.122.03

2

1

10

25.00

!!!

#!#

$$%

&''(

) ##$$

%

&''(

)# " y

py

pyy p ++

&'(!) "*89 -.+/&) *'+, "%():#;:

76.3321

)8.0(2.052.03

210

25.00

!!"!

#!#

$$%

&''(

)#$$

%

&''(

)# y

py

pyy p ++

- !"""# $% &"""'(! )*+,"""-. /0!"""# 1+2"""3 4565*"""3,7 8"""9:+ /""";<9=>:+ ?5>@9"""3. !"# 2% 45A*!B 5>!C D5C53>:+ /!2539> E5FG>:+ F5H*:+ $2@9 $% 5A!C F,9I! . 89:+2 890!"#@ F5"H*:+ $!B D5C53>:+ J2539 5A!C F,9I! 89:+2 K2,;:+ L+7=935B ?5>@93.+

AG!>- ,B9GM9 4/!;<=:+ K2,;:+2 /!>5>N+ K2,;<: $92!* /!727':+ ?!O>9: /;<9=> P+,2# 5)E7""!'2:+ ( /!:5>@9""3.+)E5""FG>:+ F5""H*:+ /""Q2>->B ,"">9 8""9:+ J%( 5A""3;* . $R""C S:T"":2

U"F=:+ /0!#E +T"6 V"FGM!2 4W,"=N E,2"# $"> X"<9=9 . ,2"#:+ YT"6 $"> J% 8"C)*+,-. /!727' /3+,7 7*Q 56,@T KB3 89:+2 4/!:59:+ /Z(G:5B UF=:+:

)!1()(

.)()()()()1(

10 !!!"

nf

xxxxxxxEn

nn

#


239


!"#(6-4):

x 0.6 0.7 0.8 0.9 1.0

sin x 0.564642 0.644218 0.717356 0.783327 0.841471

/>!Z ,!7H9: /!>5>N+ K2,;:+ /0!# L7=93+sin x 7*Qx = 0.63 ./-!9*:+ /Z7 77'2.

$$%&': 5"">@ K2,"";:+ ?27""- $[2""@* 2""68""<! 5"">!C $!""B> . /"">(G:+ DCT"" M' 7""HC /:2A""3<:2K2,;:+ $> /!,IG:+.

x sin x $ 2$ 3$ 4$

0.6 0.564642

79576

0.7 0.644218 -6438

73138 -729

0.8 0.717356 -7167 69

65971 -660

0.9 0.783327 -7827

58144

1.0 0.841471

/>!H:+ $N P+,&*x = 0.63 T=U* 4?27-:+ /!+7B \,Z ]H96.00 "x . $% 5>B2h

= 0.1 263.0"% pxx $Q /!>5>N+ K2,;:+ /0!# 8C 2G* S:T<C 4p />!H:5B:

3.01.0

6.063.00"!"

!"

h

xxp

p

S:T@2n = 4 )F5H* _>= VFG>:+ ?27-:+ 8C 5*7*Q:(

!&&!&

& "%'

)006438.0(21

)7.0(3.0

079576.3.0564642.063.0sinpy

)000729.0(321

)7.1()7.0(3.0 !&

&&!!

589145.0

)000069.0(4321

)7.2()7.1()7.0(3.0

"

&&&&

!!!

/!,IQ L5Z,% /93: S:T2.

V<Q ?#'* UF=:+ ,!7H9 /0!# L+7=935B2:

!5)(

.)()()()5(

404

#fxxxxxE !!"

$% `!'2: f(x) = sin x

!"#!" $%& '()% '!*+!" ,-#.!"xxf sin)( (*/0!" 1)2 #03.


240


S:T<C:

6-

4

)5(

)5(

10 0.3

!51.)0.163.0()9.063.0(

).8.063.0()7.063.0()6.063.0()63.0(

1)(

cos)(

&(

!!

!!!)*

)*

"

E

xf

xxf

U"""F=:+ ,5"""I9*+ $"""> P5"""#5= P5"""Q2* &"""'(* K2,""";:+ ?27"""- ,"""#5*Q \5"""3' 7"""*QD5FF=>:+ 2% ?2+7-:+ YT6 a!>! .b2"*:+ +T"6 c"d2! 8:5"9:+ ?5O>:+2 . ,";*^ 5*!7": $%

J25""39 `""!' E7""'+2 +7""Q F5""H*:+ ]""!>- 7""*Q P+,;""# J25""39 /"":+7 + . \""3'* $% 5""**@>!-8<! 5>@ K2,;:+ ?27:

8"O<O> ?@"I 8C ,I9*! UF=:+ $% &'(*(triangular pattern) /Q,"3B ,"B@!2 .

S:T"""@2– D+,5"""Ie+ ?5""">6RB– $!7"""':+ D+T D(>5"""G> ,2"""A& &"""'(*(binomial

coefficients) ?27"""-:+ 8""""C /""";<9=>:+ E7""">QN+ 8"""C . $R""""C /"""B9,:+ /"""!:5Q K2,"""";:+ 8"""C25"""F=% ,"""!f P5"""32><> P+,"""!09 E,"""!09> 2% /"""B<H9> L!"""Z ?@"""I 8"""C ,"""A&9 E75"""9G>:+ \"""!,H9:+ 1

L&9"*>(irregular fluctuations) . ?27"- L+7=9"35B E,"!B@:+ 15"F=N+ X5"I9@+ 5"**@>!25"**56T% 8"C /HB5"3:+ E,2"#:+ ,5d'93+ ]> 4 K2,;:+ . 5"> ]"d2> 8"C U"F=:+ X"I9@M!C

/"":+7:+ L!""Z ?27""- 8""C)+7:+ L!""Z W7""'g J%/"": ( 8""O<O>:+ ?@""I:+ /""&'(> K""!,F $""Q–

Error Propagation QD)« —UA²½«


241


]d2>:+ +T6 h3%, JT:+– K2,";:+ L!Z L5&9*+ L7G*! JT:+2):5Q /"#5="!/"B9,:+ / ( 8"C +T"""6 i,5"""= E7""">QN+ 1+a"""-% 8"""C K2,""";:+ L!"""Z 8Z5"""B L&9"""*9 5""">*!B h"""<=+7 E7""">QN+ 1+a"""-%

8O<O>:+ ?@I:+) LZ, /:U3>:+ P(O> ,&*+14-4.( =% ,5I9*+ E,2# c!d29 $@>!2 15F \!,H9:+(propagation of round-off errors) 5*"d,9C+ JT:+2 48:59:+ K2,;:+ FF=>B

D+,!09 ,B@% h!C(variations) 2">* ?7G> %23% 8:59:5B2 4$!9GB599> $!9>!Z $!B /*@>>(worst rate of growth) \!,H9:+ 15F=N.

K2,;:+ 15F=N+

16+ -4+ + 8+ -2+ ,

-16+ 4+ !+ -8+ 2+ ,

16+ -4+ + 8+ -2+ ,

-16+ 4+ !+ -8+ 2+ ,

16+ -4+ + 8+ -2+ ,

-16+ 4+ !+

5*B""3' 5""><@ U""F=:+ XQ5""d9! 7""Z /""*@>> /"":5' %2""3% 8""C h""*% W,""* ?27""-:+ $""> L"Z, 72">G:+ V":g ?"#* $!"' S:T"B2 4?27"-<: 7"!7- 72>Q ?5=7g ]> J% 4P+7!7- P5Z j,kCj

J% K2,"";:+ \5""3' 8""C\5""3'j$ D""0!Z $2""@9+.2 j `""!'+ ?""O>!>!Z/ 8<#N+ UF=:+.

()*+&' !#*,)-':

P5HB5"""3 5"""*,@T 5""">@– )*+,"""-. /"""!727' _,7"""* $"""'*2– ?5>@9"""3.+ /:U"""3> $R"""C D5""*5!B:+ ^GB"": P.27""- 5""*!FQM% 7""Z 5""**% 8""*G9 8""3@G:+x, y 5""*> \""<MF L""O "">!Z 75""-!g/

)(x />2<G> />!H: /<B5H>yy " E5FG>:+ D5*5!B:+ ?27- 8C D3!:.

!"#)7-4( :

/"""!5A*:+ /"""FH* 8!O+7"""'g K2,""";:+ F"""F=>B /*5G9"""3.5B 7"""-2% 8:5"""9:+ ?27"""-:+ $"""> /:+7<: W,0#:+y.


242


x 0.6 0.65 0.7 0.75

y(x) 0.6221 0.6155 0.6138 0.6170

%&':

x y(x) -, -2, -3,

410!& 410!&

0.6 0.6221

-66

0.65 0.6155 49

-17 0

0.70 0.6138 49

32

0.75 0.6170

/:+7:+ $% 5*: $!B9! E5FG>:+ D5*5!B:+ $>y $!9FH* ,=l $!B W,0# /!5A* 5A: .

Q ?""#'*C /!:5>@9""3.+ /""!727':+ V""<y(x) ] 2%y(p)[ K2,"";<: $92""!* /0!""# L+7=9""35B,;"""#:5B 5A9H9"""I> J25"""3* L"""O 4/"""!;<=:+0)( ". xy /""">!Z V"""<Q ?2"""#'<:pxx % 8"""9:+

/!5A*:+ YT6 />!Z 7-2* 5A*>2 4/:+7<: W,0#:+ /!5A*:+ 567*Q)( pp xyy %.

92!* /0!#,5B9Q5B /!:59:+ /!727':+ 5*!FG9 /!;<=:+ K2,;<: $70.00 "x:

6120426.0

7423.0

05.08469.70.0

8469.00

)12(2

0049.0017.0

104921

)1(10176138.0

8469.min

0

44

""

"

& " "*

"*".

! !".

&&&!

&!"

"

!!

pp

p

p

p

yy

phxxp

py

py

pppy

K2,";:+ L!"Z D"*5@ +Tg h"*% 5> m/:+7 D5*5!B $> K2,;:+ ?27- $!2@9 7*Q &'(M! E7>QN+ 7'% 8C)( n$ 7:+ YT"6 ?"!O>9 $"@>! h"*% 8"*G! +TAC 4 P5B!,H9 /9B5O /"!727'B /":+

/"""!*2*:+ /"""-,7:+ $""">)8:5"""9:+ 72""">G:+ 8"""C K2,""";:+ L!"""Z $% `"""!'1 $n P+,5;"""#% $2@9"""3 P5B!,H9(?5>@93.+ ,0: /B3*:5B /:+7<: P+7!- P5B!,H9 $2@9 /!727':+ YT62 4.

!"#)8-4(:

!:5>@9""3.+ /""!727':+ /""-,7 ,[7""Z2 4/""!:59:+ D5""*5!B<: K2,"";:+ ?27""- $[2""@ 8""9:+ //!2*G>:+ L5Z,N+ $> VkFG>:+ 77G<: /'!'# /!:5>@93+ P5>!Z 8FG9.


243


x 1.0 2.0 3.0 4.0 5.0

f(x) 1.5709 1.5713 1.5719 1.5727 1.5738

6.0 7.0 8.0 9.0

1.5751 1.5767 1.5785 1.5805

%&':

x f(x) $f $2f $3f

1.0 1.5709

4

2.0 1.5713 2

6 0

3.0 1.5719 2

8 1

4.0 1.5727 3

11 -1

5.0 1.5738 2

13 1

6.0 1.5751 3

16 -1

7.0 1.5767 2

18 0

8.0 1.5785 2

20

9.0 1.5805

72""">G:+ L!"""Z $% &"""'(*$2 $"""> /!:5>@9"""3+ /"""!727' $R"""C S:T""":2 4P5"""B!,H9 /"""9B5O

<#9 /!*5O:+ /-,7:+ /:+7:+ \!,H9: cf.

.!/010%&!2 (3/45,&' 6/78,&' Piecewise Polynomial Approximation

L+7=9"""""35B /"""""H<0> D+,"""""9C V"""""<Q /"""""!,5!9=.+ ?+27""""":+ \"""""!,H9 KB"""""3 5""""">!C 5*"""""3,7D5!727':+ . 2E,"!O@ D.5"' 8"C c<"#9 /"H!,F:+ YT"6 $2"@9 D.5"':+ ^"GB 8"C $"@:2 4

5"""""">% D5""""""BHQ S5""""""*6L 9<: c<""""""#9 . 7""""""Z2 56T""""""!;*9 /""""""!BTBT9:+ /""""""G!BF:+ \B""""""3B K""""""!BF(oscillatory nature) /"!:5G:+ /"-,7:+ D+2T D5!727"'<:(high-degree polynomials)

P5BT"BT9 $% /!"#5=:+ \B3B2(fluctuation) E,"9;:+ $"> ,!0"# 1a"- 7"*Q(interval) 7"Zh<@ W7>:+ V<Q E,!B@ D5BTBT9 V:g J7n!.


244


?5>@9""3. ]""B99 $% $""@>! W,""=% /""H!,F S5""*6 L""[3kHM* 2% o[a""-* $% 8""62 ?+27"":+ /"""!pa-:+ D+,"""9;:+ $"""> /"""Q2>-> V""":g /"""!<@:+ E,"""9;:+(subintervals) 7"""-2* L"""O 4 /"""!727'

/""""!pa- E,""""9C ?""""@: /!:5>@9""""3+ 5A""""dGB $""""Q /"""";<9=> D5!727""""':+ YT""""6 $2""""@9 P5"""">2>Q2 4^GB:+ .T"62 4/";<9=>:+ /"!pa-:+ D+,"9;:+ V"<Q /";<9=> D5!727'B /:+7:+ \[,H* 5**% J% +

D5!727':5B 8p!a-9:+ \!,H9:+ V>3! \!,H9:+.

\""""!,H9:+ +T""""6 b+2""""*% F""""3B%($$$$3/45,&' ($$$$9:&' !#*,$$$$)-' (piecewise linear

interpolation) D5*5!B:+ F5H* $> /GB599> /Q2>-> $!B />!H93> P5F2F= ?#* `!'

/ 0/ 0 / 0/ 0 / 0/ 01 2nn xfxxfxxfx ,,,,,, 1100

?@I:5B $!B> 26 5>@.

!2 ?+27:+ /3+,7 7*Q /F3B> E,2#B ?5>@93(: /H!,F:+ YT6 ]B99 $% $@>/!O<O>:+2 /!>9!,5f2<:+.

D5!727"""""""':5B S:T"""""""@2 /"""""""!F=:+ ?+27""""""":+ YT"""""""AB ?5>@9"""""""3.+ \"""""""!Q2 /"""""""!G!B,9:+(quadratic polynomials) !"""#$ %"""$& '(""")$ * !"""$$+ ,"""- ."""/0!11( !"""#$ 23"""4 '5"""0

."""""""""""""""5!6$78(endpoints) ."""""""""""""""59:;78 <8="""""""""""""""1>?7(subintervals) .5?""""""""""""""")!>178 @"""""""""""""""A# BCB1(differentiability) ."BDE? BD ,+ F!"D?( .57!(G1D*8 .78%78 ',G1 '+ H!5D%$- I$/1 J178, K

(smooth) !"""#$78 LM"""- %"""$& . '"""( H!"""0,? ( ="""N78 8M"""- ',"""G5 O%!"""&, ,="""N78 ."""5C!$.7P""D(?7 .""59!5:5>78.""?0!Q ',""G1 '+ R"";5 .57!(G1""D*8 .""78%78 '+ J""$/5 8M""-, K S"")!>1?7

=8=(1D!0(continuously differentiable).

'+ T="">$],[2 baCf '51#1""N(78 '""( 3""G '+ U+ff !!!, .?""V1(, O%,"";,(O=1>78 JW],[ ba'+ T=>$, K:


245


bxxxa n "###" 10

X:""""";1 ,"""""-(partition) O="""""1>78 LM"""""67],[ ba ."""""78%78 '+, Kg(x) ."""""5 Y78 ."""""5%,%C78 J"""""-S(G1D1 J178 .595:;178 f !#$78 %$&nixi ,,1,0; " . ."78%78 '+ <!"04Z 'G(5"Wg(x)

.78%?7 H!05=#1 J /1f(x) . #$ U+ %$&],[ bax J7!178 I?&[8 %C78 :,!;15 * P Y0:

,)(max.)()(8

2$

$fxgxfE h !!%&'

25C: ( ) ,max 110

iini

xxh &" *&%%

'+ U+h .59:; O=1W S, + J- + ,ba, $

S!(G1"D*8 !#$ %%& O%!5:0 P Y?7 I?&[8 %C78 .(5Q =5\V1 'G(5 ]$+ ^)8, .(5Q =5\V1,h =DG$(78 Y78 '+ *Z(broken line) ."78%78 S4(5 UM78g %"/_1 UM"78,I7,[8 ]1#1N(g! .?V1( =5` .78%(discontinuous) ]1D3D S#1, L=!DG$8 %8%:5D.

S!(G1"D*8 @"5= '"& .57!(G1"D*8 ."78%78 .D3D, .5?)!>1 .?GN( SC 'G(5,.505/G178 .C5=N7!0 S!(G1D*!0 I(D5 !( ,+ K.505/G178 <!5%,%C7!0.

'51 #$ SG '50 .505/G1 <!5%,%C a%Y1D5 UM78 <!5%,%C7!0 J95:;178 R5=#178 .(%Y1""D(78 @="" 78 S"")W+ '""( %""/5, K .""505/G178 .C5=""N7!0 H*!(G1""D8 I(""D5 '51/0!""11(

a,578 . I?& S(1N1 .(!/78 .505/G178 .5%,%C78 '[ H8=b$,c .5W!G .$,=( !$5%?W K<08,47!0 S!(G1D*8 .#5= JW .?0!Q .57!(G1D*8 .5%,%C78 ',G1 '+ '()$7 .505/G178 .C5=N

'+ d""7M I""7Z .W!"")e!0, K.""5%,%C78 !""673Y %"";,1 J""178 O=""1>78 I""?& =8=(1""D!0 S"")!>1?7O="1>78 LM"- I"?& .?"V1( d7MG .5$!478 .#1N(78 ',G1 . ."505/G178 .C5="N78 '5,"G1 '+ *Z.57!(G1"""D*8 ."""5%,%C78 <!#1"""N( @""">11 '+ T="""1>5 * %"""$& I"""1C K."""78%78 <!#1"""N( f"""(

%B#_/78 I(D1 J178, O! /(78 !#$78(nodes).

!"#$%&'() !*!+%&") ,!-.") /!-+& (Cubic Spline Interpolant):

.78% !$5 &_+ 8MZf O=1>78 I?& .Wg=B/_([a, b]%8%&+ .&,(;(, K

bxxxxa n "###" 210

Cubic Spline Interpolation WO³OFJ²�« W×¹dA�UÐ ‰ULJ²Ýô«


246


""505/G178 .C5=""N78 'h"W K H8%""B#_& I("D1 .57!(G1"D*8 .S .""78%?7f @""#C1 .""78% !"6$P0 ig=""B/_1.57!178 ,=N78:

S .505/G1 .5%,%C– :(=7!0 !67 :(=$jS - .59:; O=1W SG I?&

1,,2,1,0;],[ 1 &"* njxx jj

S .78%78 a5Q f( !6(5Q @0! 11f '+ U+ K %#/78 SG %$&:

njxfxS jj ,,2,1,0;)()( ""

O%#& U+ %$& '5159:; '51=1W I#1?( J-:

+( '+ U+ K'51=1>78 '51!- I?& '51505/G178 '515%,%C78 !1(5Q @0! 11:

2,,2,1,0;)()( 111 &"" *** njxSxS jjjj

R( '+ U+ K'51505/G178 '515%,%C78 '51!- J1#1N( !1(5Q @0! 11 !(G:

2,,2,1,0;)()( 111 &"!"! *** njxSxS jjjj

j( '515%,%""C78 '51!""- '""( S""G7 .""5$!478 .#1""N(78 !""1(5Q @0!"" 11 d7M""G, U+ K'+: 2,,2,1,0;)()( 111 &"!!"!! *** njxSxS jjjj

%,%C78 ,=N '( '5157!178 '51&,(;(78 '( O%C8, @#C11(boundary conditions):

+( .C5=N?7 .5$!478 .#1N(78 a%/$1S %$& O=5Y[8, I7,[8 '51%#/78 '( SG K'+ U+: 0)()( 0 "!!"!!

nxSxS

=N78 8M- I(D5, : =C78 %C78(free boundary).

R( .C5=N78 .#1N( !1(5Q k,!D11S ."78%78 .#1"N(,f '51%"#/78 '"( S"G %"$&'+ U+ KO=5Y[8, I7,[8:

)()(;)()( 00 nn xfxSxfxS !"!!"!

=N78 8M- I(D5, : <g0B4_(78 %C78(clamped boundary).

,=""""N0 .""""505/G178 ^98=""""N78 i""""5=/1 '""""G(5 ]""""$+ b""""C35, '+ *Z k=""""Y+ %,%""""CJ>G1 H!#0!D O=,GM(78 ,=N78 !$- ];!1C$ !(7 .0D$7!0.

/!-+& : +!*0") ,!-.") (Natural Spline)

U+ K="""C78 %""C78 ="""N @""#C1 J"""178 .""505/G178 .C5="""N78 J""- ."""5/50 78 .C5=""N78 =N78)4 (– .505/G178 .C5=N78 i5=/1 JW H!#0!D =,GM(78 +.

- 1

- 2

-3

- 4


247


/!-+& : &1*234$") ,!-.") (Clamped Spline)

J""- .""1g0B4_(78 .C5=""N78 U+ K <""04(78 %""C78 =""N @""#C1 J""178 .""505/G178 .C5=""N78 =N78)4 (– .505/G178 .C5=N78 i5=/1 JW H!#0!D =,GM(78 R.

S"5, R5")Q LM"YP5 UM"78 SG"N78 ]0"N5 ."5/50 78 .C5=N78 I$C$( '+ bC35, '=() .(?G ]5$/1 !( ,- 8M-,spline ( '"( ." #$ S"G0 L= l="(_$ 2"5C0 ]"5W aGC1"$ !(%"$&

078 !#$<!$!5:

( ) ( ) ( ))(,,,)(,,)(, 1100 nn xfxxfxxfx

H!""(,(&, S(1""N1 !""6$[ d""7M, @%+ <!""05=#1 I""7Z U%m""1 <""g0B4_(78 %""C78 ,=""N 'h""W.78%78 '& =4G+ <!(,?/( I?& . '+ R";5 %,%"C78 ,="N '"( n,$78 8M- @#C17 ]$+ *Z

'51(5#78 '51!67 H!#5Q% H!05=#1 ,+ .5!6$78 J1 #$ %$& .#1N(78 J(5Q !(Z i=/$.

"VC?7, O!"" /( .""78%7 .57!(G1"D*8 .""505/G178 .C5=""N78 I""?& S,f @""50 1 !""$$G(5 K78 ."""""5%,%C78 I"""""?& i"""""5=/178 J"""""W O=,GM"""""(78 ,="""""N78 <"""""08,478 '5"""""5/17 ."""""57!178 ."""""505/G1

jjjj dcba ,,,:

1,,1,0

;)()()()((*) 32

&"

&*&*&*"

nj

xxdxxcxxbaxS jjjjjjjj

(56: =N78 '+ ^)8,)1( .\5V7 !$=!51Y!0 @#C1(jS .505/G1 .5%,%C !6$+ 25C.

#!7#3: =N78 @50 1)2( I7Z U%m5:

1,,1,0;)()( &""" njxfaxS jjjj

-,<08,478 %%C1 .Q3/78 LMja.

#3"#3:

) +( =N78 @50 1)3( – + ] .;51$ a8%Y1D8 f() H!5$!4([ I7Z U%m5:

1,,1,0;)1( 321 &"***"* njhdhchbaa jjjjjjjj

25C:

) H!>5=/1( jjj xxh &' *1

) H!>5=/1( )( nn xfa '

) R( .?)!>(0)(xS j - .Q3/78 JW .W=/(78 (*)– I7Z .0D$7!0x I?& SVC$:


248


1,,1,0;)(

)(3)(2)( 2

&""!-

&*&*"!

njbxS

xxdxxcbxS

jjj

jjjjjj

=N78 @50 10,)3( – I?& SVC$ R:

1,,2,1,0;32)2( 21 &"**"* njhdhcbb jjjjjj

25C:

) H!>5=/1( )( nn xSb !'

) j( .?)!>(0)(xS j! I7Z .0D$7!0x I?& SVC$:

jjj

jjjj

cxS

xxdcxS

2)(

)(62)(

"!!-

&*"!!

=N78 @50 10,)3( – I?& SVC$ j:

1,,1,0;3)3(

622

1

1

&"*"-

*"

*

*

njhdcc

hdcc

jjjj

jjjj

25C:

) H!>5=/1( )(2

1nn xSc !!'

<!""Q3/78(1), (2), (3) <""08,478 '5""0 0=""1jjjj dcba ,,, '+ 2""5C, <""08,478

ja <08,478 .5#0 %!;5Z S,!C$W .(,?/(.

<"""08,478 H*,+ %;,$"""Djc <"""08,478 0="""1 ."""Q3& %!""";5Z @"""5= '"""&jjjj dcba ,,,

'( SG iMC0 d7M,jj db , '( 23478 <!Q3/78(1), (2), (3) .

)3()()3( 13

1 !&". * jjjhj ccd

JW .;51$78 LM- T5,/10,(1) I?& SVC$:

/0

123

4&&&&". ** )(

3)(1)1( 1

22

1 jjj

jjjjj

j cch

hcaah

b

'+ U+:

)1()2(3

)(111

!*&&" ** jjj

jjj

j cch

aah

b


249


<""08,4?7 '5""1Q3/78 '51!""- T5,""/10,jj db , .""Q3/78 J""W(2) ) f""), %""/0j-1

'( H*%0j (I?& SVC$:

"*&& ** )2(3

)(111 jj

jjj

jcc

haa

h

211

1

1111

1

)(3

3

2)2(3

)(1

1

&&&

&&&&

&

&*

***&&&

jjjj

jjjjj

jjj

hcch

hccch

aah

,I7Z =V1Y1 .Q3/78 LM-:

)2(1,,2,1;)(3)(3

)(2

11

1

1111

!&"&&&"

"***

&&

*

*&&&

njaah

aah

chchhch

jjj

jjj

jjjjjjj

'( J Y a!b$ 8M-,(n – 1) JW .7%!/((n + 1) S,6;(:

njc j ,,1,0; "

<08,478 a5Q '55/1, a!b$78 8M- SC %/0,jc<08,478 I?& SVC$ K:

1,,1,0; &" njb j

.Q3/78 '()1( !<08,478, K:

1,,1,0; &" njd j

.Q3/78 '()3( ! ..505/G178 <!5%,%C78 '5/$ d7M0,:

1,,1,0; &" njS j

i5=/178 a8%Y1D!0.(*)

(;( '( U+ <##C1 8MZ ]$+ I7Z =5N1 .57!178 .5=b$78, , %,%C78 JB o=BN J1&)+ (

,+)R ( =""""N78 J""""W '51=,GM""""(78)4 (78 'h""""W K.""""505/G178 .C5=""""N78 i""""5=/1 J""""W a!""""b$ S5-!;(78 JW =,GM(78 J Y78jc %5C, SC ]7 ',G5.

!-87:

<$!G 8MZf O=1>78 I?& .Wg=B/_( .78%[a, b].78%78 LM- 'hW K:

9(56: O%""5C, .""5/50 .57!(G1""D8 .C5=""N !""67 ',""G5(unique natural spline

interpolant)=C78 %C78 JB o=BN @#C1 O%5C, .57!(G1D8 .C5=N U+ K:

0)()( "!!"!! bSaS


250


!" #:

! !" #$%&$'()*+,-". /0". 1234 5607 8/&0$ 9&" -%7:. 90&34 ;":

)()(;)()( bfbSafaS ! !

!"#$%&:

#&7"/ <<=- <<>&2=7 <<-!&"? 3 <<4-". /$/<<0". 12@3<<*4 17A$<<-B- #<<- C <<&D #D ')+><<: E&F B-". 1G #&7&G H?jc ;"I) >&/" J)K&G L(n + 1) 1G 9"/ =-(n + 1) E$!B-

)( jc /&0$ E0 M" N7 >". 12O". P Q>". .IF #D$ L.

#D R3> 9&3Q>". E)S 83$%I-". NT 7>".$ U$-3". P./O7: ):

CV$D:

30". /0". 12@3*4 5&)27)9&=&)2". 90&34".:(

) W&3=7 #-nc(

00)(62

0)(0)(

)(62)(

)()()()(

00)(

)(

00000

000

0000

300

2000000

21

!"!#$

"! "! %

#$! %

#$#$#$!

!"! %

!

cxxdc

xSxS

xxdcxS

xxdxxcxxbaxS

cxS

xSc

nn

nn

#D XB& 9&=&)2". 90&34". #D YD#&234". 5607:

0,00 !! ncc

'V/ <<<=-". 9<<<A$-B- Z<<<- # <<<7"/ =-". # <<<7 F)2( C <<<&2O C <<<- Q> /<<<"$7 bAx !

[&0:

!

"

#######

$

%

&&'&&

'

'

(

1000

1)12(2200

0

2)21(210

1)10(20

0001

nhhnnhnh

hhhh

hhhh

A


251


!

"

######

$

%

&&&

&&&

'

&&&&&

0

)()(

)()(

0

b

212

311

3

010

3121

3

nnnhnnnh

hh

aaaa

aaaa

!

"

####

$

%

'

nc

c

c

1

0

x

!"#$%&'A ()*+#,% *+-. /+0! /&1+2&13" +-456&' +-*56&' (*5-7&' 5*8 96:2 /5;&' <1=,&'"Ax = b )-:" >: ?&.

!"#!$: @3AB%&' ):&' /5C*D8 9-352E:

()')()') )()()()( 0 afxSafaS

0)( baf ')

()')()') )()()()( bfxSbfbS n

) F-*42 G%nb( nbbf ') )(

)2(3

)(1)( 010

010

)1(

0 cch

aah

baf *&&')+)

)()(3)(32 010

1000 Iafaah

chch )&&'*(

H&IJ": 2

11111

)2(32)( &&&&& **')

nnnnnn hdhcbbbf

)(

2)2(3

)(1

11

1111

11

)3(&)1(

&&

&&&&

&&

)

&*

**&&

nnn

nnnnn

nnn

cch

hccch

aah

)()(3)(32 11

111 IIaah

bfchch nnn

nnnn &&

&&& &&)'*(

G12&)14%&'),( III 4%&' K"%L% M% EN)1)2( ) O1-5; O1%1=, )&"2Ax = b P-::

!

"

#######

$

%

!

!

!

"

1000

1)12(2200

0

2)21(210

1)10(20

0001

nhhnnhnh

hhhh

hhhh

A


252


!

"

######

$

%

&&'

&&&

&&&

'&&

(

&&

&&&&&

)()(3

)()(

)()(

)(3)(

b

11

3

212

311

3

010

3121

3

010

3

nnnh

nnnhnnnh

hh

h

aabf

aaaa

aaaa

afaa

!

"

####

$

%

(

nc

c

c

1

0

x

!"#$$$$%&'(A )*$$$$+,'( (-$$$$.! /'*$$$$0'*1" $$$$2345'( $$$$2645'( 7642$$$$8'( 46$$$$9 :$$$$5;0 )25' <2;" =; >' /4?'(nccc ,,, 10 !.

! @$$?A,2'*&B0$$8C( $$2123B0'( ;26$$9'( D$$& =$$B E$$AF ="$$%;'( G("$$4? /$$A2 *$$&2 &"A3& '(<' 7<2;"'( 23214'($2;"'( 2'*&B08C( 2123B0'( ;269'(" H /4I6$J9 :$5;0 /$0'( 7<

G$$1K&'( <$$;'( /$$4?'( )*$$+,'( DL $$+;M& N$$& H– =$$2O*P&'( /$$!jc - >$$2AF *,A$$%; Q-$$'('( D$$$& =$$$B /$$$! $$$2645'( /$$$KMK )*$$$+, D20'*$$$; =$$$2A;0 $$$52641 >A;,$$$8" H(factorization) (-$$$O

:1*8'( =%#'( /! *O*,6B- /0'(" H )+,'( D& R",'()S'*K'( =%#'(.(

$$'(<'( DL T6$$#,f <(<$$FU( <$$,F $$! V6J3W&nxxx ))) !10 <*$$P2X Y"$$A4&'(" H 2123B0'( ;269'(S D2469'( :5;0 /0'(:

0)()( 0 ('(''nxSxS

NZ:

nixfa ii ,,1,0;)( !((

Y8;(:

1,,1,0;1 &(&( * nixxh iii !

Y8;(:

+ ,

1,,2,1

;)(3

1

11111

&(

*&&(&

&&*&*

ni

hhhaxxaha

ii

iiiiiiii

!

-

Natural Cubic Spline Algorithm WOFO³D�« WO³OFJ²�« W×¹dA�« WO�“—«uš

- 1

- 2

-3


253


) 2645'( /KMK [*24? [*&*+, =;0 2'*0'( G("4?'((

NZ:

0,0,11 000 ((( z.

Y8;()25A' /A2 *&: 1,,2,1 &( ni !

,)(21 1111 &&&* &&( iiiii hxx .

iii h 1/(.

iiiii zhz 1/)( 11 &&&( -

NZ:

nnnn zcz ((( ,0,11

)25A' /A2 *& Y8;(: 0,,2,1 !&&( nnj

1*&( jjjj czc .

3/)2(/)( 11 jjjjjjj cchhaab *&&( **

)3/()( 1 jjjj hccd &( *

9'( 2123B0'( ;26 23214'(/O 1"A4&'(:

1,,1,0

;)()()(

],[)()(

32

1

&(

&*&*&*(

/( *

nj

xxdxxcxxba

xxxxSxS

jjjjjjj

jjj

!

!

* * *

!"#)9-4(:

/'*0'( ="<P'( G*,*21 :\!"0 /0'( 23214'( 2123B0'( ;269'( D\"B:

x 1 2 3

f(x) 0 1 4

$%&:

1

4)(

1)(

0)(

2,3,2,1

10

22

11

00

210

((

((

((

((

((((

hh

xfa

xfa

xfa

nxxx

+ ,

6)]13(14[13

)(31002102

101

(&0&(

*&&( haxxahahh

-

0,0,01 000 ((( z.

-4

-5

-6

-7

-8


254


4/11/ 111 h!

2/34/)06(1/)( 10011 " " zhz #

0,0,11 222 cz

2/32111 " czc !

23/)30.(11/)14(

3/)2(/)( 1211121

$""

""" cchhaab

0

2/13/)2/30()3/()(

1000

1121

"

" " "

czc

hccd

!

2/13/)02/3.(11/)01(

3/)2(/)( 0100010

$""

$"" cchhaab

2/13/)02/3()3/()( 0010 " " hccd

3

300

2000000

)1(21)1(

21

)()()()(

"$"

"$"$"$

xx

xxdxxcxxbaxS

32

311

2111111

)2(21)2(

23)2(21

)()()()(

"""$"$

"$"$"$

xxx

xxdxxcxxbaxS

]3,2[)2(21)2(

23)2(21

]2,1[])1()1[(21

)(32

3

%"""$"$

%"$" &

xxxx

xxxxS

!!!!"#"$%&'( )"*!!!!+'( ,!!!!-..)& /0 1!!!!2" 3!!!!&'( 45*!!!+'( !!!!6,% /!!!!7 8!!!!.)&'( 9-!!!!:'( /!!!75 "$"#4'( :/7 9%610 , SS ;'<%5 = "#"$%& ">5>):

S(1) = 0 = f(1)

S(2) = 1 = f(2)

S(3) = 4 = f(3)

?,@"05:

3)2()2(

2)2()2(

1)2()2(

01

01

01

!! !!

! !

SS

SS

SS

?(*"A05:

0)3(,0)1( !! ! SS

4)13(2)(21 00021 " "" #hxx


255


Clamped Cubic Spline Algorithm W²³¦*« WO³OFJ²�« W×¹dA�« WO�“«—uš

!!!!'(>'( /0 B*!!!!CDf >(>!!!!EF( >!!!!DE !!!!6 G*H$I7nxxx $$$ 10 !!!!J47'(5 >,!!!!2"K 15 "#"$%&'( )"*+'(S /"4*+'( 8.)& 3&'(: )()(,)()( 00 nn xfxSxfxS ! !! !

L@:

)(;)(0

,,1,0;)(

0 n

ii

xfFDNxfFD

nixfa

! !

1:)(: 1,,1,0;1 " " % nixxh iii

11

0010

/)(33

03/)(3

""""

""

nnnn haaFDN

FDhaa

&

&

1:)(:

' (

1,,2,1

;)(3

1

111111

"

%""

"

""%"%

ni

hhhaxxaha

ii

iiiiiii

&

) "*4.'( 3MNM ?,"4A ?,7,OD 9)& "',&'( P(54A'((

L@: 000000 1/,5.0,21 &# zh

Q".J' 3J" ,7 1:)(: 1,,1,0 " ni

iiiii

iii

iiii

zhz

h

hxx

1/)(

1/

)(21

11

11111

""

"""%

"

""

&

#

#

L@:

nnnnn

nnn

zhz

h

1/)(

)2(1

11

11

""

""

"

"

&

#

"

3R #5J47'( &G#SMI7'( "#"$%&'( )"*+'(:

],[,)()( 1) %xxxxSxS jjj

- 1

- 2

-5

!"#: - 3

-4

-7

-8

-9

$%&'( )'% *+ !"#:

0,,2,1 ! nnj

)3/()(

3/)2(/)(

1

11

1

jjjj

jjjjjjj

jjjj

hccd

cchhaab

czc

!

" !

!

"

""

"#

-6


256


1,,1,0

;)()()( 32

!

" " "!

nj

xxdxxcxxba jjjjjjj

!"#)10-4(: ,%(*-(# .*/*%0(# 12345- )-(# ,-60785+(# ,%0%9:-(# ,"%;<(# =24:: x 1 2 3

f(x) 0 1 4

1)3(,0)1( ! ! ff

!"#$: ) !"#!" $%& '(#)f *+,-!" ',./!" '(#) 0-12 (3.(

!"#: *+,-!" ',./!,+ ,/4:

1,1,4,1,0

2

10210

hhaaa

n

FD0 = 0 , FDN = 1

6]0)13(4[3

])([3

61/)14(313

3031/)01(3

10

10021021

2

0

!""

!""

" ""#

#""

hhhaxxaha

$

$

$

7517/221

7/95.3/5.45.3/)2/36(

7/2)5.3/(1

5.35.0)13(21

2/3,5.0,21

2

1

1

1

000

"

"

""

z

z

%

%

]2,1[)1(43)1(

41)(

433/)

41

25(

03/)21

25(1

41

25

21

23

493/)

25

417(

320

0

0

0

1

&"!"

"

!"

#"

" ""

xxxxS

d

b

c

d

]3,2[)2(49)2(

25)2(

4111)( 32

1 &"""!"! xxxxxS

%+%546!" 7%89!" ,:;;76 <= >)% ?6!" @(89!" A,4 </ *;76!" ':-%( 6B+C.D/!".


257


!"# $%&'#()4(

(1-4):

) ( !"#$%&' (")& *+,-. /0,1xxy )( (")& 23#4)5&'x 2"&4%&': x = 1 (0.05) 1.15

("6 *4# 07)5y 67 2898&2"7:; (4 .2"<%=5&' !,7>&' ?,-. /0,1 @&4%&4#,.

)A( 25"6 ?51%B'03.1 C&D,:

E- @$F&' ?451%B+' ('-F%B4#.

G- H"$%B% 45 ?IJK#.

2&4L ?1 @J K$F&' -., ,.

!"#$%&' ("6 MN# @$N" @%O' ?,-.&'y = x cos x - .52 /""7:; /5"67&.

L' (")&' PDQ ('-F%B4#2&-4N53& A.,5 7D. 7RS /""7:; /"567& AB x cos x = 0.52

x 0.9 1 1.1

y 0.04 0.02 -0.02

) ( !"#$%&' A"7)%& 2N#'7&' 2.7-&' /5 2"&451%B' -,-L T7"81 / 0,U1:

xxxf 24 sin)( ! "

T7%>&' ?9F "## x0

-VV=; !VV"#$%&' /VV5 W4VV="; DVVFK# CVV&D, ?451%VVB+' -VV#5 *45-F%VVB5 $4VV)= XVV5F -,-L&' W'7"81# W4J4B5&' Y,4B%5&'(equally spaced polynomial interpolation)

) A( -VV=; !VV"#$%&' 2VV5"6 A4VVBL& PDVVQ -,-VVL&' T7VV"81 (-F%VVB'6

" x 70-VV6, Z

C%."%= @J K$F&'.

(1-4) s¹d9

(2-4) s¹d9

(3-4) s¹d9

lÐ«d�« qBH�« �UM¹d9

lÐ«d�« qBH�« �UM¹d9 ∫lÐ«d�« qBH�«4

258


(4-4):

!"#$%& '!( )*" +#*! +%,$%& -./0%&y = x sin x – 1 1!!234 1!5(2%.

+678*%& -,58$79& :,7; '&/<$7,"(inverse interpolation) 1!5(26% :67;& =%/,*5>% :0.5 2?0 2@AB =5!( 1!!234x sin x = 1.

x 1 1.1 1.2

y -0.16 -0.02 0.12

C52%& 1,8 &?D 1B E"F,G =!5,5H& .2I%& 2FJ5 K%D C52!: )(.)()(.)())(.)(( hxgxfxgxfxgxf ! ! "

ELM,I$>% =>",N5%& =0!$O%&. =0!$O%& P?Q 1!" R",3$%& S;9..

(6-4):

=!%,$%& E,O,!">% T/./;5%& .2I%& -./0 1U.V8:

x 0.0 0.1 0.2 0.3 0.4 0.5

y 1 1.32 1.68 2.08 2.52 3.00

/./6;%& T26!F8 =@!6A /,60!W =6!5,5H& .26I>% 1$.6!O 1.O,( U"# 'F) =6%9/"x (

" 25$ +$%& =5!( :,7;% =@!A%& P?Q '/<$7&. X #,NO%& P?Yy /O4x = 0.02.

666!"#$%& 1,6668 &?D R666OB '.666>*5%& 16665f(x) =66602/ 16665 /./666; T2666!F8 -666F5!n 1Z666G =665!( +66#*! 66!"#$%& &?66Q #,66NO )66*" '&/<$667," :.667;5%& T/./66;5%& .266I%& -./660

.2I%& '!( :7;! [?%& /.5*%& 2A,O4 -8 /O4 =$",Fn [?6%& /.5*%& 1ZG +%,$%,". X

.2I%& '!( :7;!1! n \&2,IAB +#*!.

) !!"#$%: =6602/ 1665 /./66;%& T266!F8 -66M,I$ 1B =66N!N; -66",N! &?66Qn //664n 1665 //4 ,Y>M,I$ ,5O!" X \,$",F \&2&/N5 +#*! E&25%&n+1 \&2IA +#*! E&25%& 15.(

(4-4) s¹d9

(5-4) s¹d9

(6-4) s¹d9

(7-4) s¹d9


259


!"# $%"&'( )"*+,p(x) -%"./*0 1"/2 34,5x = -1, 0, 1 -".6 7,1"8 9!,":; "< =-%"""./*0 >+"""?* 7"""(%.@*0 A1,1"""'@*0 B,C"""D*0: """< = 1,"""@2 C"""E%/2 1"""/2 F""";(%<*0 !""".*0 F"""(%;

<*0 B,CD*0% F!/2 !# $%&'* 7,18*0 FG@:;,p(x) F(,G-@*0 -%./*0 1/2.

:(8-4)

B!(-;*0y = f(x) F"!*%;*0 -%./*0 1/2 HI-JK@ :(7,3), (8,1), (9,1), (10,9) 1"8,5 F@!#y 1/2x = 9.5 %@:;&L* M/0C84 FN!E 01O;&%(7.

B!(-;*0 !# PJ( Q-J! Q*%;*0 7,18*0xxy )(: x 1 1.05 1.10 1.15 1.20 1.25 1.30

y(x) 1 1.0247 1.04881 1.07238 1.09544 1.11803 1.14017

) ( !" #$ %"&' #("$)*+(, -./ : 32.1,28.1,01.1

%&012 3('0 %*"45 657/.

) 8( ! 392 :7;05.1 x %"&' -./ <x =>:0.+ %?&0@, #("$)*+(,.

x f(x)

0.001 0.54483 33726

0.002 0.55438 29800

0.003 0.56394 21418

0.004 0.57351 08675

0.005 0.58308 91667

->2 A&,@)5: %"&' -./ x = 0.00180 -B,C&@)*) (" A.

A&,@)5: 3&' D@E& D5()5: #/-.5:y : x y

0.01 98.4342

0.02 48.4392

0.03 31.7775

0.04 23.4492

(8-4) s¹d9

(9-4) s¹d9

(11-4) s¹d9

(10-4) s¹d9


260


%"&' -./ y ("->2x = 0.0341 ) F)G H0I&; %J&K 3-4)*:2!.(

(12-4):

A&,@)95 L("&' D@E& D5()5: #/-.5:: " # $

xt

e dtyt

%"&' -./ y ("->2x = 0.0378

x 0.00 0.01 0.02 0.03 0.04 0.05 0.06

y #% 4.0379 3.3547 2.9591 2.6813 2.4679 2.2953

A&,@)5:y & f(x) %&5()5: @(?>5: ->2 M"&' 3/9E":

x 0 2 4 6

y '% 7 61 211

) ( D9& (" 3&' -N-G #/-.5: !"/ < %&O)>"5: A/0I5: #/-. ! N/P$:

)3(),2(),3(

),6(),0(

32

22

fff

ff

!!!

()

) 8( !Q" %&5("$)Q*+: %&-/-G5: -./ %&"("R: A/0I95 !)/&> %J&K 3:-4)*(,%S5(S5: %.0-5:.

) T( %"&' -./ y ->2x = 1.

%QQ5:-5: 3&QQ' U-QQG; DQQV BQQ@4 FQQ92 W/QQ)G& D5(QQ)5: #/-QQ.5:f .QQ2 :7QQX CQQY/" !N&MGGK/ B@45:.

x f(x) x f(x)

3.60 0.112046 3.65 0.152702

3.61 0.120204 3.66 0.160788

3.62 0.128350 3.67 0.168857

3.63 0.136462 3.68 0.176908

3.64 0.144600

(12-4) s¹d9

(14-4) s¹d9

(13-4) s¹d9

)4 F)G H0I&; %J&K 3-4)*:!.(


261


:

%5:-5: #("$)*: ->2 Z-G& ! !$"& B@4 0,$R W/9E5: -G5: %"&' 3$:

" ** x

xdttxf0

3

20;cos)( +

%),(S [/@4, L(&\&].) L(&@4 L+("$)*:h = 0.1 ^

(16-4):

%5:-5: 3&?5 L+/-. ! _0I>:

+** xxxf 0,)cos()(

(O>&, ("&V H(V(*"5: %&/(*)"5: @(?>5: 3:-4)*(, 8*G&*:

,2,1;)1( $ ihixi

%),(S %"&?5h . D@4 #("$)*: #"E, 3/?>* (>> _0I>/ D\&].)(piece-wise linear

interpolation) %QQQQ5:-95f . %QQQQ),(S5: %V(QQQQ*"95 %QQQQ>$"" %QQQQ"&' 0QQQQ,$ DQQQQX (QQQQ"h + ZQQQQ&G, < B@4 0,$ U-E)&0.0002^

(17-4):

%5:-5: 3&?5 L+/-. (a> a/P$ :7;:

10,)( ** xexf x

@(?>5: ->2:

,2,1;)1( $ ihixi

(V(Q*"5: %&/(Q*)" @(Q?>5: b7QX Z&G !&)&5(Q))" !&Q)@?> W !&Q, %V(Q*"5:/ < (QO>&, (Q"&V H W/(QQQ*)h %QQQ5:-95 L(QQQ&\&].) L(QQQ&@4 L+("$)QQQ*: (QQQ>9"2 3QQQS <f =)QQQ>& BQQQ@4 0QQQ,$ %QQQ"&' (QQQ"V <

3:-4)*(,h = 0.01^

:(18-4)

%Q&>(S5: %Q.0-5: !Q" %Q&-/-G5: -Q./ )(2 xp %Q5:-5: #"$)Q*) DQ)5:f(x) = sin x

@(?>5: ->22/,4/,0 ++ x !)/Q&> %J&QK 3:-4)Q*(, 657/ < . %Q"&' DQV BQ@45: 0N-Q')x(p2 %@?>5: ->2x = 1 .B@495 %&9EI5: %"&?5: C" B@495 [0a-P?"5: %"&?5: !0(' 3S.

(16-4) s¹d9

(17-4) s¹d9

(18-4) s¹d9

(15-4) s¹d9


262


(19-4):

%5:-5: ! _0I>f(x) D5()5: #"($)5(, %Va0PE":

" ** x

t xdtexf0

2

10;)(

-' M> /H(V(*"5: %&/(*)"5: @(?>5: ->2 %5:-5: b7X 3&?5 #/-. a- c2 :

,2,1;)1( $ ihixi

Z&Gh !&)&5())" !&)@?> W !&, %),(S5: [/@45: 3.G #S"& . BQ@4 0,$ %"&' DX ("V%5:-5: #("$)*: ->2 =)>& f $)*:Q@4 L+("Q 3:-4)*(, L(&\&].) L(&h = 0.01 ^

(20-4):

) ( ?5 #/-. #c"2%5:-5: 3&:

21,ln)( ** xxxf

H(V(*"5: %&/(*)"5: @(?>5: ->2:

,2,1i;h)1i(1xi $,

%QQQQ),(S5: [/QQQQ@495 %QQQQ"&' 0QQQQ,$ DQQQQX (QQQQ"h #QQQQ"2 -QQQQ>2 (X7QQQQ4B> ! !QQQQ$"& DQQQ\&].) DQQQE&,0) #("$)QQQ*:(piece-wise quadratic interpolation)

%5:-95f B@4 0,$ %"&' ]/(.)) + Z&G,510$^

) 8( %5:-95 #:d*5: #G -2 :

10,)( 2 ** xexf x

(21-4):

%5:-95 H(&-/-G5(, L(&\&].) L+("$)*: (>9"2 :7;: xxexf $ )(

[0)I5: DV H(V(*"5: %&/(*)" @(?> ->2 [0, 1].

) ( L(&E&,0) #("$)*+: !($ :7; B@4 0,$ %"&' (" %Q),(S5: [/Q@45: %Q"&' HQ>($/ <h

= 0.01 ^

) 8( [/QQ@495 %QQ>$"" %QQ"&' 0QQ,$ (QQ"h ]/(QQ.)& + (QQ>-0 / L(QQ&@4 #("$)QQ*+: !(QQ$ :7; < B@4 0,$ 610$ ^

(19-4) s¹d9

(20-4) s¹d9

(21-4) s¹d9


263


%&5()5: H(>(&,5: #"$)*) D)5: %&E&,@5: %&,&E$)5: %G&015: -./ :

ix 0.8 0.9 1.0 1.1

if 1.0 1.2 1.1 1.5

! _0I>S(x) %&5()5: H(>(&,5: #"$)*) D)5: %&E&,@5: %&,&E$)5: %G&015: DX:

ix -1 0 1

iy 1y 2y 3y

! _0I>/S(x) &5()5: %J&K5: 74B)H(>(&,5: b7O5 %,*>5(, %:

10)(

012)(

)(32

2

3

312

1

**,,,

**$,,$

xdxcxbxaxS

xxxxxS

xS

) ( 3&' -./ a, b, c, d.

) 8( %.&)> 3:-4)*(,) ( !" #$ %"&' 0N-')5.0(),5.0( yy-.

) T( 3&' -./ 321 ,, yyy.

) -( #"$)Q*) DQ)5: !)/Q&> %J&QK DQV %Q&>(S5: %Q.0-5: !Q" %&5("$)*+: %&-/-G5: -./ %5:-5: H(>(&,y.

3&' DX ("a, b, c, d %5:-5: #E.) D)5::

]1,0[

]0,1[311

)(32

32

.,,,

$.,,$

xdxcxbxa

xxxx

xS

-P?E95 %,*>5(, %&,&E$) %G&01-1, 0, 1 -G5: @01 A?G)/ <2)1( -S.

(22-4) s¹d9

(23-4) s¹d9

(24-4) s¹d9

) !" #$%&' ()&* +,-!./ 4 .(


264


!"#!" $% &'():

2061)(

01)(

)(32

2

321

!"!#

"!!!#

#

xxxxxS

xdxcxbxaxS

xS

*!+,!" -+)+*.!" /01,2, 3,!" *4*.5!" *.*41,!" 6*'7!" 38:

x -1 0 2

y 1y 2y 3y

$0 /1 0*9 38 +0:a, b, c, d ;

$0 /1 0*9 38 +0<321 ,, yyy ;

=*9 38 +0a, b, c, d, e /4>, 3,!":

]1,0[

]0,1[2

)(32

32

43

$!!"

"$!!!

#

xexdxxc

xbxxax

xS

#?4@! .2)!+. *4*.5 *.*41, 6*'7-1, 0, 1 ;

!"#!" $% &'():

32

202

012

)(32

3

412

43

3

412

43

!!!

"!"

"!!"

#

xdxcxbxa

xxxx

xxxx

xS

*.*41,!" 6*'7!" 38 *!+,!" -+)+*.!" /01,2, 3,!" *4*.5!":

ix -1 0 2 3

if 1f 2f 3f 4f

A'*BC" *.*41,!" *#<#6!" 38 +0:– D*'4, 3:S – /01,2, 3,!" f A'E,(!" 3E:[2,

3] . EEEEEE0*9 '*#EEEEEE?,! EEEEEE*#<#6!" FGEEEEEE8 =#B,EEEEEE2"f(2.5) EEEEEE0*9<)5.2(f % .=*EEEEEE9 H,),EEEEEE2"<

4321 ,,, ffff.

(25-4) s¹d9

(26-4) s¹d9

(27-4) s¹d9


f�U)« qBH�«

ÍœbF�« q{UH²�«

Numerical Differentiation

�U½UO³�« ‰Ëbł ◊UI½ bMŽ �UI²ALK� W¹œbŽ mO� º

�U½UO³�« ‰Ëb−Ð X�O� ◊UI½ bMŽ �UI²ALK� W¹œbŽ mO� º

�U½UO³�« ‰Ëbł ◊UI½ bMŽ �UI²ALK� Èdš√ W¹œbŽ mO� º

ÍœbF�« q{UH²�« mO� W�œ vKŽ ¡UDš_« dOŁQð º

f�U)« qBH�« �UM¹d9 º

267

f�U)« qBH�«

ÍœbF�« q{UH²�«Numerical Differentiation

! "#$$$%&f(x) '$$$()* '$$$+,+,-.(analytic) /$$$01+2 34$$$,51– 64*$$$7 8$$$9– *$$$&:/$$0&+; /$$1+9 </9/$$=1() '+4/$$=.1() >/$$?&() "$$5; . '$$1+2 @$$=-& ! A) /$$&1 @4$$,>1()4

'?.$$B1f /$$1C/$$D#+E '$$>?& 4! 64*$$7() >/$$?& F*$$-C *$$&: .$$(4G) <$$&/H I$$9 '(J$$=1(/9 K6/1H.=L)4 6M/%.() 1 N/7+O1 *5. '(J=1(/9 F#PG) <&/H Q)4 '.-; 6M/%. '(J=1.

'?.$$$B1,( '$$$+;+#?. N/$$$1+2 8$$$>5. 8$$$.() '$$$+**5() R+$$$S() "$$$5; T.&.$$$=& 8$$$,+ /$$$1+9 K$$(4G)f '$$+&/U() '?.$$B1()4f '$$()*,( !!"#$ %!!&'() *+,#!!-./)0f 1+!!,2 34!!56 4!!2'7048/).

9"+!:/) 7;</) =. >&?2) !@1A/) *B.+!?./) =@!" *+!CB?/) 704!8 DBE!. $!F2) <&#A./) (=G:

)1log( !"# hDU

H!!!!!@5D 7!!!!!I+<#/) 7!!!!!.+?. 0!!!!!J)/( dxdD # 0 !!!!!</) 7!!!!!.+?. 0!!!!!J !!!!!@.+.() 90$)( 1 iii yyy $" !.

)1log(1 !"%h

D

%&' 7;52 $0&@+# K0L<. >)4A#:+"0:

]31

21[1 32

$ ! $ "h

D

!!!/)4&/ %!!!/0() ,#!!!-./) M!!!1?# N@!!!;/) OP!!!QRf !!!@.+.() 90$!!!</) !!!/S4"k T U01A/) >850h.

�U½UO³�« ‰Ëbł ◊UI½ bMŽ �UI²ALK� W¹œbŽ mO�

)1(3312

21)()( )()()(

1

!"

#$%&'( )*+*)* xfxfxf

hxfxDf

268


8@V0 @WL$./) 90$</) /S4" %/0() ,#-./) 4+k& !CB?/) >4A#!:2 =G +22L.@ @/+#/))9"+:/) 7;</) MR D+I@G +J+2$LP M#/)0:(

' (

dU

dUU

dUd

U

U

41

2sinh1

21.cosh2

2sinh2

2

2

2

&

&

&

!"

!"

")

"

' (

$!$")

$*

*!$"

!"$

53

42

2/12

6403

241

162123

21

4211

41

&&&

&&

&&

U

ddU

/) ,#-./) 7.+?. %&' 7;52 CB?/) OPJ X@"$#"0 @2+E2D:

N@;/) =G F5B2(2) 7!.+?.&/ !@4$</) 30!,/) !/S4" %!/0() ,#-./) M1?#&

U+!!!!1?./) 1+!!!,2/) 34!!!56 4!!!2' ,#!!!!-./) OP!!!J 4+!!!8@V 9!!!@"1#&/ U4!!!!@<. $!!!@Y K/P!!!" M!!!J0)U+1?./) *+2+@"/) 7048 MR U$0LP./) ( 90$!</) 704!8 =(– 4!?" Z!2@0L#– M!1?@ S

1+!,2/) OP!Q/ !&"+,./) @4$</) @WL$./) 90$</) !@80W/) !@WL$./) 90$!</) M!1?@ +!.2 [)0 T .

N@;/+R K/P/0(2) !1,2 4!2' %!/0() ,#!-./) \+!:5 4!2' 1!,R 4@<# =@!" 1!:0/) M!R 704!!8/) M!!R =@!!#1,2)=@#4!!,]' ^G .( !!1,2 4!!2' %!!/0() ,#!!-./) 4+!!8@V +!!.G 704!!8/+"

V _+#52R $E .&/ N@; 4+8@D 7.+?.&/ @80W/) 30,/) /S4"&.

M&@ +.L N@;/) OPJ 4+8@6 +22L.@0:

CB?&/ =.@() a$1/) \$I2(2) $)4,./) MR:

' ( 2/121

4

$!&+

)2(5640

332411

!"

#$%& )+) ,,,

hD

)3(856016

9014

1212

212

!"

#$%& +)+) ,,,,

h

D


269

ÍœbF�« q{UH²�« ∫f�U)« qBH�«

) 0+:@ $)4,./) )PJ01 9"+:/) 7;</) =. >&?2 +.L($)4,./) MR Z"$I2 ^G T:

,,-

.//0

1+ $!$ &

*

*&

16

4

21

2

3

2

1

4

2

2

11

' ( $&!&$+# 4128

32811

%&' 7;52R:

' ( 2 32 3

!&$&!&$+&

$&!&$&$&!&$

"

+"

6

140

14

30

12

6

1

4

640

32

24

14

128

32

8

1

1h

1..1h1D

=L/0:

)( 1

21 $$" EE+&

=G ^G:

)yy(y 1k1k21

k $! $"+&

2

3 )4()(140

1)(301

)(61)(

21

16

16

14

14

12

12

11

!$$$!

!$$$"

$!$!

$!$!

kkkk

kkkkk

yyyy

yyyyh

Dy

&&&&

&&

3$AG U$0;" 0G:

2

3 )4()(140

1)(301

)()(21

16

16

14

14

12

12

61

110

4!$$$!

!$$$"

$$

$$

yyyy

yyyyh

Dy

&&&&

&&

!"#(1-5):

=G b$<2: ]3.1,7.0[,sin)( 5" xxxy

/)4/) >@,/ DS048 =c0Ly – / "0:5. @$-' >+C$G #:– @0+:#./) 1+,2/) 42'U$0LP!!!!./) U$!!!!#</) M!!!!R *+R+!!!!:./) U0!!!!1A/) >!!!!85 $+!!!!"#'+" Th=0.1 704!!!!8 =c0!!!!L Z!!!!2.0 T

90$</) .=@#,#!-./) =. 7L .@C 4+8@V 7048/) )PJ >4A#:) >Eyy 444, !1,2/) 4!2'x

= 1 =@#.@,/) =. 7L MR d1A/) \:5)0 T.

ÍœbF�« q{UH²�« ∫f�U)« qBH�«5

270


$%&:

x y &6 2& 3& 4&

0.7 0.644218 73138

0.8 0.717356 -7167 65971 -660

0.9 0.783327 -7827 79 58144 -581

1.0 0.841471 -8408 85 49736 -496

1.1 0.891207 -8904 87 40832 -409

1.2 0.932039 -9313 31519

1.3 0.963558

=G b$<20.10 "x =!. 7!L !.@C \+:5 \0&1./) =0L@R0,0 yy 444 . 9!@"1#" CB?/))4( 4 %&' 7;52:

2

3

000002.0||

540302.01cos)1(

54030.0

)000079.0000087.0(301

)007827.0008904.0(61

783327.0891207.01.02

1)1(0

"

""4

"

$$!

!!$$

$*

"4#4

Error

exacty

yy

CB?/) 9@"1#"0(3) %&' 7;52:

2

000041.0||

8414711sin)1(

841512.0)000004.0(901

000085.0121008408.0

)1.0(

1)1(20

"

$"$"44

$"789$$*$

!*$$"44#44

Error

exacty

yy

'($)#:

,#-./) \+:5 MR M:+:() 45/)y4 CB?/) =.)4( 4 0J:

]11[21

0 $$:4 yyh

y


271


=G ^G:

)5()]()([21)( 000 hxfhxfh

xf $$!:4

1,2/) 42' e+../) 7@. =G ^G))x(f,x( 00 0G $#0!/) 7!@. D+!"@$,# 0+!:@ >@,#!!!!!!!!!!!!!:./) 1!!!!!!!!!!!!!A/)AB !!!!!!!!!!!!!#1,2/) =@!!!!!!!!!!!!!" 7!!!!!!!!!!!!!;)0/) =@))hx(f,hx( 00 $$

0))hx(f,hx( 00 !! ]7L!!!-/) $!!!F2) [OP!!!J %.!!!:#0 T N@!!!;/) !!!1,2/) U4!!!'+C 1:0#./)(mid-point rule).

,#!!-.&/ 1!!:". !!@"@$,# 3$!!AG N@!!; K+!!2J0f 4 =!!. +!!Q@&' 70!!;5/) =!!L.@ CB?/) =. ,#-./) \+:5 MR M:+:() 45/)(1):

)6()x(f)x(fh

)x(f)hx(f0h

10

00 $!" :4

>@,#:./) 1A/) 7@." D+@:42J 7fE.]# MJ0CB 7L-/+" +.L.

,#!!-.&/ 1!!!:". !!!@"@$,# !!!E/+E N@!!!; K+!!2J0f 4 =!!!. +!!!Q@&' 70!!!;5/) =!!!L.@M!!! g1?# !!!CB'f 4 !!!@<&A/) 90$!!!</) !!!/S4"; !!!CB?/+" Q@"!!!- T(1) %!!!1?# M!!!#/)f 4

@.+.() 90$</) /S4" MJ 1:"./) E/+E/) N@;/) OPJ0 T:

)7()()()(

)0( 000

h

hxfxf

h

xfxf

$$"

;:4


272


>@,#:./) 1A/) 7@." D+@:42J 7fE.]# MJ0AC 9"+:/) 7L-/) MR.

h@;/) OPJ %&' 70;5/) =L.@0 T $0&@+# K0L<. =. K/PL HBE/))PLJ:

)9()0()0()0(

)8()0()0()0(

!4$"$

!4!"!

xfhxfhxf

xfhxfhxf

+J i$1"0%&' 7;52 b?"/) +.Q@I?" =. =@#CB?/) =@#:

)10()0(2)0()0( !4"$$! xfhhxfhxf

1!!:"./) N@!!;/) M!!1?# U$!!@A() !!CB?/) OP!!J(5) =+!!#CB?/) +!!.2@" T(8), (9)

=@#1:"./) =@#N@;/) \@#$#/) %&' =+@1?#(6), (7).

N@!;/) M!R =!.@() a$!1/) !@+Q2 +!Q2d" af$?]# ,#-./) =G F5B@0(6) +.4!2'# 70h $<;/) %/6!?# %!/6 \$!CG N@;/) OPQR K/P/0 T! ,#!-./) a@$ N@!;/) =G S6 T

(5) +!!!Q@R d!!!1A/) =( T =@#@$!!!A() =@#N@!!!;/) =!!!. 7!!!L =!!!. 94G 4!!!?]#) N@!!!; M!!!R ^G W!!!L$./) 9$!!!</)(5) ( !!!"#$ =!!!.2h N@!!!;/) =!!!. 7!!!L M!!!R d!!!1A/) +!!!.2@" T(6) ) 9$!!!</)

M.+!!.() ( N@!!;/)0(7) )M!!<&A/) 9$!!</) ( !!"#$ =!!.h . %!!1?@ d!!1A/) =G *+!!"E6 =!!L.@0 @/+#/) *+CB?/+":

)i( @WL$./) 90$<&/:

)11()()3(62

)()()0(

200 <fh

h

hxfhxfxf $"

$$!$4

)ii( @.+.() 90$<&/:

)12()(2

)()()0( 00 <fh

h

xfhxfxf 44$"

$!$4

)iii( @<&A/) 90$<&/:

)13()(2

)()()0( 00 <fh

h

hxfxfxf 44"

$$$4

!"#(2-5):

HBE!/) 1!:"./) h@!;/) >4A#:)] !@WL$./) 90$!</)(5) !@.+.()0(6) !@<&A/)0(7) [ ,#-./) 4+8@Vy4 42'x = 1 MR 7+E.(1-5).


273


!"#:

58148.01.0

783327.0841471.00)7(

49736.01.0

841471.0891207.00)6(

539400.02.0

783327.0891207.00)5(

! "#

! "#

! "#

y

y

y

!"#$%&' %()&*# (++,&' -()&' ./0 12*)%#":

540302.01cos)1( "Exacty

3&"4' %()&' 56 +71– (892%&' :"2;&' <(= 5% 7>*1&'– ?@+6 A0 *?(B C!D&'" *E"*F(: E = 0.000902

)>GGH%I& !GGF#% J(GG= 3GGIK *1IGG=L *GG%9"y" 3GGIK M"GG=L&' N&/GG9 *GG119%(B O )>GGGH%I& !GGGF#% <(GGG=y "" 5GGG% 5GGG%(4' P2GGG!&' AGGGB M"4' +GGGL&*# Q*GGG;>9R*# NGGG&/" O

@S,&'(3):

1021)10()01(02

2/12/10

021

02

0 2

!$! !!!! %

!!

&'""

yyyyyyyy

yyy

yh

yDy

(

(

(

)14(2

0 2

101

h

yyyy !$!

&""%

CG!DI& 2(+G)> +*G7(T 5G9%("(error estimate) !GF#%&' G(#(2)>&' <(G=&' ./G?&)*GGG02(U" )#*GGGF&' GGG(#(2)>&' J(GGG=&' MGGG9& V*GGG%"%K" (2"I(*GGG> IGGGFIF>% -'+D>GGGF*# . VSWGGG%B

X2(D4' <(=&' AB C!D&' 2(+)>&) (1*W&' )>H%&'y "":(

)()4(!4

)0()3(!3

)0(!2

)0()0()0(1

4

32

)fh

xfhxfhxfhxfhxfy

$

$""$"$ $'

)()4(!4

)3(0!30!2001

432

)yhyhyhyhyy $$""$"$ %

MW%&*#":

)()4(!4

)3(0!30!2001

432

*yhyhyhyhyy $!""$"! !


274


5(>2(D4' 5(>@S,&' Y%7#":

))()4()()4((!40

20211

4

*) yyhyhyyy $$""$ !$

56 Z2;#")4(y X2>;&' 3IK I=>% &'+]0,0[ hxhxI $!'

% +7"(I+, 56 [(L#:

)]()4()()4([21)()4( *), yyy $

)()4(120

20211

4

,yhyhyyy $""$ !$%

)()4(12

20

2

2

101 ,yh

h

yyyy !

$!!""% !

!GGF#%&' GG(#(2)>&' <(GG=&' AGGB CGG!DI& \"GGI!%&' 2(+GG)>&' GG@S,&' ./GG0 AGG!,>" (1*W&' )>H%I& )#*F&'0y "".

S,&' Z,# 5]' 3>L :#F *%(B *17>1>F' )>GH%& G(++K V*%(@ A!,> A>&' ^*@ GG&'+ f ] 3GG&"4' )>GGH%&'f " GG(1*W&' "6f "" )>GGH%I& -(GG%,>&' 5GG9%(" O)(kf GG#>2 5GG%

k _`GG& YB2&*GG# Ok – SWGG%– AGG!,> GG@SK E6 AGGB2!D [ X+"GG7"% GG!)1 +GG1K NGG&/"*G1*(#&' M"+G7 AB3G!,%&' ^ . +G(;> AG>&' ^*G@S,&' ZG,# a' 5"G,# b>1>GF1 AGI( *G%(B"

M"+GG7&*# X+"GG7"% 2GG(U GG!)1 +GG1K GG;I>D%&' ^*)>GGH%&' 3GGIK M"GG=L&' AGGB . '/GG0 AGGB" :#*GF&' MG=;&' AGB *G?(IK *1IG=L AG>&' M*%9>FR' J(=# +(;>F1 ++=&' *0*1%+D>GF'"

M"+GG7&*# ^GGF(& !*GG)1 +GG1K GG&'+I& -(GG@ 3GGIK M"GG=LI& .IGG$*B '/cGGB X+GGK J(GG=&' ./GG0 *1 GGGGGGGG;I>D%&' ^*)>GGGGGGGGH%&' 3GGGGGGGGIK M"GGGGGGGG=LI& GGGGGGGG#"I!%&' J(GGGGGGGG=&' *GGGGGGGG1(!,> *GGGGGGGG?1cB ^'2GGGGGGGG%

),)3(,,( !fff """ M"+7&*# ^F(& A>&' !*)1&' +1K.

(%*%4' :"2;I& 5>"(1 <(= A0 (&*>&' @S,&' VSW%B:

�U½UO³�« ‰Ëb−Ð X�O� ◊UI½ bMŽ �UI²ALK� W¹œbŽ mO�


275


!

!

!

$-$!$-!$-$

$-..!!

$$-.!

$-$

$-//0

1223

4$-//

0

1223

4$-//

0

1223

4$

03)2233(

61

02)2(

21

00

03

321

)2)(1(0

221

)1(00

03

30

2

20

10

ypppyppypy

yppp

ypp

ypy

yp

yp

yp

ypy

[(L:

phxxpxpxypy $ '' 0,)(

AB2! *1I$*B '/cB @S,&' ./0)5>"(1 <(= ( 3&T #F1&*#p56 dLS% Y% O:

pyh.)p(xyh(x)yhdx

dy(x)h

dp

dy(x)

dp

pdy

dxdh

dpdx.

dxd

dpd

dphdxphxx

" "'"

#$ 0

AI( *% 3IK *1I=L:

!$-$!$-!$- " 03)2623(

61

02)12(

21

0. yppypypyh

5 67 8 )15()263(10

32

61

02

21

0 !!$-$!$-!$- "% yppypyhpy

@S,&' ./0 I$*;%# MW%&*#"3IK M=L1 O:

5 67 8 )16(110

30

2

2!!$-!$- "" ypy

hpy

IK M=L1 X2(D4' @S,&' ./0 I$*;%# N&/9"3:

7 8 )17(10

3

3!!$- """ y

hpy

$%%%&!'(: ^GGG1*9 '/Tpx 5cGGGB M"+GGG7&*# GGG!)1p=05cGGGB M"+GGG7&*# 5GGG9> -GGG& '/T *GGG%1(# O

09p.

)*((3-5):

56 Z2;1y (&*>&' !*)1&' +1K *?%(B -"I,% &'+:

x 4 6 8 10

y 1 3 8 20


276


-(@ +7"6yyy """""" ,, !)1&' +1K:

)6( x = 4 )\( x = 4.5

!"#:

ix iy - 2- 3-4 1

2

6 3 3

5 4

8 8 7

12

10 20

)6 ( !)1&'x = 4 M"+7&*# !)1 /DC140 x . 56 E6p = 0

5 67 8

21]4[

81)4(

41]43[

41)4(

12114

623

212

21)4(

"""

! ! ""

.$.!$ "

y

y

y

)\ ( G!)1&'x = 4.5 M"+G7&*# ^GF(&G& G!)1 \2G@6" OGG7&*# *?G AG0 M"+x = 4 N&/G&" /DC140 x5"9> N&/#" O

41

245.4 ! p

5 6 5 67 85 67 8

7 8214

81)5.4(

041413

41)5.4(

484142

46

163

613

21

412

21)5.4(

"""

.!$ ""

$.$!$.!$ "%

y

y

y !

)*((4-5): I& V*<(= +7"6^*)>H%yyy """""" ,, (&*%9>FR' 2;(T <(= I$*;%#.

!"#: 56 -GI,1 2G;(T <(G= AGBpq ! 1 G&R+# *G?I9 <(G=&' VR"6 \G>91B Op *?I$*;> M?F(&.

!

!!

$!!!

$$!$

$!$

$//0

1223

4 !$!$$//

0

1223

4 $$

02

6

))(1)(2(1

26

)1()1(0)1(1

02

3

20)1(1

2

3

11

yppp

yppp

yppy

yp

ypyp

pypy

((

((


277


:;<$$!!

!=>? !$! "%

!)02)3232(

61

12)1

2)3(61

011

ypppdpd

yyppdpdyy

hpy

(

((

7

:;<$$!!

!!$!

!02)2362(

61

)13(11

22

61

01

ypp

ypyyh

(

(

:;<

=>? $! """

:;<

=>? $!$ ""

!

!

02

121

02)1(1

21

3

2

yyh

py

ypyph

py

((

((

+

:#*GGF&' MGG=;&' AGGB *GG1(62)M*%9>GGFR' ( ^GG1*9 '/T eGG16p(x) 5GG% GG(+"+L&' AGG0 72+n &'+&' M%9>F> A>&'f(x) *G?%(@ -"I,%&'))(( ixfif ' ("*GF>%&' !*G)1&' +G1K

^*B*F%&':

bnxxxa @@@ !10

[(L:

1,,1,0,1 ! !$ nihixix !

&'+&' 5(# @S,&' 5cBf(x) (+"+L&'")(xnp A0:

(*))()()( xnExnpxf $

C!D&' [(L)(xnE @S,&*# 3!,(:

(**)!)1(

)().()1)(0()(

)1(

$!!!

$

n

fnxxxxxxxnE

n ,!

@S,&' *1I$*B '/cB (*) GG& #F1&*# ^'2% X+Kx G!D&' +GL ^*)>GH% *G1I%06" O C)(xnE GGG(+"+L&' ^*)>GGGH% 5cGGGB O)(xnp GGG&'+&' ^*)>GGGH%& GGG(#(2)> V*<(GGG= +GGG,>f(x) .

!*G)1&' 5G% G;I>D% V'+'+GK6 *1/D6 *%I9") -@2GI& G;I>D% V*G%(@ E6n ( J(G= 3GIK *1IG=L GGGG;I>D% GGGG(++K . GGGG;I>D%&' J(GGGG=&' ./GGGG0 5GGGG% MGGGG9 AGGGGB CGGGG!D&' 2(+GGGG)>& #GGGGF1&*# *GGGG%6"

^*)>GGGH%I& GGG!)1&' ^GGG1*9 '/T *GGG% GGG&*L AGGGB -"GGG%,&' 3GGGIK VS?GGGF V'2GGG%6 _(GGGIB Ox ^GGGF(&

�U½UO³�« ‰Ëbł ◊UI½ bMŽ �UI²ALK� Èdš√ W¹œbŽ mO�


278


54 NGGG&/" O M"+GGG7&*#)()1( ,$nf AGGGB GGG%"I,% 2GGG(U GGG&'+x ) 56 dGGGLR, 2GGG(U GGG&'+AGGGB GG%"I,% x (*?>IGGG$*;% Y(!>GGGF1 SGGGB . ^GGG1*9 '/T *GG%6x f*GGG>L1 SGGGB M"+GGG7&*# GGG!)1

,%&' 2(U &'+&' ./0 I$*;%& 2;G= AGB \2G$(F '/G0 *?I$*;> 54 %"I] +GL6 \#GF#_'"@4')( ixx ! 54x M"+7&*# !)1 [AI( *%% g$>( *%9:

)(*)()()(

)()()(

""$" "

$

xnExnpxf

xnExnpxf

/DC#n = 1 :

!)1(

)(.)]()[(

.)]()[()(

)1(

0

!)1(

)()1(

0

$!!$

$:;<

=>?!! "

$

$

$

n

fxxxx

xxxxxE

n

ndxd

n

nf

dxd

nn

,

,

!

!

!)1&' ^1*9 '/cBx !*G)1&' h+GLT AG0nxx ,,0 ! O 3GHS>( M"4' +GL&' 5cGB3IK M=L1":

!)1n(

)(f.)]xx()xx[()x(E

)1n(

n0dxd

n $,

!! "$

!

*12#>K' '/T VSW%Bn = 1 5*>!)1 *1(+& E610 x,x 5cB O:

!2

)(f.)]xx()xx[()x(E 10dx

d1

,""!! "

)(f2h

01 )x(E ,""! "

)()(211 ,fxE h "" "

6 CGG!DI& 5*GG>2(D4' 5*><(GG=&'",&' AGGB V*)#*GGF *GG%?(&T *12GGH 5(GG>@S(12) " (13) .

*12#>K' '/ i'"n = 2 !*)1 [SW *1(+& E62,1,0 xxx 5cB:

)()(

!3

)(.)]()()[()(

6

2

12

)3(

2102

,

,

fxE

fxxxxxxxE

h

dxd

"""! "

!!! "

@S,&' AB *?(&T *12H6 C!DI& <(=&' ./0"(11).

(+"+LI& #F1&*# *%6")(xnp (%*%4' :"2;&' <(=# VSW% *?#>91B:


279


)(1)(

)18(002

20

10

).0()(

xnpdpd

hxnp

fn

n

pf

pf

pf

hpxnpxnp

"

-//0

1223

4$$-//

0

1223

4$-//

0

1223

4$

$'

!

5 6 )19(00212

21

01 :

;

<=>

?-//

0

1223

4$$-!$- fn

n

p

dpdfpf

h!

*%+1K SW%Bn = 1 3IK M=L1:

h

xfhxf

h

fxp

)()()(1

000 !$

- "

@S,&' AB *?(&T 2*H%&' <(=&' A0"(12).

@S,&' 56 +71 '/90")(* " ] G@S,&' MG$*;> AG0 AG>&'(*) [G(L OnE GB j2k,l% @S,&*# [(**)!,> )>H%&' \m2k)l> ;I>D% (++K V*<(= *1(f " AGB C!DI& 2(+)> Y% O

<(= M9 . @S,&' *1I$*B '/T MW%&*#")(* "3IK *1I=L O:

)()()( xnExnpxf ""$"" ""

*1(!,> @S,&' ./0"– ++G,I& G;I>D% -(G@ /D6 +1Kn – \G(2)>& G;I>D% V*<(G=%&' )>Hf "" <(= M9 AB C!DI& 2(+)> Y%.

)>GH%I& G;I>D% J(= 3IK M"=L&' 59%( e1cB V*%"%K")()( xkf ) [G(Lk

= 1, 2, ... ( ++,I& ;I>D% -(@ /DC# <(= M9 AB C!DI& 2(+)> Y%n @S,&' AB:

)()(

)()(

)()( xknEx

knpxkf $

$,- ./0 1)234# 56*78 9--:"# ;)<8"# =6>

Effect of Errors on the Accuracy of

Numerical Differentiation Formulae

GGGGF*FL E++GGGG,&' MGGGG$*;>&' J(GGGG= YGGGG(%7 n+GGGGk,l>(sensitive) \GGGG(2)>&' Q*GGGG!D4(rounding errors) &'+&' -(@ ABif. 2% A>&' (&*>&' <(=&' _2+1 N&/ g($">&"

%3&"4' )>H%&' A!,> A>&'" O V*)#*F *1,f " (892%&' :"2;&' &R+#:

)()3(62

)()()(

2

,fhh

hxfhxfxf !

!!$ "

X2=>D%&' X2"=&' AB *?#>91 56 59%( A>&'": tEh

ffxf $

! "

2)1( 02


280


[(L:

)()1()2(2

1

hxfhxfxff

xx

$ $ '

'

)()3(6

)()1()0(02

,fhtE

hxfhxfxff

!

! ! '

tE MW%( o*!>@R' C!D(truncation error).

5*>!"#GGG$%&' 5*GGG>%()&' *1(+GGG& 5"GGG9( R X+*GGGK"2,0 ff 5*GGG>(#(2)> 5*GGG>%(@ *GGG%1 i'" O *1"9>&"*

2,*0 ff \(>2>&' 3IK . L(LG= +G,> 5(G>%()&' 5(>*G0 5% S9 56 *1$2B '/cB

++,&k (B C!D&' 56 E6 O (2H,&' -*@24' 5%8"*7>( R *?k!.105.056 E6 O:

2,0;105.0* !.A! ikifif

)>H%&' 5K 2(#,>&' *119%(B)1(xf " AI( *%9 Q*!D4'" (#(2)>&' -()&' &R+#:

tErEh

ffxf

!"#

2)1(

*0

*2

!"rE #!$%&'( )* +&,-'( ./0'( 12(rounding)!1 34567',8 9/7:

)]*00()*

22[(21

22

*0

*202

ffffh

h

ff

h

ffrE

!!!"

!!

!"

)20(1021105.02

21

*00

*222

1

kh

kh

ffffhrE

!$"!$$$%

&'(

)*+ ! !%

): ,-;;;;<$= (> ?(1Mxf %)()3( @$;;;;&A'( B;;;;=, -2,0 xxx. .;;;;/0'( ): +&-&;;;;C-=

$!D%& B= BEF'(f # B',&'( D"'( G1,H&! I #!$%&'(1 J,/&5I( )K LF #8C8:

3610.

21

2

Mhkh

tErEtotalE

!%

%

>2 48,&F81( D"'())K!M( N$/'( (B= 4'(DF h:

3610.

21)(

2

Mhkh

hE !"


281


4K!5 O$PQ ,KEF R-: S"6-h 4;K!5 )T=kh

!10.21 D;"'( 4;K!5 U%-;& ,;K-!8 3 D;!G&

;;;0M(;$ .;;;!5 );;;* ;;;"8-=; 4Kh ;;;H& B;;;&'(;;;;K!5 L7;4E(h) );;;FK! ,;;;K $P;;;Q:3 L;;;7H& V:0)( "# hE:

min

3/1

10.2

3

03310.

2

1)(

3

2

hM

h

Mhk

hhE

k/00

1

2334

5"6

" !!"#

$!87& B= 4K!%'( W>2 X!17&81E(h)): DH- 3:

3/13

3/2

102

321)min( MhE

k 001

2334

5

$"

4!$;;Y* Z;;<(1K 4&;;C' [6\;;K 4;;%!5D O,;;-,!8'( O;;-,F (>] R;;-: 4;;H!&-'( W>;;2 9;;-7K1) ): V:k = 6(4;K!5 O;-,F1 33M Z;51&-= 3 @D;"1'( [,;8!$%& V1,;C&– [(^,;-8 W>;2 9;E*

4H!&-'(– )M 4!$Y* Z<(1K 478$: 1"-' 4%!5D )1F& L<,A&'( 4P!Q )::

410655.0

1

3/2

102

321)min(

6

!$"

$001

2334

5

$"hE

)1;;;F& 9;;;E*M( #;;;&$'( V> L,KF&;;;CI( 9;;;E* 4;;;!-8K'( VDD;;;7'( L;;;<,A&'( _!;;;Q1#!$%&'( ^,/0M _E8: @$1Q8 4C,C" .@$1Q'( B= L<,A&'( 4P!Q O-,F (> ?(1:

7"

m

i

ifih0

1 8

: )T;;=1;;2 #;;!$%&'( #8;;C8 .;;/0 $;;8F7 !kih

10..21 8 J1;;KHK'(1 37 i8 `8;;Q!

L,KF&;;CI( 4;;8&$ ,;;-DG ,;;KEF [($;;!8F . VDD;;7'( L;;<,A&'( 4P!;;Q )T;;= L,;;\K'( L!8;;C 9;;E*1

a5$ 4'.CK'( B= @$1F>K'( )14-5( D1D;" B;= ,;b!= J,;/&5I( .;/0 )1;F!)2(hO $;8F:1 3

01;;2 #;;!$%&'( #8;;C8 .;;/:kh

!$$ 10341 .;;/0'( );;K $;;8F: (>;;21 3k

h!$10

21 V>;;'(

4567'( B= R!E* ,-EQ"(20).


282


:(1-5)

c!8/&'( a!5)(xff / B',&'( L1DH'( B= @,/7K'( /,%-'( D-* 4K1E7K: x 0.25 0.50 0.75 1.00 1.25 1.50 1.75

f(x) 1.284 1.649 2.117 2.718 3.490 4.482 2.755

) :( D-*x = 1.00 >0.81h = 0.25 a!5 DH1::

)(6),(),(2),( xfxfxfxfE 9:9;<

) #( ): X$A-: )(

)()(

xf

xfx

#"=

dx

xdx

)()(

=> "

)K LF 4K!5 #C"()1(),1( >=.

) d( 4;%!%"'( B;= B/7! c8,C'( L1DH'( ): aE* (>]c;!8/&'( a!;5xexf ")( 4"!";Q4!$Y* a,5$: 4\6\' c8,C'( GH'( B&H!&- )K LF B= ./0'( #C",= 3)#(

:(2-5)

a5$ 4'.CK'( B=(1-4) ): X$=(: ),(.)()( xyxyx #"=

dx

xdx

)()(

=> "

B;;;;;= 4;;;;;!b&-K'( c1$;;;;;A'( L1D;;;;;H O,;;;;;-,!8 a(D0&;;;;;C,8): ( );;;;;K L;;;;;F 4;;;;;K!5 #;;;;;C"()03.1(),03.1( >= ,Kb-K LF B= ./0'( #C"(1 3+e,&-'( 9E* cE*1 3.

) :( a;;;;5$ 4'.;;;;CK'( B;;;;=(13-4) );;;;K L;;;;F 4;;;;K!5 #;;;;C"(yy ###, );;;;K L;;;;F D;;;;-*1,0 "" xx.

) #( c!8/&'( ): aE* (>]y 4!E<,A&'( 4'D,7K'( c%"! 0" # ## byayx ;!" )K LFa, b )K LF 4K!5 DH1.= 3 `!"Q DD*a, b.

(1-5) s¹d9

(2-5) s¹d9

(3-5) s¹d9

f�U)« qBH�« �UM¹d9


283


:(4-5)

) :( 4!',&'( O,-,!8E' @D1D"K'( c1$A'( L1DH )f1gF.

) #( hO,-,!8'( W>2 ZK cA&& D1D" @$!\F' 4-FKK 4H$D L5( B2 ,K

) d( 4!',&'( a!%'( D,H!i O,-,!8'( W>2 aD0&C(:

)5.0(),5.0(),12.0(),12.0( yyyy ####

) D( ): X$A-y D-*x = 0.5 *: D5 O,-,!8'( L1DH B=,0'( 4;K!%'( O;!/ 4;e/3.265 .D,C'( c1$;A'( 9;&" @D1D;"K'( c1$;A'( L1DH #,C" D*:;C4 . (>,;K

h./0'( $,Y&-( 4!A!F L1" S"6&

x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y 0 0.541 1.168 1.887 2.704 3.625 4.656 5.803 7.072 8.469 10.00

DH'( a!;5 X;78 B;/7! B',&'( L1 c;!8/&'()(xyy / . 4;!-,\'( 4%&;YK'( 4;K!5 D;H1: D-*x = 3.

x y

2.94 0.18256 20761

2.96 0.18110 60149

2.98 0.17967 59168

3.00 0.17827 10306

3.02 0.17689 06327

3.04 0.17553 40257

3.06 0.17420 05379

!8/&'( a!5 X78 B/7! B',&'( L1DH'( c)(xfy /

x 1.08 1.12 1.16 1.20

y 0.295 0.295 0.290 0.280

) :( );;;K 4!',KF&;;;C( 4;;;!D1D" 9;;;E* L;;;Q"&' 4;;;!K,KM( c1$;;;AE' )&1;;;!- 4P!;;;Q c;;;8/L5: 1: 4\',\'( 4H$D'(.

(4-5) s¹d9

(5-5) s¹d9

(6-5) s¹d9

f�U)« qBH�« �UM¹d9 ∫f�U)« qBH�«5

284


) ( !""#$%&'(y )""*%+,'( -"". /"",01 )""#23+ 4""'1.08 < x < 1.20 . -#5(6""78 6""9:; /,0<'( )#23+'( )%*+),( yx. 2=#; 69:;:y )%*+'( >?@ 6+1.

) A( !#$%&'( B; CD1 (?8y )#D=2E&'( )'62<,'( !*7#:

)4(34222 !"! # $%&'

() ! xxyyxyxn F#7n 9:, G#7H 661 ),#I 69:J. Kn.

B; LME+: xxxf 3cos22sin)( #"

) ;( )"9M6'( B", )#'2,N&OP( Q+(M9P )#6:67 BR:N !"#$%&'( RM"S*T& -"&'( )"5'25'(f(x)

)*%+,'( UVW 2

0 *++ x

23+#$ 2,#. X2.2O,'( )#:2O&, %2*+ Y$M; 6+1 !#$%&D' X2+#1 ?WJ$ Z'?:.

) ( )#3&+,'( !:ME'( U:69 BR:N Y$M[( X2+#<'( %2*+'( >?@ C(6W&O2$.

) A( B, UN ),#I O7( U:69'( (?@ C(6W&O2$:

$%&'

() $

%&'

() $

%&'

()

6,

3,

12*** fff

C#*'( >?3' )D$2*,'( \2%W[( O7(:.

) 6( ),#I O7( U:69'2$ )+2<&OP2$$%&'

()

169*f \2E#&OP2$(extrapolation).

)#M(M7'( 2N#,2+#6'( -. )#*#*7'( X2#,N'( ]678 C#*' U:69 -D# 2,#.: x y

0 1

0.2 0.92

0.4 0.84

0.6 0.74

0.8 0.62

1.0 0.4-0.29

1.2 0.22

1.4 0.24

1.6 0.27

1.8 0.29

2.0 0.32

(7-5) s¹d9

(8-5) s¹d9


285


] !"#$: /+7+,y -O;M C#*&O, %W B1 M2$1 6+1 x = 1.[

) ;( ),#I MR6Iy 6+1x = 0.22.

) ( B, UN ),#I 69:;yy , 6+1x = 0.4.

%&''()*: )""'(6'( C#""I B"", :6""$# 4""+[ _(M""0+y B; `""""D,; 23N:D""""O(behaves smoothly) a;

]6""",'( -""". )D"""H&, 2"""3+;8.00 ++ x 2"""++b."""*. ]6""",'( (?"""@ C6W&"""O+" U:6"""9 6"""#':& -""". %

B; a; K !:ME'(

/D1 U,&c# P; 9# U:69'()P; -%d#( )"%*+'(x = 1.0 )"'(6'( U2"H&( C6"1 $"O$y )%*+'( >?@ 6+1.

)'(6'( C#I -%<# -'2&'( U:69'(y 23D=2E& C#I: y %2*+'( L<$ 6+1.

x y y 1.4 0.9855 0.1700

1.5 0.9975 0.0707

1.6 0.9996 -0.0292

1.7 0.9917 -0.1288

) ;( )"'(6'(y )"*%+,'( -". /",01 )"#23+ 2"3'1.4 < x < 1.7 . )",#I 6"9:;x )"%*+'$#<N&'( -ON<'( U2,N&OP2$ /,0<'( )#23+'( :; Q+(M"9P )d#H 2,8 C(6W&O2$ -

XME#8 . ),#I 69:; C5y U2,N&OP2$ )D$2*,'()Mc2$,'(.(

) ( B, UN ),#I 69:;yy , 6+1x = 1.5.

) A( B, UN ),#I 23<#%&O& )I6 /D1J$ 69:;yy , 6+1x = 1.42.

%&D' _2,#I -%<# -'2&'( U:69'(" !#$y .;c,'( ),#I 69:" )*&y 6+1x = 0.5.

(9-5) s¹d9

(10-5) s¹d9


286


x y

0.35 1.521525

0.40 1.505942

0.45 1.487968

0.50 1.467462

0.55 1.444243

0.60 1.418083

0.65 1.388686

B; LM"E+y = f(x) . )"#+25'( !:M"E'( /"&7 )"#,2,[( !:M"E'( )d#"H "&N(2, K'( /D1 UH7&' U=2.: )*&c,D' )#'2&'( )#$#M*&'( )d#H)(xfy - :

. /2140321)0()0(0 yyyh

xfxyy !#!0 - -

)*$2O'( )'JO,'( -. )#$#M*&'( )d#H'( -. J%W'( 67 69:;(11-5).

J%W'( 67 Y, )#EDW'( !:ME'( )d#H C6W&O(:

)(2

211)1( pf

phfpfphxf 1 $$

%

&''(

)!#2#"#

F#7:

)( ixfif -

B; Q&+&O&':

)(22

)1( 01 1fhff

xf #!

"

Y=n = 4 )#'2&'( )d#H'( -.] )d#H'( M0+((15)[

34

567

8,$$

%

&''(

)##,!#," 00

2)12(10

1)( fn

n

p

dpdfp

hf

hxnp

(11-5) s¹d9

(12-5) s¹d9

(13-5) s¹d9

(14-5) s¹d9


287


Y= C5p = 2 '( /D1 UH7&' )*&c,D' )#'2&'( )#$#M*&'( )d#Hf :

)438180(21)2( ffffh

xf !#!"

!"#$: #H'( >?@ U(:6D' )$O+'2$ )%:$=, )d4Pf 9 )9M6'( B, X2#6:67'( a;: (4).

:

)d#""H'( /""D1 e2""%&IP( J""%W: ""#M*&'( \2""%W; M#5J""& M6""+ )'J""O,'( >?""@ -"".M#6*&' )#'2&'(f ] )IV<'( M0+((14) 23' 72H,'( J%W'(::[

)2,0(,)()4(12

2)1(

2

2012 xxfh

h

fffxf 9!

#!" 11

+ B; LM""""""""""E4)()4( Mxf + C#""""""""""*'( B;:if B""""""""""1 6""""""""""#f# P J""""""""""%W 2""""""""""3#.k!;105.0 . ),#I M2#&W( a6g# B; YI:&T# 4+; X$5(h B, )$#MI:

4/11024

min4

$$%

&''(

) ;"

!

Mh

k

)+N,, ),#I UI; /'8 -DN'( J%W'( U#D*& /'8.

B; LM"""".(1)()4( +xf )""""'(6D'f -'2""""&'( U:6""""9'2$ )""""+#$,'( . C#""""*'( C(6W&""""O2$

U:6""""9'( (?""""@ -"""". ^2""""%<,'()"""",#I MR6""""I K)8.1(f _2,6W&""""O,)d#""""Hf -"""". M:N?"""",'( )*$2O'( )'JO,'()15-5( :%W'( C97 M2#&W2$:h=0.1)",#I MR6I C5 K)8.1(f M2"#&W2$

h = 0.2)5'25 M, C5 K M2#&W2$ h = 0.3. x 1.5 1.6 1.7 1.8 1.9 2.0 2.1

f(x) 0.9975 0.9996 0.9917 0.9738 0.9463 0.9093 0.8632

B; X",D1 (?8f(x) = sin x C#"I aJ".h )*$2"O'((0.1, 0.2, 0.3) U"=.; -"%<&)9#&+),#I -%<& a; K)8.1(f J%W UIJ$h

),#I B; !*7&h -%<& -&'( )*$2O'( )'JO,'( )9#&+ Y, !E&& >?@minh.

(15-5) s¹d9

(16-5) s¹d9



ÍœbF�« q�UJ²�«

Numerical Integration (Quadrature)

ÍœbF�« q�UJ²�« v�≈ WłU(« º

ÍœbF�« q�UJ²�« …dJ� º

WDO�³�« ·d×M*« t³ý …bŽU� º

WDO�³�« Êu�³LÝ …bŽU� º

q�UJ²K� Èdš√ mO� º

QD)« �«d¹bIð º

W³�d*« q�UJ²�« mO� º

©W¹UNM�« s� q]łR*« »«d²�ô«® ÊuÝœ—UA²¹— WI¹dDÐ ¡UHO²Ýô« º

…œdHM*« �ö�UJ²�« º

”œU��« qBH�« �UM¹d9 º

291


ÍœbF�« q�UJ²�«Numerical Integration (Quadrature)

ÍœbF�« q�UJ²�« v�≈ WłU(«

! "#$% !"#$%& "'()% *+ ,-.'

!b

a

dxxfI )(

)i( /-0%"12 1!"#$%& &314 51!(6 )"17(8 (9$1:!%& *1! ;"1'"(9+< =>1?%& *1! *<1#( )6 5(@(@9$%&(analytical methods) 9 A B)"$>!%& 5%&)%& C'"# <% D$f C&3

;"(2:' 0(:2 E@(@9$ #F) G! "!5

0

2

dxxeI.(

)ii( 511%&)@% 511(@(@9$%& 5H(11?%& *<11#$ )116f I(116 ,11>2 011JK I11@>' "11!' L&< A 511!<@>! -11(M 5%&)%&f(x) )C"'(N ( 51.@$O!%& 0"1J'%& ,1>2 )'Nniix ,,2,1,0; !

"111P(@N 111?9' E111$%& C&Q&-111J%& 111G!– 51112-7$ *111! 111!>!%& E111K– I(111J%2-P#%& -"($%&R"5'(>! 5'!S+ )'N E.

*($%"9%& *! T+ EK)+ ( A)= (T))>%& !"#$%& D%8 U7@'.

!!"#$ %&'()#$ *+(,:

5%&) )7<+g(x) *((%"$%& *(0-F%& /J9$:

)i( 5%&)@% ;"2:"'! ;"2(-J$ *<#$f(x).

)ii( ;"(@(@9$ "P@!"#$ P:%& *! *<#(.

*VK E%"$%"2< : #b

a

dxxgI )(

292


5%&)%& *<#$ B)"Ng 5()<)9))<)9 B-(G# ( 5(%"!#$:&(interpolating polynomial)

W-O+ &<) ;"X(+ Y"'4 C'"# * L&<– C"()<)9%& -(M– "P!&)O$:& *#!(.

*<#$ "!'(9<g S!-%& I)O$:': "''VK 5()<)9p S!-%& <+np "1P'+ D%8 B-"FZ% *! 5()<)9 57-)%&n . !"#$%& =(-J$ G!$ E%"$%& #F%& *! 5.@$O!%& Q&S7[& ;\G!K :

b

a

dxxf )( ) 511%&)%& ]11(9f "11!(K C"K"11:!%& 5(<"11:$!%& 0"11J'%& ,11>2 )11'N 011JK 511K-_>!

"1P'(2equally spaced base points ( 1!"#$%"2 : b

a

dxxnp )( 59"1:!%"2 1G!( T31%&<

a!%& C3O+ #F%& &34 EK< A 5@@n I(J%& :n = 1, 2, 3, 5.

!"#$%& T))>%&– b 5%&)@% 5.@$O! 5()<)9 C"2(-J$f(x)

f(x) f(x)

)x(1p f(x)

f(x)

)x(2p

a b

n = 1 n = 2

0

x 1

x 0

x 1

x 2

x

1

x 2

x 3

x

f(x) f(x)

f(x)

)x(5p

)x(3p

a b a b

n = 3 n = 5

a b

0x

1

x 2

x 3

x

1−

x x0 x

1 x

2x

3

4

x


293

ÍœbF�« q�UJ²�« ∫”œU��« qBH�«

-+./&#$ 012 *!3'4)56781#$( (Simple) Trapezoidal Rule:

5%&)%& D'9'! C9$ 52<@0!%& 59":!%& =(-J$2 B)N"J%& c34 I<J$ f(x) 59"1:!%"2 D111'9'! D111@N *(111$0J'%& *(1112 111?&<%& I(J$111:!%& 0111O%& C1119$f(x) 5111J0'! E111K-0 )111'N

]11111111(9 A 11111111!"#$%&x=a, x=b )& -11111111a'&E%"11111111$%& #11111111F% ( *(11111111$0J'%& *(111111112 T+ A$ % $ %)1(,1,)0(,0 xfxxfx . 51111%&)%& *+ T+f(x) *1111! 5(%"!#$1111:& 51111()<)92 "1111'4 =-1111_J$

D%<[& 57-)%&)(1 xp )I(J$:! 0O T+.(

d-9'!%& e2F B)N"6

d-9'!%& e2F 59":! *<'"6 /(20$2<D@N ?9':

& ' & ')1()0(2

)()(2

01 xfxfxx

bfafabI ("

)("#

* & ' abhbfafhI "!(# ;)()(2

9:81&8 *!3'4)56781#$( (Simple) Simpson Rule:

*+ ,-.':

21,2,0baxbxax (!!!

*+ T+: )] ;\G! (= h[ 0112 xxxx "!"

) #11F%& -11a'& .( I(11J%& *+ ,-11.'<)2(),1(),0( xfxfxf 511!<@>! "11P@#) "11!8+ ;\G1111! 511112-7$ C&Q&-11116 *1111! B"11110>! "1111P' 5H(1111?%& E1111K ,(<>$%"11112 "P2"1111:9 *1111#!( <+ A

5%&)@% 5(@(@9$%&.(

ÍœbF�« q�UJ²�« ∫”œU��« qBH�«6

294


51%&)%& D'9'! C9$ 52<@0!%& 59":!%& =(-J$ f":+ D@N *<:2!: B)N"6 I<J$f(x) 511('"G%& 5117-)%& *11! 511()<)9 D11'9'! C119$ 59"11:!%"2)(2 xp 11J'%"2 -11!$ ]\G11%& 0"

$ % $ % $ %)2(,2,)1(,1,)0(,0 xfxxfxxfx *(111111112 B-<11111111?9! 59"11111111:!%& *<11111111#$ ]11111111(92 *(0O%&bxx,axx 20 )!)!.

)x(f

f(x)

)*<:2!: B)N"6(

)(2 xp

0x 1x 2x

a b

"1!# 5(%"!#$1:g& 5()<)9%& c34 5H(? D@N h& *3V2 5%<P:2 <?9%& "''#!(<5!">%& 5H(?%& EK ,(<>$%"2 <+ A i2&-%& ?.%& EK "'>! -!

2210)(2 xcxccxp ((!

C2&<G%& *((>$% 5J2":%& ]\G%& 0"J'%"22,1,0 ccc.

!#*2

0

)(2)(2

x

x

dxxp

b

a

dxxpI

*N ,(<>$%& )>2<)(2 xp D@N ?9' 5(@(@9$%& "P$H(?2:

& '

221201

;)2()1(4)0(3

02 xxabxxxxh

xfxfxfhI

"!"!"!"!

((#


295


"G!:(1-6)

*+ ,-.':

"("!

!

3

1

)57223(

)(

dxxxx

b

a

dxxfI

I(11116 =1111:9&I A d-11119'!%& e21111F B)1111N"6< *<1111:2!: B)1111N"6 *1111! 1111# I&)O$1111:"25%"9 # EK E@>.%& U0O%& =:9&<.

9%&: ) #F%& -a'&(

)*<:2!: B)N"6( )d-9'!%& e2F B)N"6(

) +( d-9'!%& e2F B)N"6:

& ' 26251)3()1(2

13 !(!("! fftI

) =( *<:2!: B)N"6:

10 !x 21 !x 32 !x h = 1

1)0( !xf 9)1( !xf 25)2( !xf

& ' & '322025941

31)3()2(4)1(

31 !(+(!((! fffSI


296


3220

3

15

27

32

4

234

!,-.

/01 "("! xxxx

exactI

0

315

322026

!

!"!

SE

tE

5;.<&:

5%&)%&f(x) "G!%& &34 EK *1#!+ Y%31%< A ;"1(@(@9$ "P@!"#$ P:( ](92 C-($O& 021X%"2 U10O%& =":9< A !"#$@% 50<2X!%& 5!(J%& =":9)E1@>.%& U10O%& ( 1# E1K

&31114 f(111% *111#%< A *($J2"111:%& *(111$7($'%& *111! 111!"#$%& R"111:! E111K ;"111!<!N "1119%& <1114T))>%&.

*<:2!: *<'"6 /(20$2 "P(@N "'@?9 E$%& 57($'%& EK U0O%& *+ a9\( Y%3#5('"G%& 57-)%& *! 5()<)9 "P:.' E4 5@_!"#!%& 5(@?[& 5%&)%& *[ ;&-.? T<":(.

%&'()=# >+?@ A7B )D@N+ C"7-) C&<3 5(%"!#$:& C"()<)9 I&)O$:"2:( Other Integration Formulas (employing higher degree interpolating

polynomials):

*+ ,-111111111111.'f(x) 5111111111111'(>! 0"111111111111J' )111111111111'N 5111111111111())>%& "111111111111P!(6 I<111111111111@>! 5111111111111%&)

n10 x,,x,x f"11:[& 0"11J' D!11:$(base points) . *<11#$ *+ "11!8 0"11J'%& c3114 "P'(2 "!(K C"K":!%& 5(<":$!(equidistant) ! -(M <+ C"K"1:!%& 5(<"1:$(unequally

spaced) . !"#$%& /-0 *! *"$N<!7! ;"!<!N )7<(K e(@N ;Q"'2<:

)+ ( j(11?) *$<11('– f$<11# ((Newton – Cotes formulas) 5(<"11:$!%& 0"11J'@%"P'(2 "!(K )">2[&.

)= ( j(111?)f<"1117 ( 111!"#$@%(Gaussian quadrature formulas) -111(M 0"111J'@%"P'(2 "!(K )">2[& 5(<":$!.

E@( "!(K -?$J' d<:< D@N )">2[& 5(<":$!%& 0"J'%& j(?.

f":[& 0"J' *+ ,-.' Y%3%<nxxx ,,, 10 ) I(16 "4)1'N I<@>!%& 0"J'%& E4<f I(6 =":9 "4)'N *#!( E$%& <+f ("P'(2 "!(K C"K":!%& 5(<":$! *(12 5K"1:!%& *+< A


297


T<"1:$ *($(%"1$$! *($0J' T+h B<10O%& I179 D!1:$< (step-size) . I(16 *+ ,-1.'<f =($-$%& D@N E4 5@2"J!%&: nyyy ,, ,10

5;.<&:

] 11!"#$%& 5H(11?– "11P'(2 "11!(K C"K"11:!%& 5(<"11:$! f"11:[& 0"11J' "11P(K E11$%&–

D!:$:

)+ ( 511J@H! 5H(11?(closed) 11!"#$%& "11$("P' C11'"# &38a, b Af"11:[& 0"11J' *!11X:2!:< d-9'!%& e2F E$H(? EK "!#*<.

)= ( 59<$.! 5H(?(open) *($("P'%& W)98 C'"# &38)"!4\# <+ ( 0"1J' *! C:(%f":[&.

*($("11P'%& *V11K *($%"119%& *11! T+ E11K<a, b ,11>2 i11! *(11$J20'! "11'<#( *+ "11!8 51$2"G%& B<10O%& I179 *1! 59(9? C".N"X!2 "P'N *($9&S! <+ f":[& 0"J'h *(12

f":[& 0"J'.

&314 E1K #1F <+ E1K ;\G!K E10>( T31%&< 1?.%&4 51.@$O! 51()<)9 C"12(-J$ 5%&)@%f(x) "1!'(2 A*($9<$.! *($H(? *"(0>( #F%& EK *((<@>%& *(2(-J$%& *+ a9\'

*($J@H! *($H(? *"(0>( *"(@.:%& *"2(-J$%&.

5(%"!#$1:g& k'&-17g 5()<)9)i12&-%& 1?.%& -1a'& ( 517-)%& C&3n -1!$ E1$%&<E4 f":[& 0"J' 5N<!7!2:

2!

!n

k

kyxkLxnp

0

.)()(

)())(())((

)())(()()()(

1110

1110

nkkkkkkk

nkk

xxxxxxxxxx

xxxxxxxxxxxkL """""

"""""!

("

("

),()1()0(!)1(

)()()(

)1(

nxxxxxxn

fxnpxf

n

"""(

(!(

3

](9:

33

!)(

(

(

xdx

xfdnfn

n

1

1 )()()1(

"!#$:\% 52:'%"2(interpolation): ),0( nxx43


298


Q".($1:\% 521:'%"2 "!+)E7-"1O%& ((extrapolation): *V1K3 -H1?[& *(12 i1J$]\G%& I(J%& EK -2#[&< xnxx ,,0.

(*)

)()1)(0(!)1(

)(

)())((

)())((

1)())()((

)())()((

0)())((

)()()()(

)1(110

110

1312101

320

02010

21

55555

6

55555

7

8

"""(

(

""""""

(

(""""

""""(

""""""

!

!

!

nxxxxxxn

f

nyxxxxxx

xxxxxx

yxxxxxxxx

xxxxxxxx

yxxxxxx

xxxxxxxf

nnnnn

n

n

n

n

n

"

] !"# $% &' ()*+,-:

nxnaxaxaaxpy #$ 2

210)(

./0000001,- 2 0000003# .0000004nyyy ,,1,0 ./00000015, 600000057819,-nxxx ,,1,0 :0000005; <000000=>#, 600; 9?9 $009 @A980039,- &00B 600/CD,- @EF80039,-naaa ,,1,0 ?9,- GH00' I 600; 9

6J/=7 :KC3L/ C1B F>- <> 8M,)(xnp 6178N,-.

$% O00>En I 6/,89*P00NE- 600/F F>,- 600?)F <0049Pn + 1 C8001# FF00; <0049/(8C39,- Q8NR- . ./0S $09 69/S <*, n <0=>#– 61780N,- 60,F839,- .-FDP0N87 (*)–

600,-F5, 60057819 6J/00= :005;f ) 600/F F>5, %np( 6J/00= :005; <00=># 800MP598*97 .004 I <098*P(integration formula) . <098*P,- T/0= $09 F F0>9 )0/U FF0; F0? /B &,80P,87

69/S VN> GH'n.

WA4009 .. !"##$%n = 1 ) FF00; I:00, R- 600?)F,- $009 6/,89*P00NE- 600/F F>,- X% Q8NR- C81# =2 (:5; <=># I:

,)(1)(

1

0

1

0

E

x

x

dxxp

x

x

dxxfI ## %%

Y/>:

& ')1()0(2

)(1

1

0

xfxfhx

x

dxxp #%


299


Z)>#9,- [7\ 6J/= X%) $% O>E01 xxh !#(.

)(12

3

"fhErrorE ((!#$

Y/>" 61C#9,- &B 89 6C1# &'& '10 , xx $% X% I& '10 , xx)".

]CD,- 6J/= $8')7 &C3#N E ^- $H_7 </5S F37 GH' ()/DR-.

!"$%&n = 2 :5; <=>#:

E

x

x

dxxp

x

x

dxxfI ## %%2

0

2

0

)(2)(

Y/>:

& ')()(4)(3

)( 210

2

0

2 xfxfxfhx

x

dxxp #%

$ N79N (F;8S X%:

)()4(90

5

"fhE !$ , & '20 , xx)"

W8"/% 6J/=,- GH' ) ]CD5,E (</5S F37 ^- $H_7 8M#8')7 &C3#N.

!"$%&n = 3 :5; <=>#:

& ')()(3)(3)()( 321083

3

0

xfxfxfxfdxxf h

x

x

#%

& '30)4(

80

53 ,,)( xxfh )! ""

(F;8S 6J/=,- GH' :9NP 83 $ N79N,

(Simpson's three – eights (83

) rule)

....... -H*' ..

W8178N $/P) *H9,- ]CD,- &PJ/= &#8')7 &C3# &5/ 89/B .


300


!"#$ %$&'()* Error Estimates

+, -. :/0(12 n = 1 ) !"# $%"&'( )*+ ,%-. /01 2 )*134 567%*4 89 5:*'9;.<=4 5:%1%>*4 8? ,?@7>A9*4.( 5*%'B9*4 89 (*))C& DE>A

)()()( 11 xRxpxf !

)()()( 10!2

)(

1 xxxxxRf ""! ## $

%%

%%%

##""!!&

!

1)()()(

1)(

)()()(

0

1021

0

11

1

0

1

1

0

1

1

0

x

x

x

x

x

x

x

x

x

x

dxfxxxxdxxRE

dxxRdxxpdxxf

$

8? F>=$ /G 5*1H69 5*4% 'H<IA /0x. *4 5:%:H9.*4 8J4 K!LA<15:*'.:

3'('40*1 )3'50/6*#$ 3!7+*0#$ 30')#$ 3'&81:( (Lemma 1: Integral Mean Value Theorem)

8'.*4%""""*4 M""""A'; 4N+)(),( xvxu 5""""OLA9*4 /""""G 8:.C""""E.9' (bax ,) M""""A';1 2)(xv 5OLA9*4 /G 5.!'P $7'#+ M4N),( bax) 8QG 2:

),(;)()()()( badxxvudxxvxu

b

a

b

a

)! %% **

8J41 .. 5:9;*4 83 R47FA))(( 10 xxxx "" S:O*4 D;* 5!*'< R'9T4%: ),( 10 xxx) 8QG 2:

0x x 1x

1

0

6

3)1(

21

1

0

2

2)1(1

0

2

2)1(

021

1

0

1021

0)(

1.)()(

x

x

xx

x

x

xxx

x

xx

x

x

f

dxxxf

dx)x(x)x(x( fE

!"

#$% &''(

!

"

##

$

%&&''(

&&''(

&

&& )

)

*

*

),(;)()( 103

10121 xxxxf +&''( **

, ),(;)( 1012

3

1 xxfE h +''&( **


301


!"#!$ : !%&#'n = 2

) !"""#$% &"""'( )!*"""+ ,"""-. / 0"""12$3'4 0"""56!'4 7"""8 01'$89+""":;4 0"""1!.!<'4 7= )=7.:>8:.(

,)()()( 22 xRxpxf !

?1<:

)()()()( 210!3

)(2 xxxxxxxR

f """! ### $

6$"@( 7A 0"1B8$9+'4 0C":.+8'4 0"81D'4 0"16E2 F"1>C+ $"2- G1C+:2 ; $22= E<; 64!D8'4)()()( 210 xxxxxx """ 0DC28'4 ,H 0+>$3 I:1'),( 20 xx.

0"CD2'4 &"'( 716."<8'4 JD2>)0,( 1x J"K<2 /– 7.":>8: !"#$% L4!M+":$>–

,B1 $8 &B#:

% &

E

hffhfhdxxf

h

h

"!'"

)()0(4)(3

)(

% &'"

""!()h

h

hffhfhdxxfhEE )()0(4)(3

)()(

(!)*+,- .-/01:

)2: J""N$OE(h) 0C"":.+8'4 0""81D'4 F"">C2. I46""8 ?P""3 01B""N$O+'4 J""8$92 L""3 /I468 ?P301B8$9+'4 0C:.+8'4 081D'4 016E2 F>C2 61MA4 J8$9+'4 ,H. /(.

J8$9+ 0BN$O8' 01'$+'4 01!1Q8+'4 R$+<2:


302


2"&"3%4 2:

7$9 4S(: dyyxxay

xbyx ),(

)(

)()( *+ ' !

!!

7TH:

),(.)(

),(.)(

),()(

)(

)(

axdxxda

bxdxxdb

dyyxxdx

xd

xay

xby

***+ " ,,'!

!

!

K$M 0'$<0 :7$9 4S(

dyyxxa

xb)()(

)(

)( *+ '!

7TH:

)(.)(

)(.)()(

adx

xdab

dx

xdb

dx

xd **+"!

% &

% &

% & % & )0(34)()(

3)()(

32

)()0(4)(31

)()(3

)(.)1()(.1)(

)(

fhfhfhhfhf

hffhf

hfhfh

hfhfdh

hdEhE

""#"#"" !

""

# "#""

"""!(#

% & % &

% &

% & % &)()(3

)()(31

)()(31

)()(3

)()(32)(

hfhfhhfhf

hfhf

hfhfhhfhfhE

"## ##""#"#!

"#"#"

"## ##""#"#!##

% & % &

% &

% &)()(3

)()(31

)()(3

)()(31)(

hfhfh

hfhf

hfhfhhfhfhE

"###"###"!

"## ##"

"###"###""## ##!###

01'$+'4 01!1Q8+'4 F>C2 7U4.:


303


2"&"3%43) :2"56!74,- 208/4%,- 2%"9,- 2"*:#(

Lemma 3: Differential Mean Value Theorem

I2$9 4S(f # 0DC28'4 ,H 0BK+8 0'4!(a, b) 081% J%A4 &B# !5.1 V2TH$:

),( ba-$

7= ?1<>: )()()( $f

ab

afbf #!""

7= $2N6H 4S( 7U4.)4(f 0BK+8 0'4! 016E2> V2TH /'4 0C:.+8'4 081D'401BN$O+:

),(,)()(

)()( )4( hhfhhhfhf "-!""

"###"### $$

)(3

2)(2.3

)( )4(2)4( $$ fhhfhhE "!"!###)

7= E""""<;$ ,""""H 0""""'4!h ] G""""%4.'4 ,""""Hh./ ! ?""""1<11 00" . W'S""""9..

,H 0'4! $Q:O2h [GN2 7= $22981 W'S'. /:

)(32)(

11,)()()(

2

)4()4(

hghhE

hffhg

"!###)

00"(( ..$

7= E<; X$N1=.: )0()0(0)0( EEE ###!##!!

#

'

''

"!##"##

"!###

h

hh

dttgtEhE

dttgtdttE

0

2

0

2

0

)(32)0()(

)(32)(


304


dudttgtduuE

h uh

' '' "!##0 0

2

0

)(32)(

u

u = t

u = h

h

t

'

' '

""!

"!#"#!

!

!

!

h

ht

t

hu

tu

dttgtth

dtdutgtEhE

0

2

0

2

)()(32

)(32)0()(

dttgttyyE

y

' ""!#0

2 )()(32)(

dttgtth

dtdytgtty

dydttgttydyyE

h

h h

t

h yh

'

' '

' ''

""!

""!

""!#

0

22

0

2

0 0

2

0

)(2

)(32

)()(32

)()(32)(

71+'4!""'4 7A X46""E2.22

2)(

&)( tth

tg" 7$+B""K+8 0""'4!'4. /2

2

2)(t

th" $""Q'

6$""@Y4)+( $Q"":O28'4 ,""H 0""DC2[0, h] 0""1B8$9+'4 0C"":.+8'4 0""81D'4 0""16E2 F""1>C+>H /&B# JK<2:

71H6C'4 0B8$98>.


305


% & )(9054

23

)(31

0;)()(31)0()(

5555

0

22

//

//

ghhhhg

hdttthgEhE

h

"! ""!

00""!" '

11

11,)(90

1,)(90

)(

)4(5

5

0#0")

00"#"!

0#0#"!)

..

...

...

hfh

hghhE

) ),(..,;)(90 20

)4(5xxeihhfhES -00""! 111

;!$% :(2-6)

&""B# F"">C+ 71""< 7."":>8:. Z6""<28'4 V>""@ ,""+D16C 7""8 J""9 ,""H [""CM'4 6\!""% J8$9+'4I J$38 ,H(6-1) ?1<:

' " "!!3

1

23 572)(;)( xxxxfdxxfI

;<,- : =/> :?*<#%,- @+A 29"*0

)23(34)46(

128

31,2,)(12

3

""!""!

00!##"!

//

// hfhE

2 3311

34,

319

319

minmax!!!440 EEE

,"B]O'4 ["CM'4 0"81% I"2$9 W'S"9. /6"1>9 ["CMB' )."B]'4 !<'45678

9:

315 74. 6"1>9

).B]'4 !<'4 78 J%= G>C'$> I2$9.

!"#!$ :(/8+%8 29"*0:

2 3 !f

hfhE

:00

31,1;)(90

)4(

)4(5

!! " ##

! "##$%&'( )##*+, -##.%/,01 2!##3$.3 "45##6 27##8 9(:;##6 <!%##35 =##*>1( 2? 9(:##@ "*!$A.1($

%&'

()

3220.


306


!"#$%& '$(")%& *+, Composite Integration Formulas

"A;> . "$,: B(C -.%/, "456 D5$*, 2E F,% 1( =*>1( -50G,1(low-order

integration formula) ".5H <In J K:546 -.%/,1( K:,8 L3G % 78[a, b] B(:#,8 M1N "45####6 D####$* ! :4####6I "####5OP& K:####,8 -####/ M####0E -####.%/,1((subinterval) KQ####R M####0E .

"45###6 M.###3, "###A;> . "###$,: B(C "45###61 ::###/,.1( D###5$*,1( 2###. "###&,% 1( "45###61(!"$/:. -.%/, . ::#/,.1( D#5$*,1( 2#. "#&,% 1( "#$/:.1( "45#61( M#0E -6R )05 %.58!56 2. -/1 S:R .1( T$U "4(n = 1) 2!3$.3 "456! J(n = 2).

-./0 : !"#$%& 1#23$%& 4!5 6+, Composite Trapezoidal Formula

S:###R .1( T$###U KQ###E%G1 B%###G5$*, "###+$:I! 25###G5$*,! 9(Q###R(! 9%###G5$*, -/###U1( 25###$5

-.%/,1( V%3R1 W1C!:

*b

a

dxxf )(

I X##A(!!-##G5 =##*>1( 27##8 B%##G5$*,1( QQ##E Q(P %##.0/ T## . D:##;1( !##Y %## Y =##*>1(! M R .1( BR, "R%3.1( 25$f(x) ".Q>,3.1( B%8:R .1( Z%$UI B%R%3. [!.&.!.

% 5Q1 2I \:; 9%.!.E!r 25#$. !Y %./ J S:R .1( T$U KQE%G1 9(::/,. 9%G5$*,)1%,1( -/U1%$ . 2I <I QQE % 5Q11+r ]%3?( *%G 2.rxxx ,,, 10 .

"1%R1( ZCY )8 "5OP& K:,8 -/ -!*:

r

xrx

rabh 0""


307


, -

, -

, - )(12

)()1(2

)2(12

)2()1(2

)1(12

)1()0(2

)()()(

)()(

3

3

3

1

1

1

1

0

0

rfhrxfrxf

h

fhxfxfh

fhxfxfh

x

x

dxxf

x

x

dxxf

x

x

dxxf

x

x

dxxf

b

a

dxxfI

r

r

r

.

.

.

//"+"+

+

//"++

//"+

+++

0

***

**

"

^5R:

rixx iii ,,2,1;),( 1 1 ".

2 3, -

))()()((12

3

)()()(2)()(2

21

1210

r

rr

fffh

xfxfxfxfxfhI

... //++//+//"

+++++ "

"0#####6,.1( "#####1(Q1( 2? 9(:#####@ !f // C#####>=, 25,8:#####*,.1( %#####_5,.5H 25#####$ L5#####G1( -#####/(extreme values) ".5H Q&!, T 78 J),( ba1. %YQ E:

;)()()()( 21 .... frfff r// //++//+//

2 3, - trrt Exfxfxfxfxfh ++++++ " )()()(2)()(2

1 1210

^5R:


308


)(2)(121

)(12

)()(

)(

12

),(,)(12

2

3

3

3

3

.

..

..

fhab

fr

abf

r

abr

bafrhtE

//""

//"" //"

"

1//"

)()(12

2

.fabhtE //""

`##. V##3% ,5 =##*>1( 2I <I2

1

r . ZC##Y D##5$*, B(:##. QQ##E SE%##A, (C7##8 W1C##1!

"4561(– ".5H B;E%A, <Ir – -P 5 =*>1( 278 " 9%$5:G, " T#,.5H `$a: M1N ") 9%#$5:G, " 2?)(.f // L5H 2. 25,;0,>. 25,.5G1 %_3; )Y KQ%E 2!/, b S!3r.(

-(+3(7 : !"#$%& 8/9!$9 6+, Composite Simpson's Formula

QQE 2!3$.3 KQE%H D5$*, ::/ )/r "1(Q1( L5H "8:+.1 c%,R B(:.1( 2.f

Q E QQE 2r + 1 ]%3?( *%G 2.rxxxx 2210 ,,,, .

2I "1!_3$ B$d 2I 2/.5!:

2 3,2 3-4 5.)()()(

90

)()()()(2

)()()()(4)()(3

)()(

)4(2

)4(1

)4(5

22642

1253120

2

0

r

r

rr

rx

x

b

a

fffh

xfxfxfxf

xfxfxfxfxfxfh

dxxfdxxfI

... +++"

+++++

++++++

"

"

**

^5R:

iii xxrabh 222,

2!!" " .

"51%,1( K:!61( )8 =*>1( QR V,/ 2I 2/.5!:

)(902

)(

),(,)(90

)4(

55

5

20)4(5

.

..

frr

ab

xxfrhE rS

66""

16"


309


)(180

)(

)(.2880

)(

)4(4

)4(

41

5

.

.

fhab

fab

r

""

""

2 3,

2 3-),(;)(

180)(

)()()(2

)()()(4)()(3

)(

)4(4

2242

123120

2

0

bafhab

xfxfxf

xfxfxfxfxfhdxxf

r

rr

rxb

xa

1""

++++

+++++

"

"

0

0*

..

'(7$:(3-6)

".5H V3R( * "+"3

1

23 )572( dxxxx

)-%d. -.%/, !Y! 1-6 T3; (L(Q>,3%$ W1C! J:

)I ( S:R .1( T$U KQE%G1 25G5$*,) <Ir = 2 "$/:.1( S:R .1( T$U "456 )8.(

)V ( 4 B%G5$*,(r = 4) KQE%G01 %_3; .

'2%&:

)-/U1( :@ (.(

2 3, -

rabhfhab

xfxfxfxfxfhI rrr

" //""

+++++ "

,)(12

)(

)()()(2)()(2

2

1210

.

)I (2%G5$*, )V (4 B%G5$*,

S:R .1( T$U KQE%G1 ::/,.1( D5$*,1(


310


)I ( 2%G5$*,(r = 2):

2 3, -

, -

, - 229225121

)2(2)3()1(21

)(12

)()(2)()(2

2

1202

6++

++7

//""++

fff

fhabxfxfxfhI .

)V ( 4 B%G5$*,(r = 4):

2 3, -

2 3, -)5.2()2()5.1(2)3()1(22/1

)()()(2)()(2 321404

fffff

xfxfxfxfxfhI

++++

++++7

214

848

12598352251

41 89

:;<= ++++

>?@

ABC

8/9:#(5)+# ;+#<! =(>+)9.&) +(?3%& 8$ '@AB$%& C&#)D.& /0(

Richardson extrapolation (or Deferred approach to the limit)

^#1%d V#5:G, M#0E %#._ . -!#6R01 %#. -.%/,1 2%$5:G, %_58 LQ>,3a5 "G5:* ZCY%._ . -/ 2. -A8I 2!/5 2I -.= .

2I \:; : : I ".5G1(%. -.%/,1 "*!$A.1(.

:1rI D#$*, "#$/:. -#.%/, "456 2. "&,% -.%/,01 "5$5:G, ".5H

QQE "*53$1( KQE%G1(1r B(:.1( 2..

:2rI "###$/:.1( -###.%/,1( "45###6 2###. "###&,% -###.%/,01 "###5$5:G, "###.5H

%_3; QE%G1( D5$*,$ 2/1! QQE "*53$1( K2r B(:.1( 2..

:1r

E "5$5:G,1( ".5G01 VR%6.1( =*>1(1rI.

: 2rE "5$5:G,1( ".5G01 VR%6.1( =*>1(

2rI.

%,1( 25,1Q%+.1( V,/ 2I % /.5 )1%,1%$!25,51:

)2(

)1(

22

11

rErII

rErII

+

+

25####$ "$####3 1( %#### .0E (C7####821

, rErE )%####._5,.5H 9%5:!:####A ]5####1! ( % Q#### E X$####6I"d1%d "1Q%+.:

^5RD L!0+. QQE.

)3(12 rrEE !


311


SC####R$!21

, rErE "*!$####A.1( "####.5G1( 25####5+, %#### /.5 êd####1( BbQ%####+.1( 2####.

-.%/,01I.

;+#<%& EFG HIJ '(7$":

"###$/:.1( S:###R .1( T$###U "45###6 )###Y ".Q>,###3.1( "###$/:.1( "45###61( 2I \:###; J 2I!: 4,2 21 rr

2

2

2

1,

,)(12

)(

rEhE

rabhfhabE

DD

.

E

" //""

`##. V##3% ,5 %## Y =##*>1( 2I <I2h `##. 9%5##3/E !I2r . )##$5:G, V##3% ,1( (C##Y! 9%####.O(Q ]5####1! 2? W####1C! J*$####A1%$. 25QQ####+1 "$####3 1%$ %_####3; )####Y 2!####/, b 9%####.!.E

D5$*,1( B(:. 2. 25;0,>..

)(

)(.

2

)(

)(

1

2

2

1

1

22

2

21

1

2

.

...

f

f

r

r

f

f

r

r

E

E

r

r

////

$%

&'(

) ////

2I % A:8 (C78:

)1()2( .. ff //7//

X$658:

4 52411

2

2 41

41

21 EEEErr

E rrr $%&'

()7

%_+$: )1(!R M1N =*>1( ".5H -G, D5$*,1( B(:. QQE ";E%A.$ T I <I.

'(7$:(4-6)

D$%#31( -%#d.1( )#,&5, M0E 2!3Q:%U,5: f%;5,3( "G5:* D$*(3-6) -!#6R01 -.%/,01 DQI ".5H M0E %._ .-%d.1( W1C )8 M*+.1( Q!QR.1(.

'2%&:

)3(

)2(21

)1(22

24

1424

444

222

EEEE

EIEII

EIEII

7E7

+ E+

+ E+

D

SCR 42 , EE:


312


)(3220

623)1()4(

)4(844)3(&)2( 2

exactII

I

EI

E "

+ E

25####,.5G1( !!!!" #$!!!!%&'( )!!!!'* +!!!!,- ./0 1!!!!2345'( 67!!!!- 0 80 )!!!!9: 3;3<34!!!!2;'(#"+=;9' 15$<>"'( 1"32'(.

!"#$:

1!2345'( 67- ?/@;AB; 0 ="3) $!A/4+C;34 1!2345 ( 3;3<34!2; 3!;"3D 80 E!"

21, II #"+=;'I $!=3 0 F+34$4!> G3!' H,0 80 I21, II !" 3J;+!, %&'($ )*%+,- .

!!% K!!3<5; !!" 3J;+!!, +!!,$=3 0#!!"+=;'( 1L3 +M!!AN, . O4!!P, 0 Q!!- 1!!"+M'( 1!!52,'($ 1<A,9' 13<342; 1"3D) ( +"M3R5@ 3<.

/0(12$34 5#$6'734

Singular Integrals

33'+;'( 354C'( /&0 K2&; (7* /4N," H,* #"+=;' #+23:

)0 ( 1S9 S"+="'( 1'(/'()+M;+2;C" $0 ( ?3D " 4T=0 $0 1"3D /,: 13U+M, V W<%;x QX'( Y/"#"+=;.

)Z ( 13U+M, V #"+=;'( [4;X.

/4!N," 4!3\ #!"+=;'( 0 ]4;N; #%N'( (7- QX F+2<+A [4$=7"'( 13//P'( K45'($nonsingular 0 80 I:

^- 1!!!!!!!!3"+_, 1!!!!!!!!9 S"+="'( 1!!!!!!!!'(/'((regular) ) 1!!!!!!!!3M;," 80finite #!!!!!!!!>+N;9' 1!!!!!!!!9<+D$differentiable 19!!!!%;"$continuous #!!!!"+=;'( Y/!!!!" Q!!!!X +!!!!M93T"; !!!!="3 80 I

"< 4$93+; 19A9A;Taylor series Y/"'( (7- QX(.

`- 13M;," #"+=;'( [4;X$(finite).

H!!93$&; /!!P< /4!!N,"'( #!!"+=;'( 1!!"3D Z+!!A&' 1!!3//P'( K4!!5'( 67!!- K!!3<5; !!="3$/4N," 43\ #"+=; )'* .#3$&;'( (7M' ?/@;A; 0 ="3 Q;'( K45'( ]P< Q93 +"3X$.


313


89,- : :,234 ;$ 5#$6'734)-:( ) ?3D ]P< /,: 13U+M, V 19"+="'( 1'(/'(x(

/a4N;'( #b0(singularity) 13'+;'( K45'( Y/&c<:

1- <*,=734 (Substitution)

#+T":

#"+=;'( !"1

0

cos dxxxI /,: /4N,"x = 0

]3$P;'( ?/@;A( 2tx "

!! ""1

0

2

1

0

2cos22.cos dttdttt

tI

/4N," 43\ #"+=; (7-$.

2- >?>?7$34 @,'1$(Series expansion)

#+T":

K<+A'( #"+=;'( HAN,

!"1

0

cos dxxxI

dxxxx

x

dxxxx

xI

!

!

#$

%&'

()*)*"

#$

%&'

( )*)*"

1

0

2/112/72/3

1

0

642

!6!421

!6!4211

!

! !

+,

-./

0 *)**"

"##

$

%""

112

1

*

12

1

0

2/112/72/3

11

0

2/1

0

1

2/1

0

!6!422

22/1

limlim1

dxxxxI

xdxxdxx

/4N," 43\ #"+=; (7-$.

!"#$% &'#()*$% +,-.* / 0 12345 !"# ,6 &'#()* 7$8 45:


314


3- ABCD34 E$6'734(Partial integration)

#+T": K<+A'( #"+=;'( HAN,

dxxxxx

dxx

xI

)sin(.22/1

.cos

cos

1

0

1

0

2/1

1

0

**#$%

&'("

"

!

!

!)"1

0

sin21cos2 dxxx

/4N," 43\ #"+=; (7-$.

86*26F : :,234 ;$ 5#$6'734)G(

<*,=734(Substitution)

#+T":

!3

*

*

)"

0

2 1dx

ex

eIx

x

]3$P;'( ?/@;A(:

xettx *"*" ,ln

!! *"*

)*"

1

0

2

0

1

2 ln1

1.1).ln( tt

dtdtttt

tI

+=; (7-$/4N," 43\ #") . 0 _&V0lnlim0

"2

ttt

.(

H00=34 E$6'734 I>J $6J !"#$:

Z!!!342;'( Z<!!!A< F+!!!,+3&0) //!!!P'( Z!!!342;=0.1234 )!!!'* FdT!!!"0.123 e(4!!!J* 3!!!&134C: ?+D40 1TdT< .+<+A&'( ( Q9PN'( R5@'( 0 f;,3(actual error) 3< K4+N'( $-$'($ #!!!"+=;9' 15$<!!!>"'( 1!!!"32'( F+3//!!!: 1<$!!!A&"'( 1!!!"32 ( 8$!!!9P'( /!!!&'( !!!" 4!!!<=0 $!!!=3

(upper bound) 84!!!!!!_,'( R!!!!!!5@9') #!!!!!!T" 1!!!!!!34_,'( g3!!!!!!%'( !!!!!!" Z!!!!!!A&B3 87!!!!!!'($

)()(12

2

4fabh 55*.


315


)i( #!!!!"+=;'( 1!!!!"3D 4/!!!!D Q'+!!!!;'( .+!!!!,+3<'( #$/!!!!J !!!!"!5

1

)( dxxy ?(/@;!!!!A+<

$A<"A [/:+D. x 1 1.5 2 2.5 3 3.5 4 4.5 5

y(x) 0 0.41 0.69 0.92 1.10 1.25 1.39 1.50 1.61

)ii( 1!J3;, Q!X h+!5;DV( R!5@ 1!"3D ZA&((i) 12<+!A'( .+!,+3<'( 0 ?!9: (7* 1'(/'( .+,+3< Q-y(x) = ln x.

)i( #!"+=;'( 1!"3D Z+!A&' $!A<"A [/:+D$ O4&,"'( H<C [/:+D " Fd= ?/@;A(:

!2/

0

)(

6

dxxf

13'+;'( .+,+3<'( ?(/@;A+<:

x 0 126

1226

1236

1246

1256

26

f(x) 0 .25882 .5 .70711 .86603 .96593 1

)ii( +3< Q- 12<+A'( .+,+3<'( 0 ?9: (7* 1'(/'( .+,f(x)=sin x R!5@'( 1"3D ZA&+X QX Q9PN'((i).

)iii( [$!5@'( ?!J& 4/D O4&,"'( H<C [/:+2' h+5;DV( R5@ 43/2; 1L3% ?(/@;A+<h ) 1'(/'( ?3D //: 4/D $0f ( #!"+=; 1!"3D )!9: #$!%&9' 1!<$95"'(f 1&3&!%

134C: ?+D40 1A"@' . $A<"A [/:+2' 1<A,'+< #(iA'( #& ?T.

4N, 0 ]: !*

""1

0

2

4)(,)(

xexfdxxfI

(1-6) s¹d9

(2-6) s¹d9

(3-6) s¹d9

”œU��« qBH�« �UM¹d9

”œU��« qBH�« �UM¹d9 ∫”œU��« qBH�«6

316


!!!!!!"#$%&' () *+!!!!!!,-.I /0!!!!!!1& 2!!!!!!0$+"&' 34!!!!!!5#6&' 7'48%!!!!!!9#0 /!!!!!!%":; <!!!!!!9=%9>+!!!=-"&' . +?4!!!;– @#!!!A%;B' C!!!A8 2D:!!!E 7'48%!!!9#0– <.!!!FA"&' .!!!A&'h 2G#!!!9"F& 2"H%-"&') <.FA"&' 44I&' .)n 2&'4&' 7:6&f ( 2!":; JF5 .E=F&I (:";+!& 2=:=!E(::+15 . <9=' 7K (". "#$%&' 2":;I (::+15 (:";+& 2=:=E.

"#$%&' 2":; <9=' !

1

0

11 dxx

'L8%" (.90"9 345#; 7'48%9#0h = 0.25 MFI,&' CA8&' 2":; <9=' 7K N $ 2":;. NO+H-&' CA8F& MF,9&' 4=&'. O.FI&' 4=&' (".

() *+,-:

"2/

0

)(

#

dxxfI

2!!":; P!!F0% 7!!$ 2G#!!9"&'h <!!!9=- () #!!-4+) 'LQI 2:+!!15 7#!!;+) 2!!I0+R S#=:=!!E 2!!&'4F& Nf(x) = sin x T

"#$%&' <9='I 2:+15 7#;+) 2I0+R S#=:=E@#A%;B' CA8& 2F0#6"&' 2":6&' 4U.). N.

)i( 2G#9"&' 2":; MV #"h <#9=& 2"WX&'

$$$

%

&

'''

(

)"

2

1

2lnxdx

9#0 2:+15 7#;+) 2I0+R S#=:=ET (.90"9 345#; 7'48%

)ii( 2A.0!!!!Y"&' /!!!!%":; SX"#!!!!$% () *+!!!!,-I (.!!!!90"9 34!!!!5#60 S#:44!!!!5 <!!!!9 Z= 4!!!!;

7'48%9#0)) ( 2G#9"h )< ( 2G#9"2h . () *+,-.J .K #!"V <!:%+%&' JF5

(5-6) s¹d9

(6-6) s¹d9

(4-6) s¹d9


317


!!"#$%F& (#%0.!!9="&' (#%:44!!I&' (#!!%":6&' . [#,:%!!9X& (.!!94+#1%:+ 2!!6:+A \!!0A+:4!!6% J!!F5 .!!E=F& +!!!8] I !!"#$%F& (!!!5 +!!0I% 2!!;X5 [#!!!A5^0 _!!&L. NI

2&B40J .K.

)iii( L8C0 ""2

1

5.0& hxdxI

7:; 4U.) IKJ ,, .+:46%&' MG MFI,&' CA8&' <9='.I `" (+#;. N(i).

)i( ' 2":; PF0% 7$ 2G#9"&h <#9=& 2"WX&' $$$

%

&

'''

(

)"

2

1

2lnxdx

T >+=-"&' /01 345#; 7'48%9#0 2:+15 7#;+) 2KXK& S#=:=E

)ii( "#$%&' 2":; <9=' "2

1

xdxI

>+=-"&' /01 345#; 7'48%9#0 S'L8%" N:

)) (h = 0.5 )< (h = 0.25

2!:#a-&' (!" ! bUc"&' <'+!%;B' 26:+A \0A 7K):%!9'(.!94+#1%:+ [#, ( .!E=F& 2":6& +8] +:46% JF5I 0#6"&' MFI,&' CA8&' 4U.). N . `" (+#;(i).

"#$%&' 2":; 2:+15 7#;+) 2KXK& <9=' 2/

0

sin

#

dxe x

"#$%&' 2":; 2:+15 7#;+) 2KXK& <9=' *

1

0

21dx

x

e x

(8-6) s¹d9

(9-6) s¹d9

(7-6) s¹d9

”œU��« qBH�« �UM¹d9 ∫”œU��« qBH�«6

318


)i( %&' 34.4="&' d#-#:0&' (" "#$%&' 2":; +4; 2:&# "2

1

)( dxxyI

>+=-"&' /01 345#; 7'48%9#0 x 1.0 1.2 1.4 1.6 1.8 2.0

y(x) 1.0 0.8333 0.7143 0.6250 0.5556 0.5

)ii( 2&'4&' e8% d#-#:0&' fLV d-#$ 'LQx

xy 1)( " MG MFI,&' CA8&' <9=#G(i).

)iii( %&' d#!!!-#:0&' 2!!!5."U" #!!!-:A5) 'LQ!!!Yg#0 2!!!:&#260#!!!9&' d#!!!-#:0&' 2!!!5."U" J!!!&Q 2G# N "#$%&' 2":; <#9= 45CGI 3+G.%"&' d#-#:0&' $ S#"48%9" >+=-"&' /01 345#60.

x 1.1 1.3 1.5 1.7 1.9

y(x) 0.9091 0.7692 0.6667 0.5882 0.5263

)iv( M!!!%U:%- 7'48%!!!9#0(i) .(iii) .!!!E=F& (.!!!94+#1%:+& [#,:%!!!9B' 2!!!6:+A \!!!0A6% JF5 "#$%F& 4:4U <:+I .T2:+15 7#;+) 2I0+R h:=E .V V

)v( !"#$%&' 2!":; <#!9=& (.90"9 345#; \0AI 3+G.!%"&' d#!-#:0&' !$ S#"48%!9" N>+=-"&' /01 345#60 #a:F5 dFE= M%&' 2U:%-&' `" 2U:%-&' (+#;..

"#$%&' 2":; <9=' +

1

22

1sin1 dxtt

:

"#$%&' 2":; <9=' ME;#-&' (elliptic integral) *2/

0

2sin411

#

dtt

(.90"9 345#; 748%9' , -16#"h

)i( 2&'4&'f 2:&#%&' A#6-&' 4-5 #a":; 7.FI": x 1 1.2 1.4 1.6 1.8 2.0

f(x) 2.718 3.320 4.055 4.953 6.050 7.389

(10-6) s¹d9

(11-6) s¹d9

(12-6) s¹d9

(13-6) s¹d9


319


&' 2":; <#9=& (.90"9 345#; \:0A% #--$": V "#$%: "2

1

)( dxxfI

T 3#AI"&' d#-#:0&' 7'48%9#0 _%0#UQ d-#$ (Q : 2":; <9=#G 7I-I 34!5#6&' fLa0.

'L#"& i+1#G j+8R' d-#$ ( k'. 2":; <9='. NI #aG+I% j+8) 26:+A OC0.

)ii( 2&'4&' 7:; S#:FIG MV 260#9&' d#-#:0&'xexf ")( .FI,&' CA8&' 2":; 4U.) MG M 2":6& _+:46%I MG(i).

)iii( #al%:A5Z) 4; 2:&#%&' 34:4U&' d#-#:0&' () *+G'

x 1.1 1.3 1.5 1.7 1.9 2.1

f(x) 3.004 3.669 4.482 5.474 6.686 8.166

!!!!"#$%&' 2!!!!":; <!!!!9=' 2

1

)( dxxf d#!!!!-#:0&' !!!!$0 S#-:I%!!!!9" (.!!!!90"9 34!!!!5#60

(m' _:4& 3+G.%"&' .;+R' 445 4U.). 2=:=E&' 2:+1I&' 7#(N) _%0#UQ MG.

)iv( 2G#!!!9"F& #!!!a0 h"!!!9: 2!!!":; +!!!0$) 4!!!U.)h !!!"#$%&' 2!!!":; <#!!!9= (!!!$": n!!!:=0I

44!!!IF& S#=:=!!!E >+!!!=-"&' /0!!!1 34!!!5#60N /!!!9,- M!!!G #!!!"$ 2:+!!!1I&' 7#!!!;+R' (!!!"(iii) . 2":; `" (+#;h MG (.90"9 345#; 2&#= MG 2"48%9"&'(iii).

:

)i( =' 2:&#%&' d#-#:0&' ("2":; <9 "2/

0

)(

#

dxxfI

(.90"9 345#; \:0A%0.

x 0 12#

6#

4#

3#

125#

2#

127#

f(x) 0 .25882 .5 .70711 .86603 .96593 1 .96593

)ii( M!!!G 2!!!0#Ug' M!!!G 2=:=!!!E&' 2:+!!!1I&' 7#!!!;+R' 44!!!5 7!!!$(i) .4!!!U&' () 7!!!F5 'LQ 2&'4&' d#-#:0 MAI: \0#9&'f(x) = sin x T

)iii( !!"#$%&' 2!!":; <!!9=- () #!!-4+) 'LQI .!!V #!!"G N >+!!=-"&' /0!!1 34!!5#; \!!:0A%0 3.!!A8&' 7!!U=h 2!!":; (.!!$% M!!$ <.!!FA"&'I 44!!IF& 2=:=!!E /!!9,- (!!" 7#!!;+R'

MG #"$ 2:+1I&'(ii) T 2":; `" (+#;h MG 3#AI"&'(i).

(14-6) s¹d9

”œU��« qBH�« �UM¹d9 ∫”œU��« qBH�«6

320


/%FU5 dF ?UZ9. *+R' (" o.+#E \FAZ)a ("W!&' 4-5t – 2!H=& (!" S#!9#6" \X!!Ag'– 2!!:-#K (:I0!!9 .) X!!8 7:!!; M!!AI: M&#!!%&' .4!!U&'. Na 2FU!!9"&') 2!!9#6"

+%"&#0/2:-#Kp ( 7:; 4-5t 2,F%8"&') n:=t 9#6"2:-#K&#0 2.( t(sec) 0 10 20 30 40 50 60 70

a(m/sec2) 30.0 31.7 33.6 35.7 38.0 40.7 43.7 47.1

4U.):

)) ( o.+#E&' 25+9v (" $ 4-5 t = 10 .t = 50.

)< ( (:-"W&' (:0 o.+#E&' #aIA; M%&' 2G#9"&'t = 10 .t = 50.

:

() *+,-: dttexexy

x

*"

0

22

)(

)9% 2&'4&' fLV (Z9.Z4 "#$% J"Dawson's integral.(

(i) ))( 3.!!!A8F& 2!!!":; +!!!0$) M!!!V #!!"h 2!!!&'4&' 2!!!":; <#!!!9=& #!!!V+#:%8' _!!-$":y

#"4-55.0"x ]2!":; <#9=& O) )5.0(y [ (::+!15 (:";+!& 2=:=!E>+=-"&' /01 345#60.

)<( 2!!":; <!!9=')5.0(y !!E /0!!1 34!!5#; 7'48%!!9#0 (::+!!15 (:";+!!& 2=:=>+=-"&'.

)q( M!!!!G _!!!!%0#UQ M!!!!G C!!!!A8&' 2!!!!":; <!!!!9=')< ( 2!!!!":; () 7!!!!F5 'LQ)5.0(y

.'4!U&' M!G 34.!U." M!V #!"$ 2:+15 7#;+) 2I09& 2=:=E 2A.0Y"&' O.#9% 2:9#:6&'0.4244364.

(ii)4+,-"&' "#$%&' 2":; <9='

25.0

0

dxx

ex

(15-6) s¹d9

(16-6) s¹d9


321


:

!"#$$%&'() *$$+),-"'( !"#$$%&'( .$$/0 #$$12345& 6$$%"3 7$$"#%&/' *$$3,,8'( 9:$$5'( *"3; ,#+3< 7"#%&/' *3,,8'( 9:5'( =,>? 9345&' @!A"B C *"#0 D,,8&"'(

!"

32

2

3

2

)(

x

x

dxdyyxI

)'( 9:$$54 *>3>$$E'( F$$&"3; G#$$H> 6$$%"3 IJ$$'()*$$"32'( K$$583 L$$3> *3$$H#32'( 7$$"#%&

55.790"exI (D:)E'( ./0 7"#%&'( G&%M:

(*))(

3

2

" dxxgI

L3>:

(**))()(

32

2

!"

x

x

dyyxxg

*$"3; G#$H>' *$3,,0 *$23:5 IN 9345& ,M8BI 9$345&'( O3$; *$B:8"' P#&>M$Hg

7$$"#%&'( *$$25M" K$$B 5#$$2M'( Q$$84 ,$$M0[2, 3] 7$$"#%&'( O,R&$$HM O3$$2'( SJ$$T *$$B:8"') C :3RU((**) ."B *"3; *B:8"' @!Ag *52M'( ,M0x = 2 7"#%&'( O,R&HM

!"

16

2

)4()2( dyyg

*4)/5"'( *3,,8'( *23:5'#4 7"#%&'( (JT G#H> 6%"3).

!"#$%&:

V- 7"#%&'( *"3; G#H>I *$3M#A'( *+:,'( .M>M"4 G3:2&'( *23:5 O(,R&H#4) D,$0#; 6)H4"H– 53H4'( 6)M#2'(.(

]'()*$: SJ$T O,R&$HW& O,R&$H& X) 7$>'( Y#$MAN :$1Z3 7$"#%& IN [$" *$23:5'(*3H#32'( 9:5'(.[

\- #13/0 /E> K&'( *+3&M'( KB ]5R'( :(,2" G#H>.

(17-6) s¹d9

”œU��« qBH�« �UM¹d9 ∫”œU��« qBH�«6

322


'+",-$%& ./0$12&!-"% '3!"#$%& '03/4)%& ./0"$5"% '03026,%& 7&789& :!7;

'0$12&!-%& <!=/6%&

>?/02%&

>3026,%& 775%&

'03/4)%& ./0"$5"%

# " )%8" ,#+3? *B)_E) ,#+3?Z( 1$" AZ 3

34 n

# *B)_E" G:` × *B)_E") ,#+3?C( C = A.B 3n # *B)_E" G:` × F+&"),#+3? y( x.y A" 2n # #13/"#0 .'? *B)_E" 73/>&),#+3? UL,( A = L.U

3

3n

# K5R O#ZM 7>) ,#+3?x( yx)(x "% LUA 2n # *3B)_E" *',#8" 7>) ,#+3?B(

) 7> aB#%&n K5R O#ZM( CBLUAB "% )( 32. nnn "


o‡‡‡‡‡Šö*«

W�UŽ �«—U³²š« Ã–U/ ∫ (1) r�— o×K*«

‰uBH�« �UM¹d9 WÐuł√ ∫ (2) r�— o×K*«

W�UF�« �«—U³²šô« Ã–U/ WÐuł√ ∫ (3) r�— o×K*«

W¹eOK$ù«Ë WOÐdF�« �U×KDB*« qO�œ ∫ (4) r�— o×K*«

325

!"# #$%&#:

!!!"# !"""#$% &'() *"""+, -$"""./0 12#3"""4 -/5'"""6) 7!"""4, 10'"""86) -9'""":6) ;"""<$ -)7+46) -/ '0=8 >? @'A6) B7)$5 1CD/ -E 1$+D/6)– 7"+46) )F"% !2#/$– -", GH"#H

) 17,I J)7K!4 1)LH6M0/NOP) )F"% Q"+DR S'"048 Q!,'"O4 '0"O6)$ Q)'!0"O6) S7!0T >?/ >""? 7""+46) 77"", M""+T$ U""H!T'A V!""OH) 1""W' @'""A6) B7)$""5 M4""O# >""? 16!""D6) X$7 M""/7:

U#!8O.

M#04/6) Q)$#O6) >? 7+46) >? @'A6) B7)$5 77, >AD0 >6!H6) X$7R6):

M#O6) 1977 1978 1979 1981

@'A6) B7)$5 77, 19234 22846 24558 24884

)E ( M#O @'A6) B7)$5 77, '7T X$7R6) )F% Q!#!04 1)7YHO!41980.

)J ( M#""O B7)$""56) ZF""% 77"", -E !""#/+, )F[1980 \""+4 7""T25432 3""AY6) M""/0T !""/? >? '7:/6) 77D6) >? >4O#6))E (]

!"#$% &%'($% : >AY6) 1!C#6) !#076 -E &'^#:

78.46130.6291.5

17.5914.59003000.

!

"

yx

yx

) E( -$7$ -0H67!"""D/6) X07"""4H -$7 ;F"""5+6 _$!"""R M""":0'A4 1!"""C#6) )F"""% X"""5 7"""R$E/$ >LR6) L!8H'I) ab)'R[ 1!T'E MD4'( M4':/ Q!4!O56) a0/R.

) J( $""% >""AY6) 1!""C#6) )F""26 A$4""c/6) X""56)x = 10.00, y = 1.000 M""/0T !""/? >? d77D6) X56) >? 3AY6))E ( ]

) e( a""/$ >"" LR6) L!""8H'I) -$7$ -0H67!""D/6) X07""4H 7""D4 >""AY6) 1!""C#6) X""5 7"",E6) a""0/R b)'""R[M0'""., 1!""T'E M""D4'( M""4':/ Q!4!""O55)$ J""O )F""% >""? 3""AY6)

X56)MR0H#6) *+, @+, 1= .

(1) r�— —U³²š«

(1) r�— o×K*«W�UŽ �«—U³²š« Ã–U/

326


)$"#$% &%'($%: >AY6) 1!C#6) !#076 -E &'^# :

x – 9 y + 2 z = 1

8 x + 2 y + 3 z = 30 2 x + 3 y + 6 z = 31

) E( # X"%"cM":0'A J'!":H -/G – S X"56 !"%!#:4A )F[ d'!"0HY) X"5 d3"4 -0`7"H4/ ] QI7!D/+6 >AD/6) J0H'H6!4 1!C#6) )F% ] )F!/6$

) J( M":0'A @"04AH6 M4"O!#/ S'$"< >"? QI7!D/6) JH8)G–S !"24'!:H -$"80 B"054-""""8/0 !""""/ V'""""OE. X""""56!4 !`7""""H4/ M"""":0'A6) ZF""""% -""""/ Q)'""""0'8H M""""=N= @""""4A 1""""=

>40':H6))( #$#%$#%M0'., 1!T'E MD4'( M4':/ Q!4!O56) a0/R b)'Rf4$ .

) e( H6) 7"""D4 GH!"""#6) X"""56) >"""? 3"""AY6) M"""/0T !"""/ X"""56) -E !"""#/+, )F[ M"""=N=6) Q)'"""0'8 $% 1!C#+6 A$4c/6): 4 z 1, y 2, x

*+%,$% &%'($%: X/!8H6) -E &'^#

&!

1

0

2

4dxeI

x

;'""5/6) M4"".6 M""48'/6) S7"",!:6) 1)7YH""O!4 U""H/0T J""O5HO. '7""T– Mg0""< 1)7YH""O!4 V!""AHTI) 3""AY– S$""AY6) 1""R5h T *""+, X$""<5+6 J$""+A/6) M""/0I -0/T'""6 M505""<

-00'.,. X/!8H6) M/0T JO5) 1= -/$I -00'., -0/T'6 M505<.

-."/$% &%'($%:

J""451N""OP) )'$"""8F 7I$() h'!""4$ X"""O#6) S'""=8 >"""?!""=!# i)$ " 1"""84 Z!""4/ >#f"""?M/!0:6) 1$0 1/() " S!"056) >? !2H6!O'4 M/() 1$:H6$. 7"075H *"6[ M"/!D6) S$,7"6) 7"DH$

-""/ X"O#6) M""/() b!"#4E 7)7"",E X"0+:H6 S'8!""/6) M5+"O()M0/N"OP) -A$""6) >"? M""<!Y$ >4'D6).

/O# -$0+/ J'T( >4'D6) -A$6) -!8O 77, >AD0 >6!H6) X$7R6)M X810 Q)$#O:

M#O6))t( 1950 1960 1970 1980 1990

-!8O6) 77,)P( 77 95 127 166 200


327

) E( 1)7YH""""O!4""""6) -!8""""O 77"""", '7""""T Q!""""#!046) ZF""""% M#""""O >""""4'D6) -A$1975 M#""""O$1984.

) J( -!8""O6) 77"", -E Q""/+, )F[ -077""D6) -0F""% -""/ X""8 >""? >4""O#6) 3""AY6) J""O5) \+4 7T147 M#O !#$0+/1975 \+4$ 186.785 M#O !#$0+/1984.

) e( >"4'D6) -A$"6) -!8"O 7"0)LH X7"D/ '7"T X$7"R6) Q!"#!04 1)7YHO!4)dP/dt( M#"O1990 .

-0"($% &%'($%: MO/Y6 7R$E M67!D/6) 'FR M0'., 1!T'E

xe x !

-H$0# M:0'A4– M040':H6) M/0:6!4 j!`7H4/ -$O?)'x = 0.5.

W�UŽ �«—U³²š« Ã–U/ ∫(1)r�— o×K*«

328


' (1,0)x.

) J( )A Q)'0'8H -E Q4= M"H4!=6) MA:#6) M:0'ii xx cos1 " 'F"R6) $"5# J'!":HHr

M0`)7H4) M/0T d34 -0`7H4/' (1,00 )x.

) e( 'F"R6) 7!""R0P M"4$+A/6) Q)'"0'8H6) 77", 1"8r M""/0:6!4 -0`7"H4/ -0/T'"6 !505"<10 x ]

) 7( 'FR6) 7R$Er M/0:6) ZF24 !`7H4/ -0/T'6 !505< :10 x.

!"#$% &%'($%: M67!D/6)* + 01)( 2 !!, xeexf x 'FR !26r = 2.

) E( M:0'A -/ -0'0'8H @4AN-R M/0:6!4 !`7H4/ M67!D/6) X5610 x.

) J( 'FR6) ;,!cH 7R$Er.

)e ( 'FR6) )F% $5# !0D04'H J'!:HH M0'0'8H Mg0< k'HT).

)7( 7""5)$ )'""0'8H @""4AM""/0:6!4 !`7""H4/ M""5'H:/6) Mg0""<6) ZF""% -""/ )10 x J""O5)$ M/0T >? 3AY6)x !20+, Q+<5 >H6).

)$"#$% &%'($%: -0H0#K) -0H67!D/6) X5 J$+A/6) -E &'^#:

),()2.0(sin

),()2.0(cos

2.0

2.0

yxGyey

yxFyex

x

x

,

,

M:A#/6) >?

* +- .20;20, ////, yxyxD

M0'0'8H6) MH4!=6) MA:#6) M:0'A4.

) E( 4=)-0H6)76) -E QF , G M:0'A6) ZF% J'!:H l>mA l'm. -!::5H :


(2) r�— —U³²š«

&12% &%'($%:

) E( M""67!D/6) -E Q""4=)0cos)( 2 !, xxxf A4""c6!4 7""5)$ 'F""R !""26 "r " >""?S'H^6)

329

!

! DyxGG

DyxFF

yx

yx

"#$%

"#$%

,1

,1

) ( !"#4 $%&'()" )*+'", -&$%#(. /01 2, 3.%$%4'2,0 00 && yx.

!"#$" %"&'$": 25 6%78:

'''''

(

)

*****

+

,

-

&

'''''

(

)

*****

+

,

&.

4

1

4

0

y,

2520

5200

0252

0024

9:#;(. <):=8(. >? @A#,(. 25 6%78@y x &A B@):C -:&$%# <.+;':D)"–

>+$E.

) 5( >:::? FG:::" 2$*+:::'", 9:::;(. <):::=8(. >:::? @:::?8 -:::&$%#(. /0:::1 %):::&' 2,:::H8 >:::1I.0),(@ I 9*.+'".

) ( 9::J >.K::DA( L::'")CM 3::8)4 2M)5 :( !::"#J N <::O83 -::&$%#(. /0::1 2::, 3.%::$%4' P%7::Q(. >::?()" )*+::'", !

0x 0 &L::'")CM 3::8)4 2 R.@ N : !::"#8 S::$4 T%'U)::J NV !:"# <:W N):X"%)&' 2),:H Y, -&$%#(. /013 3.%:$%4'– L:?.%'U. Z@:H 9:J–

F%7Q(. >?()" )*+'", !0x 0 &.

()*+$" %"&'$":

) 5( -J@7Q,(. >[A?'''

(

)

***

+

,

-

-

-

&.

20179

24812

633

2$:$WAW,(. ):X$A,)\ ]:(ML, U 25 ^:$?"A = LU -:$,E%.@; <.+;':D)" L:(0@ NS0?A( B@)C.

) ( ;'D..01 <+ 9#;(. <)=8(. >?( >$A?'(.yx &A ^$?'''

(

)

***

+

,

-

&

5

20

3

y.

) _( 25 3"W. :yyxx 1-- .&/ ttt A

W�UŽ �«—U³²š« Ã–U/ ∫(1)r�— o×K*«

330


]^$?yx &A NE,%(.@ t %@a+, ](M %$b$)transpose( -J@7Q,(.[

-,$U )D?( -UcO(. /01 <+;'D. <Wyy 1-- .tt A

) +( T%::b.9::#;(. <)::=8(. >::? L::84,$ S::$4yx &.t >::$A?' L$+::( 2)::4 .0MLU

-J@7Q,A(A) . -=?c, :-"@A#, %$d >?(. >$Q)7'.(

(,*'$" %"&'$":

) 5( -(.+A( -$X'8,(. !@%7(. >@+C 2e@4y >@+C()" -8$",(. #)&8(. +8\ -,@AO,(..

6 4 2 o x

211 61 7 1 y

) ( ),4':DV. -$+@+?(. +C@5 !@%:7A( 2'@:$8 -f$:Q <.+;':D)" -:W()W(. -:C%+(. 2:, -$(-$,),g..

) _( 9',$U +C@5yy 000, +8\x = 0 <$U L(04@ Nyyy 000,, +8\x = 1.

) +( h\ .0M -::(.+(. 25 <::Ay -$A::H)7'(. -::(+)O,(. !::&?'0&%0%00 byyayx >::4 ^::$? 2,a, b >4 -,$U +C@GJ N i$?Q ++\ 2,a, b.

!*'$" %"&'$": 25 6%78:

1&23

1

2 ln dxxx

) 5( >,)4'(. -,$U D?.I 2@D",D j+\)U <.+;'D)""" SI 0);')"@5.0&h.

) ( G#;A( F@AO(. +?(. -,$U D?.max

3 -,$&(. /01 9JSI.

) _( -:::::,$U :::::D?. 9:::::AO7(. G:::::#;(.actual

3 -:::::,$&(. 9:::::JSI -:::::,$&(. 25 3:::::,A\ .0M91 >,)4'A( -#@"H,(.6.9986217Iexact & . Y:, 9:AO7(. G:#;(. -:,$U 2%)U <WG#;A( F@AO(. +?(. -,$U.

) +( -J)::D,(. -::,$U k::A"' <::4h >::,)4'(. ::D?8 25 )::8+%5 .0MI <)::U%. -::WcW( )?$?::Q I S%?8,(. l"b j+\)U <.+;'D)" -$%b\

2@D",D j+\)U <.+;'D. +8\ -&")D(. m*)'8(. Y, 2%)U .


331

4 53,5.10 "x.

5 ( 2$%$%4' !"# -$"$%&'(. -,$&()" )*+'", -&$%#(. /01 2,20 &x.

2- n. 2$'(+)O,(. >?( 2'@$8 -&$%# 2, j+?.@ j@#; !"#82$'$:

04

03

&%-

&---

-

y

x

eyx

yex

) 6%J.000 && yx(

3– -(+)O,(.0)1()( 2 &--6 xeexF x +8\ %0C )X(x = 2 .

5( %# 2, 2$'@#; !"# -,$&()" )*+'", -(+)O,(. >?( 2'@$8 -&$10 &x.

( %.%4' 3.%, ++\ +C@5) h\)H' F5 Smultiplicity (%0C(. .01.

_( j@:#; !:"#@ N i$?:Q(. %0C(. @?8 )$O$"%' )"%)&' %)&'' -&$%# T%'U. -$"$%&'(. -,$&()" )*+'", -&$%#(. /01 2, j+?.@10 &x.

4– 5 ( 9#;(. <)=8(. >? +C@5bx &A ^$?

777

8

9

:::

;

<

&777

8

9

:::

;

<

&

10

8

4

b,

320

332

222

A

-J@7Q,(. >$A?' !$%# 2\ L(0@A 2$$WAW,(. )X$A,)\ ](ML, U.

( h5 +::U L::85 6%::78 c$::A?' 3::$#\LU -J@7::Q,(A . >::?( -::&$%# T%::'U.9#;(. <)=8(.bx &tA ) ^$?tA -J@7Q,(. %@+, 91A.(

o- lC',(. )D?( -$()\ jZ)74 3.0 -&$%# T%'U.x a#O$ F0(.@-UcO()" 9:

! yx 1 CDBA -& -

2, >4 ^$?A, B, C, D :J@7Q,::O"%, -: -nn= 2:, >:4@ NA, B -J@7:Q, @ Nj+%78, %$dy lC',1=n .

,O(. ++\ <4 I )X'?%'U. 9'(. -&$%#(. 9J -"@A#,(. -$")D?(. 3)$A

(3) r�— —U³²š«

- ! ("#$%&'()" #$*+,)" #$+-.* /".$-.0% 1! /&(ii xx ln31 !"

2.'$+%%) 3$4)" 5$4,6-47)" ( #-&-.+% #-8"9%&" #:-; <=

W�UŽ �«—U³²š« Ã–U/ ∫(1)r�— o×K*«

332


6 - #$-*>)" /?9'$@:)" A$B, 3$4) #$-.-.0%)" C.$*)" 2.'+%) #-D'D=" #-.B,)" E,% F.57)" GH #-.-.0% #I-7 2.'+%) GH'0)"5 <.5.J)" *.K)" 1! LMN :

# $ # $ 0c,,2,1;cx.x 1 %!"!" kT kk

9$$$-45)" 3$$$4)" L$$$)O :cx.x "! T $$$%&:G8"9$$$%&" 2$$$-.+% <P$$$& 1-89# $ nR&0x

5$$Q# $ 1'T( R$$-4)(T( #H5S$$7:M) G$$S-*)" .$$*+)" T$$7, 5$$QT <! U1! <! U#H5S$$$$$$$$$$$$$$$7:M) #$$$$$$$$$$$$$$$-%"V)" A-$$$$$$$$$$$$$$$+M) #$$$$$$$$$$$$$$$+M*: #$$$$$$$$$$$$$$$:-; .$$$$$$$$$$$$$$$&0! 5$$$$$$$$$$$$$$$Q

)(max)(1

TT ini)(

**!.

W! 9$$; X$$,! Y.$$S, #$$-.-.0% C.$$* RZ$$( /$$-*Na, b, c 3$$4) /?9'$$@:)" A'$$B, #-*>)"A x = b/'H5S7:) #-%"V)" A-+)" 1! Y.S,5 U .-.0%)"T GQ RZ()":

!( #H5S7:)"aT: 5.03 !) 1.02 !) 9.01 !)

2( #H5S7:)"bT: 6.03 !) 4.02 !) 2.01 !)

[( #H5S7:)"cT: 2.13 !) 8.02 !) 4.01 !)

W- '\-!5 ] A'B,)" 34) '\:"9>%D" 10:- RZ()" C.*)" <! S ] '\:"9>%$D" 3$J ] "V':)5

_ – ! (; 9$$$$$ 5! 3$$$$$:'0%)" #$$$$$:-+ "!1

01011 dx

xI $$$$$D'& T.$$$$$4,:)" a&$$$$$K F9$$$$$N'; A"9>%

#&0.:)" 3':@%D'&5 Uh = 0.25 .=" 9$4)" 9 5!P$*>M) LMN 1-$&5 a$,-& 1.'$;5 UGM@S)" P*>)".

b - #)"9)" 1! Y.S,

# $# $# $# $ 32

2041

432

0141

432

323

322

321

**""",

** " ,

** "" ,

!

xdxcxbxaxS

xxxxxS

xxxxxS

xS

#-)'%)" /','-&)" 3:0%D% G%)" #-@-&*)" #-&-@0%)" #4-.K)" GQ:

ix -1 0 2 3

if 1f 2f 3f 4f

)" GQ ':94 59 #-&-@0%)" #-3S 3:0%D% G%)"f F.%S)" GH- .3,2 9>%D" ] cV$Q A #:-; .-9+%) #-9594)"f (2.5) #:-;5# $5.2f /. A-; d%,%D"4321 ,,, ffff.


333

xxexy

x sin

2

!

) !( #:-; 2D4"y 9,Nx = 0.005 *+H A';.! #(Z() #&.+: /'&'D4)" e-:` f".`g& .

) 2( X%&'Ò 1.';*5&J:)" 2"5`)'&exacty 2'D4 GH G&D,)" P*>)" #:-; 9 5!5 Uy.

) [( ] .-I7 A! .-&0 G&D,)" P*>)" 3Q

)i( h-$$&H ".$$-&0 1'$$0 1OX$$)V G$$H 2&$$D)" 1 2'$$D4) #$$+-.* i.$$%;" A$$( Uy #$$;9&.$$$&0! #$$$:-; 2'$$$D4) #$$$+-.*)" cV$$$Q C$$$&*5 Uy ) fj$$$`)" 3$$$4 9$$$N! <!)! (

\%4.%;" G%)" #+-.*)" ':9>%D:' .(

)ii( ".-I$$$7 P$$$*>)" 1'$$$0 1 k"51h-$$$&H X$$$)V G$$$H 2&$$$D)" "V$$$Q 1'$$$0 1O 1h-$$$& A$$$( U A-; e-:` LMN 'J-! C&*,-x R-41''x ? A! .

!"#$% &%'($%: 1! Y.S,1ln2)( , xxxf

) !( #)9'@:)" 1! /&("0f(x) ! 94"5 .V` '\))*+$"* F.%S)" GH- .4,3.

) 2( #$)9'@:)" 34 25M*:)" 1! Y.S,0f(x) ! #$*+,)" #$+-.*& .V$`)" "V$Q 9'$`-l#%&'()")(1 ii xgx !" C-&*%M) 6M7% #-)'%)" m-7)" <PH U)2.'+%% <! ( 1-89%&:

#*+, <P&0x R-4- .4,30 &x ]

(i) 1ln2)( "! xxg

(ii) 21

)(

!x

exg

(iii) 1ln22)( ! xxxg

) [( C$$$&*4 #$$$-&.'+%)" #I-$$$7)" 1$$$: /".$$$-.0%) 1'$$$0 1O '$$$&.'+% n.$$$D=" #I-$$$7)" 5!F9$$4"5 #$$$-&.'+% #I-$$7 1$$$: .$$(0! X'$$,Q ( '89$$$%&: #$$)9'@:)" .V$$$` L$$MN 35$$$74M)

#:-+)'&30 !x.

) 9( %)" 99N .h9;A';.! #%D mM&% #;9 L)O 3575M) #&5M*:)" /".-.0 .

(4) r�— —U³²š«

!"# #$%&#: ! "#$%

W�UŽ �«—U³²š« Ã–U/ ∫(1)r�— o×K*«

334


'&()&# #$%&#: &'()'* &+,'* -(.%'* (%/0' ! "#$%:

6115

7136

21

21

!

!

xx

xx

) !( 1#2334'* &335 &33+,'* -(33.%'* *6337 8339yx A +#33:'* 0033; <33=/> 033?2! -33@ A

BC=(D='* <52$4='A&'()'(E2 A:

)i( 2$4='* B%(F *6G (= #F6* <5A +#:'* <H/H; -! +#:'* <%IJ.

)ii( &3333+,'* -(3333.%'* (3333F *6G (3333= #3333F6*yx A K3333/H; -! +#3333:'* 3333IJ

+#:'* .

) L( -(>#! <@C@ M'G <E#N= B(E(IJ'* O*#?PE:

)i( <52$4='* KHJA (Q/H=(; M'GL , U R/JA = L U .

)ii( &+,'* -(.%'* KJ 0?2!yx)( LU

)iii( *2 *#33/#F) S33E+ T633'* K33J'* /33IJ)' <33/#/#F)'* </$334)'* <33N/#+ 33= *033J

U/H; BH4J.

)iv( &+,'* -(.%H' +2E9='* KJ'* 8= /IJ)'* 0DE KJ'* #(>.

*+#,&# #$%&#:

= CF ! "#$% A, B, C <DE#= <52$4=)( nn" = CF !2 A<=2HD=A,

C !2 A 10#3$%= #3/V <52$34=y U3?)=)1( "n U3?)='* L(3IJ L23H+='* !2 A-23HD=

x R/Jyx 11 ## CBA

U3333?)='* M3333H; K233334JH' (3333)N/#+ &3333H/ (3333=/5x &3333E/#N)'* 003333D'* L(3333IJ L23333H+='*2 A

K95! (=Q/! (/E2 A /)N/#+'* = KF &5 <E2H+='* </E(IJ'* B(/H=DH' .

-&!"# ./0,1&#:

)W(LIJ* 1# AD

)X(LIJ* 1# CE

)Y( LIJ* F = D B E

)Z(LIJ* yx F


335

.02()&# ./0,1&#:

)W( KHJA //@H@= /H=(; M'GAAULA

)X( KHJC //@H@= /H=(; M'GCCULC

)Y( &+,'* -(.%'* KJyz C T!yz CCUL

)Z( LIJ*zv B

)[( &+,'* -(.%'* KJvx A T!vx AAUL

.3456:

x \ L2H+='* KJ'* 27 ])(%'* U?)='*:

yzvx 1111 #### CBABAA

76(8&# #$%&#: S/E+)'* -/>$ %xfy & +(N%'* 0%; <=2HD= </'()'*:

x 1.08 1.12 1.16 1.20

y 0.295 0.295 0.290 0.280

) !( 3333= </'(=F)3333I* <3333/020J K3333; K33334J)' <3333/=(=\* S2#3333$H' )23333/% <^/33334 S3333E+

K>! 2! <@'(@'* <?#0'*.

) L( S/E+)'*y <N+%='* &5 M=.; </(Q% U'1.08 < x < 1.20.

M=.D'* </(Q%'* <+N% &/@*0JG 0?2!(x, y) .(9/! 0?2!2y '' <+N%'* _67 0%;.

79(%&# #$%&#: ! "#$%$ % xxxfdxxfI ln)(; 2

3

1

(

) !( K=(F)'* <=/> LIJ*I *6,)= A 2IE=I 10;(> -*0,)I(Eh = 0.5.

) L( `+,H' T2HD'* 0J'* <=/> 0?2!)T#.%'* (maxE.

<J/J4'* -(>#\* 00; #a0> &'()'(E2N <=/> &5I J &)'*(Q/H; BH4 .

) b( K=(F)H' +2E9='* L*2?'* ! B=H; *6GI 27:

6.9986217Iexact

&HD$'* `+,'* <=/> LIJ(5actualE . <NE(3I'* <3?/)%'* 8= <?/)%'* _67 S$)) K72

maxE Oc?'* &5)L (d

W�UŽ �«—U³²š« Ã–U/ ∫(1)r�— o×K*«

336


) 0( K33333=(F)'* ! (%333339#5 *6GI 33333;(N'* -*0,)33333I(E U33333)=/> L33333IJ)I UE33333:' <33333EF#='* 10

#a033333> Ae#33333J%='*- f(33333+)>g* `33333+,' <33333/#.%'* <^/333334'* -*0,)33333I(E– K233333+'*

L233H+='*h K33=(F)'* <33=/> M33H; K2334JH' <33=.)%='* <5(33I=H'I 0033DH' <J/J334

N A 233IE=I 1033;(> S33/E+) <33'(J &335 (33Q/H; BH334J &33)'* -(33>#\* 33= U33I$%

<=/> 8= #(>2h &5 1(+D='*)!.(


337

x

xexf

1)(

# 0%;x = .0195

-(>#! <@C@' L/#N)'(E -)) B(E(IJ'* ! "#$E .

) L( +2E9='* L*2?'* 8= <?/)%'* #(>.

) b( 333 h+D) *6(333=' i33392<^/3334 &)(xf &3335 1#2F6333='*)! ( (333=2=; <333N/>0 #333/V (333=/>

-/N' <EI%'(Ex 1#/^4'*1) <<(x .

) 0( <'*0H' j#,! <^/4 0?2!f -/N' <EI%'(E S0! ]k()% O(+;l iH4)x 1#/^4'*.

)37 (L(IJ' j#,\* <^/4'* _67 -0,)I*$ %xf 0%;x = .0195 .

:2()&# #$%&#:

) !( )2/% <^/4 SE+– /#/#F)'* 2I5*#<'0(D=H' #6? 0(?/l B*#= m=, <

02

)1(ln)(2 !#!& xxxxf

(k0)E= <=/N'(Exo = 0.25.

) L( &33333337 #63333333?'* *63333333Q' <+2E33333339='* <3333333=/N'* ! -3333333H; *6Gx=0 `3333333+,'* L3333333IJ(5 A

B*#/#F)'* _67 ! BE@*2 A <I=,'* B*#/#F)'* = #/#F) KF <?/)%' LJ(4='*

(/+, L#(N)))(/D/E#) m/'2 (#6?'* *67 2J% .

) b( a/3333E LE3333I'* ) <3333'*0H' <E3333I%'(Ef ( &3333+,'* L#(3333N)'* &33335)&3333D/E#)'* 3333= g03333E (

#6?'* *67 2J% B*#/#F)H' .

) 0( <'Ì='* _67 &5 L#(N)'* 8/#I)' </#/#F) <^/4 n#)>*.

)37( a$%<J#)N='* <^/4'* = *0J*2 *#/#F) 6 A <=/N'(Ex0 = 0.25.

'&()&# #$%&#:

) !( a$% m2(? <N/#+ = B*2+, RC@ 6– &+,'* -(.%'* KJ' K0/c : 02 # yx

+#,&# #$%&#*: <'*0'* ! "#$% :

5.125.1 !# yx

$ % 32

(E (k0)E= &k*0)Eg* &E/#N)'* KJ'0,0 )0()0( yx.

$ % 0132 ))#!!! xdxcxbxaxS

(5) r�— —U³²š«

!"# #$%&#:

) ( !"#$ %&'(

W�UŽ �«—U³²š« Ã–U/ ∫(1)r�— o×K*«

338


) %( )*+,*( -#-./*( !0#/1 23 456*( %&'(.

) 7( -#-./*( !389:" ;08 SGT .

); ( <;===$'>* !===?8>5"*( -===#-./*( @(-===" ;;===A - B+===$- !===#1+",* C#'===: D===' E===>A D8===:) -+#F"*( B;6/&(!l .(

'#(&# #$%&#): !*(;*( G H-91 :

" #

" # 2061

01

)(32

2

321

$$% %&

$$ %%%&

&

xxxxxS

xdxcxbxaxS

xS

!#*+/*( @+1+#?*( D"./&/ 2/*( !#F#?5*( !#?#F./*( !'#-I*( 2J: x: -1 0 2

y: 1y 2y 3y

) ( G" D. !"#$ ;08 a, b, c, d.

) %( G" D. !"#$ ;08 321 ,, yyy.

*+,-&# #$%&#: !*(;*( B#$ 25F# 2*+/*( D8;0*( y B#$ HF? ;1A x.

x 0 0.2 0.4 0.6 0.8

y 1 0.92 0.84 0.74 0.62

) ( <8.@+1+#?*( KLM* N;8;'"*( O8-9*( D8;0 G .

) %( !P/I"*( G" D. !"#$ -Q;$y' !P/I"*(8y '' ;1A x = 0.4.

*.,%&# #$%&#: G H-91 : ( &&2

1

1)(;)( xxfdxxfI

) ( !3+====&"*( !===="#$ R====>?/ B====.h %+====&'* !===="ST*(I !#-====IA B+====$- !====,T,* +'#'====: U V-'1"*( W?I N;A+$ B(;6/&+?

) %( D"+./*( !"#$ %&'(I (L6/" X V-'1"*( W?I N;A+$ B(;6/&+?5.0&h.

) 7( 23 D(Y&*( D' ;A )% ( L+6/( Z"25.0&h.

) ;( 2==/0#/1 B(;6/==&+?)%( X)7( D=="+./*( !=="#P* O; -==6[ -#;==P/ E==>A D==:'(I X \0Y"*( %(-/$]( !P#-5 O#?5/? !#+M1*( G" D)G8&;-+I/#- ^+9#/&(.(


339

22

maxmaxmaxmax

105.2105.05

3

!"!!"

#$#$#"# zyxf

2

max105.0

2

2105.2

543233

#!"

!"

" $!" $"

f

f

zyxf

(ii)

% &22

max

22

108125.05.1105417.0

105417.041

31

21105.0

5.14

32

!"!!'#

!"$$!"

# #

$#'#

# #

$#"#

"!

""

f

zz

yy

xx

ff

zz

yy

xx

ff

zxy

f

(iii)

% &

40cot.

40

401

40cot

40sinlnlnln

40sin

yyxx

ff

yy

xx

ff

yxf

yxf

#$#'

#

#$#"#

$"

"

( )

yyxx

yf #$#"# .

40cos

40.

40sin

(1-1) s¹d9

32

max

22

max

10629.01498.01042.0

1498.00749.02075.0sin2403sin2

1042.0075.0cot401

21105.0

!

"#""$!%

$"$$$

"$&"$

f

f

f

f

!"#$: !"# $%&' ()!" f !"#$% &%'()*+,:

‰Ë_« qBH�« �UM¹d9 WÐuł√

(i)

(2) r�— o×K*«‰uBH�« �UM¹d9 WÐuł√

340

W�UŽ �«—U³²š« Ã–U/

-!

x

x

xxfxxf

!

"#!

"$ %"!

06144.0

)14.1(6

.)1(6)1()(

5

56

x

x

xxfxxf

!

#!

&$ %&!

"

""

0130819.0

.)4.2(6

.)1(6)1()(

7

71

61

x

x

xxfxxf

!

#!

&$ %"!

24.0

.)2.0(6

2.)23(3)23()(

2

22

32

x

x

xxfxxf

!

#!

&$ %&!

"

""

00530199.0

.)8.5(6

2.)23(3)23()(

4

43

33

xfxxf $ %"! 707099)( 44

x

x

xxfxxf

!

#!

#&$ %&!

"

""

00180370.0

)197(70

70)7099()7099()(

2

25

15

"####$%&' "####$()*+' ,- .####/*$ 0+1####(25f ,& "####$()* 3####/4- 2####55f 6####5789%&' :$; "<=-if >' ?@ABC9.

!"#$:

,& D'"EF: 4142135.12 !

8F/2A '1G4: 4.12 !

,- 6F)$ '1H4: )015.0(

0142135.0

'

!

x

x

,- 2-: 55 1056368.2 "#' f

,G4 0+1(2: 55 106.2 "#' f

(2-1) s¹d9


340

(3-1) s¹d9

!"#

$%&'

!"#

$%'

1

221

1

221

lnlnln

.

((

((

kpp

ppk

0303.0

)007.00075.0(4.101.0

.1..1.

1

1

2

2

2

2

1

1

1

1

2

2

2

2

121

2211

1

2

2

1

1

'

&&'

)*+

,-. /0/&

/1

/

)*+

,-. /0/&

/'

)*

+,-

./0&/&

/'

/

((

((

((

((

((

(((((

kpp

pp

kpp

kpp

pp

xg

xxg

xf

xxf

xxf

/1/

0'

/&'/

/&1/

&'

0

0

0

56

5697)(

.)32(4

.)2(4

)2()(

max

5

max

5

4

98.10135)32(4

56

56

5

max

max

maxmax

'&'/

/

/1/

f

g

xg

-../,#)$% &%'()..*% 0..1 2)+..3$% 4..5($% 06 76g 8+..,/-9) 7:+..*/10136 4..5($% ;-..< =$ >?+@<$% ABCD% -/,#)$% &%'()*% 01 2)+3$%f.

2.42.03221.091.09221

..2.1..21

...

1

/

222/3

2

422422

'222&2&222'

/&/&/01/

/33

&/33

&/33

'/

4'5

'5'

0 tztx

ztx

xztxy

tty

zxy

xxy

y

ztx

y

xtzytzxy

(4-1) s¹d9

(5-1) s¹d9

341

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

I-! )- (

9.04.29.0

1.092212.03221.03

.2/2

.2.

...

/

2

max

3

22

2

&&!

###&###&#!

& & !

((&

((&

((!

!

x

yy

uyuu

y

yyxxu

uxx

yux

))))

))

)

)J ( "###$<*K+' ,- E###LC$) ###4 M###9%+' ,###K 9###N; "(OM###( :58###N$ 6x ,& 0###+12 P 6F8##Q+' R##L+' S##K$;)

)

(( .x ) T28##N*22.4 ( "##$()* R2R##L :$##; "##(O- 6##5

64 M9%+'x.

-U

maxmaxmax

...),(

yxf

yxyyf

xxf

yxf

& ! %

! ((

& ((

!

2,K$&' V"9+' TR#L ,K 3O R8#W$> ,K DCO RW2F Px, y S+XR(a, b.

* +,

* + -bbaba

abababa

bbx

ba

axy

bab

aab

y

bby

aay

y

babaxbaxba

abyyab

"""""

& """"

!

((&

(

((!

!##&##!

"

& "

#"!

((

& ((

!

""!%"!"

""!%"!"

""

.)1)(2()2)(2(

.2).2(1).2()2(

1

..

018.010513105

16

.)2(

3.)2(

23

..

222)2(

233)2(

2

34

22max

(7-1) s¹d9

(6-1) s¹d9


342

%25.50525.0max

143

31

62

0525.00180.00345.0max

0345.0310304104569max

max ! !

" ""#" #

# !$

"%#"% !#! !

ff

f

yxf

f

bax

!"#$:

!"#$ %"$ &$ %"'() *+ ,"-. &"'/0( 123"4$f 56"'7'$ %"'() 8+x + y *". 9"4#:/4 *' !; %'() 934# 1(<=x, y >&# ,-. ?@6+) A/'() B(60/2a, b 3'CD' !; <(20/ AE (

A/'() F'7 GHx, y . %'() 3D24# IJ 3'+f A$3/$ <(20/$ *' ><K32') %$@&2 6L6a, b(

baba

abbabayxyxf

"#"

""#

"" #

232

23

22),(

B(60/2 ()A/'a, b <(20/$ IL AE AE M=N$ *OE Pf A-( 3'; 94#:(:

& '

& '

%65.1

0165.0005.030005.03max

3)2(

)1(3

)1)(32()2).(2()2(

1

3)34(

)23(3

)2(

)2(3

2).32(1).2()2(

1

..

max

2

2

22

2

!

%#% !

" "

"

"#"""""

(

(

"

"

"

"

#""""

(

(

!(

(#!

(

( !

f

f

f

ba

a

bababab

f

ba

b

bababaa

f

bb

fa

a

ff

001.)(0

,)(sin910)166(.9999995.0

)(sin)001(.61)001(.

211001.0cos

),0()(,)(sin61

211cos

32

32

))

"%#

#"

*#"

x

x

x

xxxxxx

+

+

+

++

%%!"#$: A""E *(""'($ ,""Q)+ G)<""$ 16""E =""N F""R6.166 G)<""$ I""L *+ A""D0(%(3CD @ 3' ,$J <<;(.

(8-1) s¹d9

343

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

)1))((sin(10166.0

9999995.0001.cos

9 !"

#$

% xtruncationE &

!"!#$%&' (!$&' )* +*.9999995 ,-.' /01 !#21 34'5( 67%& 8!89.

!"#$: ! "#$ %&'(&)*+ ,-+"#*+ .*/ 0-#1*(2:

cos .001 = .999999500000042

3(&456 7428 %9"*+ ! :!13 3(&1;< 3(5=->.

? -@

) !( AB)*+ %*(C D6 EBF*+ "C: 110 !k

G&1)8*+ %*(C D6 EBF*+ "C: k!"105

) G( f1(x) = .33333 , f1(y) = .71428

!"#$%& '!()%& !"$*%& #!+%& ,"-#%& .-/%& 012)%& .-/%&

Operation Result Actual

value Absolute error Relative error

yx# 11010476. " 22/21 410190. !" 410182. !" xy ! 38095. 8/21 510238. !" 510625. !"

yx$ 23809. 5/21 510524. !" 410220. !" xy % 11021428. " 15/7 410571. !" 410267. !"

-HHI J(HH&4>5*+ KLHHI DHH6 D2HH'$ EHHBF 1HH2M! ! NHHCO410267. !". PQ(HH8$ ! :!R(91! %'>F* %*-2)> "58 G('C*+ +LI.

) S( f1(x) = .71425 , f1(&) = .98765 " 105

!"#$%& '!()%& !"$*%& #!+%& ,"-#%& .-/%& 012)%& .-/%&

Operation Result Actual value Absolute

error Relative error

uy ! 41030000. !" 41034714. !" 510471. !" 0.136

wuy %! )( 11027000. " 11031243. " 424. 0.136

vuy $! )( 11029629. " 41034285. !" 465. 0.136

vu# 51098765. " 51098766. " 110161. " 410163. !"

(9-1) s¹d9


344

T1B*+ %&4>< ! NCU$uy ! EHBF*+ HM*- 1&VHW XH4B> EHBF .*/ :"Y81&2M (Z&6 D2'$*+ . 1&VW*+ ""5*+ .4< P8($*+ R ['\9 +L ]+-' 1H&2M*+ ""H5*+ DH6 G1H \= -!

& D2HHH'$*+ EHHHBF*+ 1HHH&V8& ! -" 3+1HHH&^M 1HHH2M& XHHH4B>*+ EHHHBF*+ _HHH6 . ""HHH5*+ AHHH&># +L ]+- 1HH&2M*+& 1&VHHW*+ ""HH5*+ AHH> u 1HH&2M P8(HH$*+ XHH4B>*+ EHHBF*+ _HH6 D2HH'$*+ EHHBF*+ HHM*-

3+1HH&2M `&HH* . (HH> GHH$#8$ aHH&C2 %&2(HH'C*+ J(HH&4>5*+ GHH&818 %HH&>I! bHH=-8 %*EHH'>*+ KLHHI- DHH6 G1HH=*+ -! 3+"HH# 1&VHHW ""HH< .HH4< RHH')*+ -! 3(HH2&1)8 &&-(HH'8> &""HH< T1HHB HHM>!

3+"# 1&2M ""<.

? -c

)!(

(i) 41051000

1000* !((!p

999.5 < p* < 1000.5

(ii) 4997.5 < p* < 5002.5

(iii) 9985.005 < p* < 9994.995

(iv) 9995 < p* < 10005

)G( p .1 .5 100 1000 5000 9990 10000

*max pp! .00005 .00025 .05 .5 2.5 4.995 5.

)!( x 2x

3x 26x 3x

Exact 4.71 22.1841 104.487111 133.1046 14.13

3-digit

(chopping) 4.71 22.1 104. 132. 14.1

3-digit

(rounding) 4.71 22.2 105. 133. 14.1

%B-2=>*+ %>&)*+: Exact: 149.13.141046.133487111.104)71.4( ! !)f

(10-1) s¹d9

(11-1) s¹d9

345

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

"52 AB)*+3 R(91!: 3-digit: 149.1.14.132.104)71.4( ! !)f

(chopping) ;0.14!)

R(91! %Û^* G&1)8*+: 3-digit: 149.1.14.133.105)71.4( ! !)f

(rounding) ;0.14!)

&8)&1B*+ > ,M D6 D2'$*+ EBF*+)R(91! %Û^* G&1)8*+- AB)*+= (

.04.636489.14

0.14636489.14 *!

!

)G( "52 AB)*+3 R(91!: 3-digit

(chopping) ,4.15149.71.4)371.4)671.4(()71.4( !)! !)f

DB5& R(91! %Û^* G&1)8*+-: f(4.71) = -14.6

D4& (>M &8*(C*+ > ,M D6 "&"#*+ D2'$*+ EBF*+ b2W& D*(8*(2-: 3-digit

(chopping) 0093.

636489.145.14636489.14 *

! ! AB)*+:

3-digit

(rounding) .0025.

636489.146.14636489.14 *

! ! G&1)8*+:

!"#$ %#&' () *+,- .'/, 0&12- %3/3 456 (+) 78682+"9#$ 7+: ;/+< 7- 0+=2)-( >)? (788@,5: 78:12# A8'B () (6C,#$ D!E#$ 7- FG >)? ( +62 @+', H+#G I+", 41

(++) %8J++BK$ %++"82!#$ (++) *++9:81)- .(++3/3# ?++82"9#$ %++#&' (++)@ 4++1 &++,,- .++'/, 0&++12- % () &,JB')? (()&LG M@,5: 012 HJN.

(++) ?6++C#$ &++:-@ (++) (6++C,#$ D++!E#$ 7&++B",)? ( (++) *++,N)- ( 7- H++#G ++O28) 7++: P++1 4++1 Q&6&++C'#$ 44++N4 @ ?2++L Q&++8J:N3 ?2++L (++98J:N H++#G ++:O Q&++8J:N

@3 +:O Q&8J:N . 44+N P+8J"9 (+R D+!E#$ $S+R P+8J"9# T2+!#$ U4+'G 7- A+L$@#$ 7+:@D!E#$ $SR 4#@9 (9#$ Q&6&C'#$.


346

? -d

)!( 14.2 D2'$*+ EBF*+ 4103.1 !"

)G( 1.00 D2'$*+ EBF*+ 110995. !"

?-?e 86.625, 39.375, 119.50, 71.50

?-??

%2-'C>*+ %>&)*+ %C&CW*+ %>&)*+ n Computed pn Correct value pn

0 1.000 1.000

1 .33333 .33333

2 .11110 .11111

3 11037000. !" 11037037. !"

4 11012230. !" 11012346. !"

5 21037660. !" 21041152. !"

6 31032300. !" 21013717. !"

7 21026893. !"! 31045725. !"

8 21092872. !"! 31015242. !"

?-?f n np

0 1.000

1 .33333

2 .11111

3 .037037

4 .012346

5 .0041153

6 .0013718

7 .00045727

8 .00015242

A = 1./3. = .333

B = 3./4. – A = .750 – .333 = .417

(12-1) s¹d9

(13-1) s¹d9

(14-1) s¹d9

(15-1) s¹d9

(16-1) s¹d9

347

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

C = 1./2. – A = .500 – .333 = .167

D = (4 × B – 2 × A) – 1 = (4 × .417 – 2 × .333) – 1

= (1.668 – .666) – 1 + (1.66 – .666) – 1

= .994 – 1 = – .006

E = 3 C – 1./2. = .501 – .500 = .001

Q = E/d = –.001/.66 = –.166

3–34

p(x) = (( x - 9 ) x + 27) x – 27

p(2.95) = (- 17.8 + 27) (2.95) – 27 = 27.1 – 27 = 0.100

p(2.99) = (- 17.9 + 27) (2.99) – 27 = 27.2 – 27 = 0.200

p(3.01) = (- 18.0 + 27) (3.01) – 27 = 27.0 – 27 = 0.000

p(3.05) = (- 18.1 + 27) (3.05) – 27 = 27.1 – 27 = 0.100

%B-2=>*+ J(2(#g+: p(x) = (x – 3 )3

p(2.95) = (- 0.05)3 = - 0.000 125

p(2.99) = (- 0.01)3 = - 0.000001

p(3.01) = 0.000001

p(3.05) = 0.000 125

3 - 35

p(x) = ((x – 3) x + 3) x – 1

PQ(8$*+ .4< ,WC$ R(91! %Û^* G&1)8*(2-: p(.9) = - 0.001

p(.99) = 0.00

p(1.01) = 0.00

p(1.1) = 0.001

%B-2=>*+ R&)*+ : p(x) = (x – 1)3

p(.9) = - 0.001

p(.99) = - 1x10-6

p(1.01) = 1x10-6

p(1.1) = 0.001

3 – 36

, -11

111

)

!)

xxxxy

(17-1) s¹d9

(18-1) s¹d9

(19-1) s¹d9


348

3 – 78

)!(

i) , - 01.100495.

995.00.100495. )

!)f

ii) 806700000.2)exactf

relative error = 49.5 %

iii) , -x

xxf

sinh2)

f(.00495) = 2.00

)G(

i) , - 10000001.

00999.01.01.0 )

!)f

ii) 1666.0)exactf

iii)

, -

./0

123 !)

! !!)

201

61

!5!3)(:

2

3

53

x

x

xxxxxfTaylor

! 166.20

1016101.0

4

"#$%

&'( )"

)f

) (

i) ! 1100000.1000.100 ))"f

100.09.991009990100 ")")"

ii) 0001251005.0"exactf

iii) 1

1

1

1.1)(

22

22

)*"

)*

)*))"xxxx

xxxxxf

!199

19.990.100

1

99900.100

10.100 "*

"*

"f

= 0.00502

)!(

i) ! 500.0002.0

998.0999.0002.0 ")"f

ii) 9980006645.0"exactf

iii) ! +,

-./

0#$%

&'( 111)*))#

$%

&'( 111)*)"

!21

!4!211

242 xxxxx

xf

(20-1) s¹d9

349

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

6

16xx *)"

)"#(

i) ! ! !400.0

105.2

005.000499.00.1005.0

5)"

2)

" )f

ii) 00166668.1"exactf

)$(

i) ! 2499250150 )"f

2.08.4950

8999.4950

24902500

")3

)"

)3

ii) 02.024992501 ")"exactf

18.0"45error

%900902.018.0 """4r

iii)11

1111)(

22

2222

)**

)**1#

$

%&'

())*"

xx

xxxxxf

11

222 )**

"xx

02.08.4950

2

24902500

2

24992501

2)50(

"*

"

*3

*"f

% – &%

! 03.10195.

102.10195. ")"f

009813692.1"exactf

+,

-./

0)#

$%

&'( ****" 1

!3!211)(

32

xxx

xxf

***"!3

x!2

x 21

f(.0195) = 1.00

(21-1) s¹d9


350

x = 0.58853274

0.6

xexfxexf xx cos)(,sin)( !!"#!"!!

!!!!"#$%&'($ !!!!)"*+$ ,!!!!-.&/$ ,!!!!--0 ,-!!!1.2 $3410 "x ,!!!!--52 6&7!!!!"- 67-,!!!!8 ,!!!!-*'97 : "+,&+$ ;"*+$ <=> ?@A-:

122

5403023.03678795.0

841471.03679795.01

105.0007.0584.0

478529.01

!

$

!

%&"' "

"$"

xx

x

xex

x

"

!

!"

2

7.0

70347.0

60 B.C-

01)1(

01)1(

01)0(

)(

1

2

(!"!

&!"

&!"

!"

!ef

ef

f

exxf x

*9-)+$ D2 E,F8$7 E$.3G H,-I 60 J0 (-1, 0)

?KL+,' 6"') 7I ,)K ;M.+$ 6) N.L,') H+3 O,&-&M$ 6K)"7.

ixi

ixi

ii

x

ex

exxxRN

exxf

!

!!"!

!"#

$ 2:

2)(

2

1

(2-2) s¹d9

(1-2) s¹d9

w½U¦�« qBH�« �UM¹d9 WÐuł√

351

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

!"#$%&'# (!)*+& +,-$& #./10 !x !*+%*# 0!)*# 123 +,245:

05.

0292359.04804445.04660874.1

4804445.05373532.0

7330437.02

111

1

1

"

! !#$

!

!

ff

D

exe

6!4*# 7!78% 9:;, #.*;.

7.0

7038077.00292359.07330437.02

%

!&!' x

< -=

xeyyx !% ! ln

>- ?7@,:

x

x

ex

xf

exxf

&&!#

! &!

21)(

0)2ln()(

nx

nx

nne

nx

enxxx

& &&

& !2

1)2ln(

1

(!)*+& +,-$& #./ :00 !x0!)*# 123 A45, +,,BC D:

8.08187309.0

2.005.0

2238554.0

2045686.015.012ln

2.0

12

2

1

!%!!!

!%" !(

!

!& !

yeey

xxx

x

x

x

7EF A5:

(3-2) s¹d9


352

8.08187309.02.0

105.0)()(

212922.0

21.

2ln11

)2(lnln

0

21

)2ln(11

)2ln(ln(

0))2ln(ln()(

))2ln(ln(ln

2.0

1

1

112

1

0

)

1

%!!%!%

)"#! $(

!&

!

!

&&&

&& !

!&&!

& ! !

&

eyx

xfxf

xx

x

x

x.

x

xxxx

xxxf

xyx

n

nnn

G*+H A5: )())2ln(ln(ln xgxyx !& ! !

0!)2* )1,0(1)( *"# xxg

))2ln(ln(1 & !& nn xx !7!78%*# 6!4*# I&J% 0H K&+%%(*# L+&!7)%*# )!7J M-) %&+H*# J),*# (>%;!, )!7J >( N'$&.

< -O

ix

ix

ii

x

x

e

eixxx

exf

exxf

&

&

!

&!#

!

1

1)(

)(

1

66723

3

2

1

0

105.01012.0102.1

385671431872.0

5671432.0

566311.0

5.0

)")!)!

!

!

!

!

xx

x

x

x

x

PP:$ 1PP23 A;PP452*; D>!7PP!78% Q+PP%5, >!PP,H# >!KPPR;(* PP:$ 1PP23 A;PP452* SPP,- M-

L#7!78% HTH Q+%5, UR#;( %V*: ;W 0+:7- %V* N+5!54 7.X*#;:

567143.0!x

(4-2) s¹d9

353

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

< -Y

)&+V*# *ZV(*# A5 [C 0V7*# 7\,#.

xeyyx

xyyx

!%!&

!%!

0ln

0

+, - , -

2720.027203.0

11

)(

2

2

2

2

02

2

0

%!

&& !&& !

! !

.xxx

xxx

x

x

xe

edxxeArea

567143.0

)22()(0

)2()(

21

2

2

1

!%!%

! !#!

&!

xex

exexxg

exxg

x

xx

x

X%+,*# *$+K(*#)( xex ! 0:7 A#]V*# *$+K( [W4-2 +^V@,.

72797.0)2()2()(2

2

2

1min !&!&!! xexxgg xx

< -_

)-(

%28.05014.0

014.0100

1.0.)61(2

.612.122

..

541)1(

414)()(

)(

max

3243

323

!!(

!& !

(/0

123

4 & !( !

(!(55!(

!&!

&!&!!

66

666

6

6

xx

xxx

xx

xdxd

xx

xxx

xxfxg

x

) ( )i(

1) 2331 3)(4)(4 xxgxxgxx nn !#% !% !&

*$+K(2* [&!7)% A5 $+X!a 43 &! xx:

(5-2) s¹d9

(6-2) s¹d9

(7-2) s¹d9


354

)2,1(022

041

040

043

*%789

:%

" %

" %!

!

xroot

x

xx

) >( `!7: 7.X*#x = 2(

)2,1(13)( 2 *;:!# xxxg

dvgt% !$3+&% 6!4*# 2) %&&!%&!&

331 4)(4 xxgxx nn

.)2,1(1)4(3

1)(3/2

convgtxx

xg %*;"&

!#

' !&7+)% 6!4*#.

3) 44 331 !% !& xxxx nn

&;2J(*# *$+K(*# AH(% ' b.W;

4) %&!%&!& 22141)(41xxxg

xxxnnn

, -

.)2,1(1)(

8181)(3232

dvgtxxg

xxxxxg

%*;:#%

& ! !#

' % 6!4*# !$3+&

5) %&!%&!& xxxgx

xx

n

nn

4)(4

1

.)2,1(1)(

424.

421)(

22

couvgtxxg

xx

xxxxxg

%*;"#%

& ! &!#

' !&7+)% 6!4*#

>!%6!4*# >BC c*.&;)2 ( D)5 ( d!4*#; D >+%*;&)()1 ( D)3 ( D)4 ( *;&)( 7!e.

>!%*;&)(*# >!%6!4*# >!& ,7+)(2*;)2 ( D)5:(

)2,1(1

11

)4(1.

6)4(324

5

26/13/2

5.1

*;"

""

<<

&!

&)&!#

#

x

xxx

xxx

g

g

!4*# >BC c*.&; 6)2 (ARCf# [W N+&7+)% g7Vf# M- D.

355

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

)ii(

445

5

4

3

2

1

0

3/1

105.000002.0

796324.1

796344.1

796543.1

798468.1

81712.1

2

)4(

)"!

!

!

!

!

!

!

&!

xx

x

x

x

x

x

x

xx

7.X*# (!: >BC c*.&;x [W !7h3 0+:7- K&7f N+5!54:

7963.1796324.1 %!x

)iii(

1033024.0

)4796324.1(31

)(

xin

3/2

5max

11m

!

&!

#!#!

!

xgg

xxError nnm

5102304063.0

00002.01033024.01

1033024.0

)!

) !Error

) !"#$ %&$'( )*+, -./(" 01 21(

)3 ( " %$*4 56*6+(" -./(– 0**-78 0*$4-(– %(9:;$<(043 !! xx 5=1.80.

)->-#=( 1 0 -1 -4 1.80

1.8 3.24 4.032

1 1.8 2.24 0.032

" 0#

195826.19.0

2/24.248.18.1

024.28.1

2

2

j

x

xx

$!

%&'(

)* +!$!

" ,,

- #= ?$:@(" ?+(": 20.19.0,80.1 jx $!.


356

A -B

%C*DD,(" E-:DD6F ?DD/1 0DD$)(1 nn xx / , %DD(9:;$(" ?DD+()(xx / -6F(" :=-.DD/ GDD*+ 5DDH* IF$*40 J-7(" K6+F* 01 E/* L 1)( 123 m0/

%$*4 M>:@ :$<@#m N-&1 E-:6F(" 0:@ :$<@ L-C,1

%*H-:6F -*O ". 3 2)( 1

00/ (i)

) M>:@ ".P QP0 0x(

%*H-:6F ". 3 ! 6.0)( 00/ e (ii)

%*H-:6F ".! 3 ! 2.01)(2

1 00/ e (iii)

0*FC*D,(" 0D$ ?D@ R"9'FD&" 0@$* I>1 21 )ii (#)iii( %C*D,(" R"9'FD&" ?D!ST"#(iii) 0TN-&1 :UH-:6F.

A -V

)1( xdxdg

g 4 4 .

3,...

.

2max

maxmax

!! 4

4 4

xxgg

germ

xdxdg

g

00704545.04431.0...

113216

431.001.0

431

431

418

212

max

!!

+ 4

! !

erm

g

g

xx

dxdg

)E( W-

X-Y>:

xxxg

xxg

xx

2.)3(2)(

)3()(

)3(

2

22

22

! 3

!

!

-./<( %*H*-6F %$*4 9:/*Z#x:

036)3(

021)2(

03)1(

03)0(03)( 2

5!

1!

1!

1! " !!

f

f

f

fxxxf

(8-2) s¹d9

(9-2) s¹d9

357

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

-./("6 %6J>$(" 5S 6*(2, 3) "

18)2( 5 3g

%DD("9(" 0[DDS \(.DD@g3 ]9:DD* H 9DD* Fx %DD("9(" 0[DDS \(.DD(# Lg(x) :DD>$:$F=" %DD6J>$ 5DDS %DD*98:HF)/(" %6J>$-..(

7 8 2/1

3 xx ,

7 8 2/1

3)( xxg , X-Y>:

)3,2(13

14

12

1.321)( 9:1

,

, 3 x

xxxxxg

%("9("g(x) %*H-:6F "

X-Y>:

20 x

7 8;; 3

34

323

4

3

2

2/1

1

105.0000558.

105.0006841.

1101198.2

1100640.2

1093799.2

1010029.223

!

!

+1 !

+1 !

,

xx

xx

x

x

x

x

-./(" 01 21– %*-78 R:4-1 %_`_( a:+*+,– #=x = 2.110

)3(

344 1xx

mm !!2<

) 0Tg3 $ %*,4:>F %("9x(

54

3

1049552.00000558.01

0815614.03334

1

)(

!+ +!2-

,

3

mm

xx

xgm

<

A -Wb

%(9:;$(" ?+ E#<J$(" 03)( != xexf x

;; !"#$x

!"#$x

efx

efx

fx

2

1

2 0389.162

028173.31

01010

5 ! "

1! ! "

5 ! "

)2,1(,)1,0( 21 99 xx

(10-2) s¹d9


358

)1(

3

3

)()(

1 !!

! 3! , ixeixixe

xxfxf

xx ii

iii

X-Y>

11

2

7.0

7.0

1

0

105.1006.00647769.0

6190269.0

6125493.031.27.0

7.0

!! +1+.4"! 4

!!!

x

eex

x

-C,T" 5>*&(" 5_"9+Z" 01 211x #= 9+"# 2-78 R4-( a:+*+,:

6.01 x

)E(

(i) 3

)(3

)(xx exgexg 3"

%("9(" %$*4 01 21g3 0*F*(:F(" 0*F$*6(" 0*H c#"-FF:

(ii) xxxgxxg 13.

31)()3(ln)( 3"

) 9>82:1)( xxxg 13( " 211" 3g

%C*DDD,(" 01 21(ii) #DDD+> %DDD*(:FF$(" :DDDU$*4 E-:DDD6FF2x R4-DDD( %DDD*(:F(" :DDDUF$*4 dDDD<8 ?DDD,+># L9+"# 2-78: 5.12 x

3

!46.29.0g3

e3e 2

!"#$

%&' !(

! "#$% &$' % ()&'!*+ ! ,-% .%1 /-0 2xx

17543.023

)3(

5.1

6.0

2

2

1

!)!

)! *

x

x

x

x

ex

dxexArea

12!'*+ 345- /-0:

0y&0y

(11-2) s¹d9

) (

359

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

)2(10

0

)1(ln0

110/

10/

x

x

x

eky

xkey

! "

!

!"#$%& '( )*"+),& )*)-#k, x.

./01 2!1)3"4 567 5"#84 2&9:;k 2"+),&"4 567 <=01":

# $xxxx ln10ln10

2>>&!? +)>%! @4# 2>"+),&"4 A/>>- <>0 +>,;#x 2>>&!? +)>%!B !*:;)>>9"4 !*"+)>,&"4 C+>0D '>>( E#>,1k. * F9G;';!G: !!101&6" '( H:* HI):*"4 2I:1 J +%1

2:I1&"4(5, 6) 5"D <=1 J K&! L)!;)90 E!#,*"); #J

'6! )&K 2%!*1"4 A/-:

EGM1:

)6,5(

75.0)6(,95.1)5(

ln10)(

%!

&

&

root

ff

xxxf

i

iii

i

iii x

xxx

xfxf

xxln1ln10

)()(

1 &&&

& "

& '

567 <=01 2!G!GK*"4 2N!="4 A/- &#:

x = 5.7289

2"+),&"4 '( E!#,*");#)O ("+),&"4 #J 2)P (567 <=01:

k = 0.98429 ! k = 0.98

(12-2) s¹d9

)% ( 356"4*+ 7"25' 4"8: 1)( ,( -g

96:- 4&#;!*+ <:*+ &=.

3!65*+ >-2$ 2!?$ @"A% 356"4*+ 7"25' &$6)(-g( "BC%.

)7 ( D- 1)( ,( -G

1)(1

1 ,+(+ .-. g


360

E- 4*+ 7"25' &$62!/-0 $!6 2! @"A% 356" )(-. g)!

)F ( !"#: )(0 3/33 xgexexex xxx !"!"!# ###

$%&'()* +, -'.(/): 3/

31)( xexg ##!$

2639632.03

)(3/7.0!!$

#eg %

773.07729665.0

105.00004078.0

7729665.0

7725587.0

7741418.0

7680019.0

7918897.0

7.0

345

5

4

3

2

1

0

&!

'(!#

!

!

!

!

!

!

#

x

xx

x

x

x

x

x

x

&01&2:

2639632.031)7.0()( 3/7.0 !!$#!$#& #egg %)

26.0!) 3451

xxe

xxgxGx

26.126.03/

26.11

1)(

11)(

*#!

**

*!!

))

)

7728832.0

7729283.0

7.0

2

1

0

!

!

!

x

x

x

773.07728832.0

105.00000451.0 312

&!+

'(!# #

x

xx

6708)* +# 9.:1 (**) +;, !<;= >;'? +0%;0%@( A;)B &;1C(.* D;0. E &;F=%&'( G;? *%;02@ H%I#6I,4.

J -KL

!"#: xx exgex ## !"! )(

ixii exgx

#* !! )(1

(13-2) s¹d9

361

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

545.0

607.0

5.0

2

1

0

x

x

x

571.0

560.0

580.0

5

4

3

x

x

x

567.0

567.0

568.0

565.0

9

8

7

6

x

x

x

x

567.0 ! x

!"#!$:

6.0

6065308.0)()( 5.0

"" #"$ """

%

&% && eeeg

%&'#

xxe

xxe

xxgxGx

375.0625.0

6.16.0

6.11

1)(

11)(

'"

'"

''

'

%%

%

ixx

eix i 375.0625.01 '"

'

56658.0

5.0

1

0

x

x

33

32

105.000001.056714.0

105.000055.056713.0

"

"

() *!

(+ *!

x

x

() *"&+, -!.&/ *$0$1 !2"23 &4516 7/ 8/ 567.056714.0 $ x

*5"9#16 :)( !;<'# =>1(?6 *@"&A1!B !;"C, !#C32 :916)DE!>9FG16(H I4>) J&!>@9 7>K1( *>>>"#!$16 *>>>@"&A16)*>>>1LEFG16 ( M6&>>>"&K9 E>>>FB J(>>>CAG16 &4>>>5C1 !#C>>>3( N>>>"2 &>>>"$KB O&>>></

*F<9 7G PEB A@Q *$0$.

R -ST

x = 1.76322

R -SU

:) &45C1 *2"2316 *G"@16: z = i

V9#" 0$G 7/: 000004.10000055 iz '"

(14-2) s¹d9

(15-2) s¹d9


362

R -SW

431411)(

2411

12)(

2)(

1)( 2

)!)''" #

''"

) #

" #

"

aaxg

aax

axxg

axxg

axxgx

R -SX

ii

iii

i

iii

xx

xxx

xfxf

xx

xx

xx

sin1cosln

)()(

0cosln

cosln

1

!!"

#!"

"!

"

!"#: 10 "x

50000086613.0

3029554.1

2934079.1

)2(

)2(

2

2

1

"!"$

"

"

# xf

xfx

x

x

30.1"% x ($%%!&' $%()!*)

& ' '2/

3.1

30.1

1

2/

1

sinln

cosln

(

(

xxxx

dxxdxxArea

x

x

!"

" ))

08.00775151.0 "" )$%%!&' $%()!*(

+ -,-

Area = 0.23611

+ -,.

43*ab

(16-2) s¹d9

(19-2) s¹d9

(18-2) s¹d9

(17-2) s¹d9

363

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

+ -+/

$%01 203% 245036*7 843#* 9#%:*7 9;7<=>7x = 1, x = 2 80%!%!?6*7 $6@0%# 8A%0B C0%1461@ D

8*<5E(*7 F=*: 02)2(ln)( " ! " xexxf

8(%31 $%G<61()2,1(0 +x HI' FB=#: x = 1.1462

y = 1.1462

+ -+,

x = 1.403602

+ -++

2410.2,7684.1 21 "" xx

+ -+J

x = 1.30296

) K)! 8*L:(*7 F= 5M%N !O#717-2(

+ -+P

Q!AB*7 8%5R#*7 843# <#' 0"#y

x = 0.94775

(20-2) s¹d9

(21-2) s¹d9

(22-2) s¹d9

(23-2) s¹d9

(24-2) s¹d9


364

+ -+S

T5%#=#( U<' F;(% $%1(*7 F?&*7xa ) 8"I6V( K%3* T15;I*a ( H0#=#(*7 2(xsin ) W0%=

x 81X@( .( 8(%) T<7Y 5(I? Z#N [M7@*7 $(@a 8(%) T<7Y 5(I?xa ) !%A6(*7 8(%3*x 5R:"#(.

H#=#(*7 23% 9? Z#N F?&*7 $( [M6%@xa !"#$% H0#=#(*7 C@\xsin ) C03=6% H06= 4!00&*7xxa sin, K%00) 200%(X*x 8001X@(*7( 800(%) $@00?6@ Da T00)@*7 900\ $00?(% 500( !A00BNZ00:"# D

H0000#=#(*7 $]0000\xa H0000#=#(*7 ^0000(%xsin 50000( 8000043# <0000#'– 5R(0000:7 !0000"#0x - ) H0000#=#(*7xa3 7_` C3=% F?&*7 9\ ( 8(%) TB3# @* W%=1a K%0) T0#5? Q!AB*7 8(%3*7 a_` $'xa

K%) $( !ABNxsin K%) E1*x 81X@(*7) H#=#(*7 b*_ C3=% 5(?xa2 cd;(.(

^50000(6*7 8000043# <0000#'0x 0000:66 a_0000` 500006(%) Q@5y 500006(%) Q@50000:66 50000(?y# $%0000%#=#(I* 810000:#*51xyxay sin, "" $N eN:

00

00

cos2

1.

sin

xx

a

xxa

"

"

$%*@000RX(*7 9000\ $50006*<5E( $500065 @ax ,0. f_000=# $N cd;000( $000?(%a $%6*<5000E(*7 8(000:31 cg@N HI' FB=#* 5(R%ME1 HI'00 tan2 xx " 8*<5E(*7 a_` F= <E1@– cd;( $6@%# 83%!41– $%%E6@

8(%)0x 8(%) HI' FB=#* $%6315:*7 $%6*<5E(*7 Q<=h 9\ @E#a a = 0.85124

+ -+i

)N(

5 -6 0 10 -12 1

5 -1 -1 9

5 -1 -1 9 -3 1

5 4 3

5 4 3 12

25.012/3)1(/)1()1( !"!"#" GGD

(25-2) s¹d9

(26-2) s¹d9

365

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

)j(

25.125.01)1(1)(

)(

00

)(

)(

1

" "!"!"%

!"!" #

DxDxx

xDxxx

i

iiixG

ixG

ii

+ -+k

)1(0sin " ! xyyx

FB6 5(<#'@y $]\ H(OE*7 5R6(%) H*h0"#y.

8*<5E(*7 FM5$1) !%A6(I* 81:#*51x HI' FB=6*:

0)()(cos1 " # #! yyxxyy

%@E6*51@0"#y HI' FB=# 8*<5E(*7 a_` 9\:

)2(0cos1 " xyy

$%6*<5E(*7 F l=(1), (2) H(OE*7 8%5R#*7 843# 9 m4E% c5%#n:

)2(cos/1)2(

)1(sin)1(

#!"-

#!"-

xyy

xyyx

8%!%!?6*7 83%!4*51 $%6*<5E(*7 $%65` F=*@) cd;( ( !%A6(I* 8%G7<617 8(%3* cg@N o56=#y.

K0000IE#– F7p0000:*7 $0000(– 80000*7<I* H0000(OE*7 80000(%3*7 $Ny $0000( j!3*500001 20000361"x . %@E6*500001@1"x 8*<5E(*7 9$1( # HI' FB=#:

1 = y – sin y

- sin y = y – 1

$N <X# K:!*7 $(y )F?&*51 $%%#=#(*7 24536 843# <#' ( $( 81%!)2 $N !"#\:

)5.0(

4029981.22cos/1

2,1

01

1

00

*!"$

"!"

""

yy

y

yx

N eN $]\ [%=B <<' j!)q Z#y = 2

(27-2) s¹d9


366

&'()":

5(` 8%!&' K5)!N 8;d;* H(OE*7 8%5R#*7 843# 5%;7<=h:

x = 1.086, y = 1.944

+ -+-

yyxkyxh

xyxykhx

ygxg

yyxyxg

yfxf

xyxyxf

yx

yx

!"!!

!!" !

!!""

!!"

"!"

! "

22

22

22

22

)12(2

2)12(

12,2

),(

2,12

),(

4203389.0,7728814.0

0203389.0,0271186.0

08.08.16.1

08.06.0

11 ""#

"!"#

!"!

"

yx

kh

kh

kh

- !

x = 0.7 sin x + 0.2 cos y

y = 0.7 cos x – 0.2 sin y x y

0.5 0.5

0.5111143 0.5184227

0.5161248 0.5114378

# "# $%&'()*+', -. (.," /012 340'(0.5, 0.5)$

-

2.93

-55

n = 14 , r = 1.3247

(28-2) s¹d9

(29-2) s¹d9

(31-2) s¹d9

(33-2) s¹d9

367

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

%"%" xaax 11

21)(01)(x

xfxaxf "&%"!"

)2(1

1

2

1 ii

i

iii axx

x

xaxx !"

!!"%

-56

7'8(9*:+', $&"%; 7<%=:

)()(

.)(xfxf

mxxgx &!'"

2

2

)(

)().()(.1)(

()*

+,-

()*

+,-

&

&&!&!"&

xf

xfxfxfmxg

.

.

/

0

11

2

3

&

&&&!!"

()*

+,- 2

)(

)().(1.1

xf

xfxfm

()*

+,- &&! &! !!"&&

&! !"&

4!"

!

!

)()()()(2)()1()()(

)(.)()()()(

0)(,)(.)()(

22

1

xqrxxqrxmxqmmrxxf

xqmrxxqrxmxf

rqxqrxxf

m

m

m

>*%?0& @0)A&', 5 6 %"!!!"& 0111)(m

mmrg

(quadratic convergence)

7%B=C, $&"%; 7<%=:

7 82)(

)(.)()(

)(

)()(

xf

xfxfxg

xf

xfxxgx

&

&&"&

&!'"

D)%*%?0& E%' @0)A&',. %4!"& 011)(m

rg

-5F

7%B=C, $&"%; 7<%=:

)(4

21

2

Nx

xixix

i

i !!"

(35-2) s¹d9

(36-2) s¹d9

(34-2) s¹d9


368

7'(*+', $&"%; 7<%= (1):

)1(2

21

2

Nx

xixix

i

i !!"

7'(*+', $&"%; 7<%= (2):

5 6)2(

.21 2

2

Nx

xxixix

i

ii !!"

G,0%0H& IJK', LM)&;:

N N1 N2

0x 1.5 1.5 1.5

1x 1.458333333 1.416666667 1.411764706

2x 1.436607143 1.414215686 1.414211438

3x 1.425497619 1.414213562 1.414213562

NOO&. P%.OO=', @,"OOQ', $R SOO.J;910! TOO+ UOOV&%3x $&"OO%; >&<%OO= $OO+ W/R $OO+ 7OOQ&);',$%&'(*+',.

!"#$:

7OO4(', XYOO# NOO'Z -"OO="B' NOO'Z [)OO&.; 7%BOO=C, $&"OO%; 7<%OO= UOO%?\&?20 $OO+ D](OO? D,0OO%0H&3

G,0%0H&.

–5^

5 62

1ln)(2xxxxf ! "

xx

xf !

" 11

1)('

.030608 x

.046172 x

.069843 x

.106088 x

.162129 x

0.25 x

5

4

3

2

1

0

"

"

"

"

"

"

\"?_+', 0YQ',0"9.

nnn xxError "!"' 9:

7?`;', $C )%*%?0& E%' @0)A&', 21 / nn :: 7&?)K G`%' . IO%. a)O%\b "(O?% @0)OA&', $H'"

7?`;', $R65.0/1 ; nn ::

(37-2) s¹d9

369

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

$R SOOOO.J;" 0)0( "&f 0VOOOO='," a0 E)OOOO++ G);%OOOO`', 0"OOOO.+" a c)OOOO\*+', 7OOOO'()*+B' 0YOOOOQ 7',(', N;.;+'f(x) (;2x = 0 . 0V=',"0 7'()*+B' 00H+ 0YQ .A&', T%0`& $H+%",(b&`)? @0) 3

7'(*+', $&"%; 7<%=:

5 65 6 ,2,1,0;'1 ! " ixfxf

mxxi

iii

!"# $%&'"# ()$"*: # $# $

,2,1,0;'1 ! " ixx

xxi

iii %

%

+&,# $ # $

# $xf

xfx

' %

– !"

& '32011

1)(

0613710)3(

00986102

,inx

xf

f

. )f(

(!"

)

*+!,

-

-./0"# 12 342 *,#( .56 $"*7 !8" 9) :)& '3,2.

g(x) = ln (1+x ) + 1

# $ # $ & '3,21'1

1 ./*0"

) xxgx

xg

9) :) $&;.74/ $%&'"#.

<#.&.=/"# **> .&*4/" :

# $ .1! abmn

b – a = 3 – 2 = 1 4105 !2 .

# $31

31

211

11'

maxmax 0

"

" m

xxg

# $ # $ 021 !4105131 n

# $ 021! !4105ln3lnn

7918.6 304 nn

xo = 2.5

x1 = 2.252763

x2 = 2.179505

x3 = 2.156725

x4 = 2.149535

x5 = 2.147255

x6 = 2.146531

(38-2) s¹d9


370

x7 = 2.146300 44

67 105103.2 !"!# xx

$ !"#$% &'()* $+), '-!-. )/012: r = 2.1463

– !"

f(1) = -2 < 0

f(2) = -1 < 0

f(3) = 40 > 0

$ )34125312 6 57 18'$%91 2)/0 ':!; <* <%=3 % &3,2. f ( 2.5 ) = 11.3 > 0

! "#$%&$' ()&*$' !5.2,2

nnabpp

2"#"

$%$"#" 9.192

2310 6 nn

%+#),& -.%/# 0')#)1&$' 22320 ')#)1&.

– !" – ) #

& ' & ' $()$( 2/2/

41

21 xx exgexg

& ' ! $*+,) 1,5.1 xxg "#+)%,&

& ' & ' $()$( xxgxxg 24ln 2

& ' ! $*+-) 1,5.1 xxg "#23%+&

) 4(

& ' !1,5.0,max

*)( xxgm

44

2/1

1

2/ eex

x((

(

on

n m *#*

& ' & ' 52/11055.0

4".,

ne

lnn$ & ' & '42/110ln

4",e

1139197.10 /$-$ nn

5$6 %+#),& 7%&89 -:11 "+.;<=$' ">2$' 5$6 ?@9 5&8 )#)1&.

) 7 (A.:: "B#@$' C'2D&/%+2/

21 xex ( :

(39-2) s¹d9

(40-2) s¹d9

371

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

xo = 1

x1 =.8243606

x2 =.7550534

x3 = .7293362

x4 = .7200180

x5 = .7166712

x6 = .7154729

x7 = .7150443

x8 = .7148911

x9 = .7148364

x10 = .7148168

x11 = .7148098

%EE#9%F : ()EE&*$' EEG 4.EE;<=$' )HEEI$' JEE3 "EE#23%+& "EE#9%F$' "B#EE@$' !1,5.K "EE#L'2&+A' "EE=#,$%+ %9:2EE+ 'H M'.xo = 1 /G9 2N+ ?@20 ()E&*$' EG )EDO )HI 5$6 )#)1& !5,4 $' 4)%E,&& PE#8 .E!. QE#$6 "B#E@

4.3065798 C%>): "/=D$ %8#8@

xo = 1

x1 = 1.3862944

x2 = 2.0395629

x3 = 2.8117654

x4 = 3.4539194

x5 = 3.8653137

x6 = 4.0903800

x7 = 4.2035701

x8 = 4.2581628

x19 = 4.3065741

x20 = 4.3065798

J: REEEE8A. :& ' !5,412 *+.,() xxg )HEEEEI$' .EEEE89 "B#EEEE@$' SHEEEE! 4)%EEEE,&& T$HEEEE$. K

()&*$' T;& G U>'.$'.

– !$ – ) #

f (o) = -3 < 0

f (2) = 6.41615 > 0

& ' !2,00sin4 *+.-.0() xf

1 ()&*$' G 2#8. )HI "$'2;$ !2,0.

) %(

& ' xcosxg41

21 0(

(41-2) s¹d9


372

& ' ! 2*+,#() 2,0141sin

41 xxxg "#+)%,&

& ' 25.041

max(()( xgm

mn (b – a ) < *

& ' & ' 8644.71050225.0 5 (2-2.," " nnn

xo = 1

x1 = .6350756

x2 = .7012567

x3 = .6910080

x4 = .6926510

x5 = .6923890

x6 = .6924308

x7 = .6924241

x8 = .6924252

) &(

g (x) = 5 x – cos x – 2

& ' ! 2*+-0() 2,01sin5 xxxg "#23%+&

– !

2("2(2( 0333 axaxax

f(x) = x3 – a = 0

345

678 0(""(0 22

3

1 231

3 nn

n

nnn x

axx

axxx

a = 20

xo = 2

x1 = 3

x2 = 2.7407407

x3 = 2.7146696

x4 = 2.7144176

x5 = 2.7144176

– ' – ) #(

)i (+%F$' "<,9$' ",#)< J= "@%D "$%8 2N& J&.#9 "B#@ "&x = g (x) P#8 K:

& ' & '& 'xf

xfxxg )"(

().@$' G %VNW. J1=#. & 'nn xgx (01

11 00 *0( nnx 9

nnx *0( 9

(42-2) s¹d9

(43-2) s¹d9

373

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

P#89 ,#,8$' )HI$' .! . ",+%/$' ().@$' G X#.N&$%+. :

& 'nn g *0(*0 0 99 1

& ' & ' & ' ::::0))*0)*0( 999 ggg nn !2

2

J: P#8. & '99 g( T$H;G K:

& ' & ' ::::0))*0)*(* 0 99 gg nnn !2

2

1

J: 0EE+F9 K %E#N#+)& 4)%E,&& A J&.EE#9 "B#E@ J: 0%E+FYG& ' 0;) 9g. J: PE#8r !""#

$%&'( :

! ! ! ! 0;.2 "#$ rqxqrxxf

! ! ! ! !xqrxxqrxxf %#&#$% ..22

! ! !xfxf

xxg %#$

!22

21xg

f

ff

f

fff

%

%%

%

%%#%$#$%

! ! ! ! ! ! ! !xqrxxqrxxqrxxqxf %%#&%#&%#&$%% 2222

! ! ! ! !xqrxxqrx4xq22 %%#&%#&$

! ! !

!2

.

'()

*+, %

%%$%

xf

xfxfxg

! ! ! ! ! ! !

! ! ! !2

2

22

2

42

'()

*+,

'()

*+,

%-#&#

%%#&%#&#$

xqrxxqrx

xqrxxqrxxqxqrx

! ! ! ! ! !

! ! !2

2

2

42

'()

*+,

'()

*+,

%#&

%%#&%#&$

xqrxxq

xqrxxqrxxqxq

! ! !. / !. / 0

21

2

22

"$$%rq

rqrqrg

0 &)*)+ , -). / &0,.1 ) 234 5+ (

)ii ( 6.7*(.1 68)9.1 :

! ! !xx

xxg 11%#$

! ! !

!2

.

'()

*+, %

%%$%

x

xxxg

111

! ! !. /0,00 "%$$% rrrg 11

:; <.!= :


374

! ! !

! ! ! ! ! !. /xqrxxq2rx

xqrx

xf

xf2

x%#&#

#%

$$1

! !

! ! ! ! 0

2$2%#&

#$ xxqrxxq

xqrx 1

! ! ! !

!

! !

!22

2

1

'()

*+,

'()

*+,

'()

*+,

%

%%#$

%

%%#%$%

xf

xfxf

xf

xfxfxfx1

! ! ! !. /

01r21

rf

rfrf2

"$#$1%%

%% ] >+&?.1 @A#.1 :((i)[

:B C)D= ! 0$% rg 2*)+ , / &0,.&E.

) (

! !nn

nnn xx

xxx

sincos1cos1

1 #&&#$&

xo = 3

x1 = 3.0763276

x2 = 3.1100733

x3 = 3.1260867

x4 = 3.1339004

o3 = .1415927

13 = .0652650

23 =.0315193

C)D &)*)+ , -). / &0,.1const !

"33 &

21

n

n :B C)D 234 FGH.=5.01 433 &n

n .

17.17"""""G% """"H( !"""""# &""""I. J&"""""3*(.1 6"""". ! ! ! 555 $$%$ xff 0,0 68)"""""9.&E <.!"""".= K&)*)+ , / &0,, 6.7*(.1.

! !nn

nn xx

xx 11%#$&1

! ! ! xx

xxfxf

xsincos1

cos1#&

&$%$1

! ! !

!

! ! !2sincos1

sin21cos11

21

xx

xx

xf

xfxfx

#&#&#$

%

%%#$%

'()

*+,

1

xo = 3

x1 = 3.1311125

x2 = 3.1415375

x3 = 3.1415920

x4 = 3.1415920

, / &0,.1 :B &GL MDN= :B C)D 2*)+5.01 433 &n

n.

375

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

! – "" – )

!yef xx 3.0cos3.01 3.0#$

!yef xy 3.0sin3.0 3.0$

!yeg xx 3.0sin3.0 3.0#$

!yeg xy 3.0cos3.01 3.0#$

hn fx + kn fy = - f

hn gx + kn gy = - g

x1 = xo + ho = 1.6392782

y1 = yo + ko = 0.1441359

x2 = x1 + h1 = 1.6343369

y2 = - 3.10462 × 10-5

) (

x1 = 1.4722812 y1 = 0.0196365

x2 = 1.6279147 y2 = 1.987 x 10-7

x3 = 1.6313391 y3 = - 3.70105 x 10-11 33

23 105103 ## 676$# xx

3723 10510987.1 ## 676$# yy

O&P B 6QRQ. 60)P7 6+=#;1 :B SB.

! – "#

f ( x, y, z ) = x + ey – y – 0.9

g ( x, y, z ) = x + x2 + sin y + z2+ 0.3

h ( x, y, z ) = cos x + ln (1+y) + 2e-z – 3.5 fx = 1 gx = 1+2x hx = -sin x

fy = -1 gy = cos y yyh &$

11

fz = 0 gz = 2z hz = -2 e-z

2$ )0,0,0(),,( zyx

1$xf 1$xg 0$xh

1#$yf 1$yg 1$yh

0$zf 0$zg 2#$zh

f = 0.1 g = 0.3 h = 0.5

hn fx + kn fy + mn fz = -f

hn gx + kn gy + mn gz = -g

hn hx + kn hy + mn hz = -h

h0 - k0 = - 0.1

(44-2) s¹d9

(45-2) s¹d9


376

h0 + k0 = - 0.3

k0 - 2 m0 = 0.5

2 h0 = - 0.2 , k0 = - 0.1 , m0= - 0.3

x1 = x0 + h0 = - 0.2

y1 = y0 + k0 = - 0.1

z1 = z0 + m0 = - 0.3

x1 = 0.0410906 y1 = 0.9680567

x2 = 0.0010642 y2 = 0.9979919

) (

x1 = 0.0132902 y1 = 0.6669445

x2 = 5.30262 ×10-5 y2 = 0.9737685

!11

21

1 2ˆ

&&

&& &#

##$nnn

nnnn xxx

xxxx

k kx

1 .6156 1535

2 .6154 7247

3 .6154 6829

8 .6154 6816

9 .6154 6817

10 .6154 6817

8

!oxxx

xxxy

"#

##$

12

212

2 2

xo = 0.5

x1 = g (xo) = .5 + .2 sin xo = .59588511

x2 = g (x1) = .5 + .2 sin x1 = .61224830

y1 = .61561534

xo = .61561534

x1 =.5 + .2 sin xo = .61549220

x2 =.5 + .2 sin x1 = .61547209

y2 = .61546817

xo = .61546817

(46-2) s¹d9

(48-2) s¹d9

(47-2) s¹d9

377

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

x1 =.61546817

x2 =.61546817

y3 = .61546817

!!"# $!!%&'() *+3 !!"# ,-!!.-/" 0!!12 3)4!!546' $7!!89% 4:!!&") ;!!<=>!!2 ;?54!!8 ;!!1540") @:!!72 A

4:&") "# ,-.-").

n nx

1 2.0394118

2 2.0020898

3 2.0001079

4 2.0000145

(49-2) s¹d9


378

! xx

5321 " xx

##$

%&&'

( ##

$

%&&'

(##$

%&&'

( !

5

12

31

14

2

1

x

x

!"#$% &$'%$ &$'% '()*#+ ,-./0:

2

3

0

1

5

12

31

14

4

13

4

1!)

!

13

8

13

41

13

84

1

10

013

10

1)

!

) 1( &$'% 2)*#+:

6153846.013

82

x

1538461.31341

13233

41

121 " ) ! xxx

) 3( &$'% 2)*#+- !"#$%:

138,

13

4121 xx

* + 1314

134det ! !,, A

-

".!$ 4*"5(

&$'%$ 65$7'% '()*#+ - 4"*8:

214

13 xx "

(1-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 WÐuł√

!

"##$

%&

!

"##$

%&

!

"##$

%

'&

12

5y,x,

14

31

2

1

x

xA

!"#$:

!"#$ %&'()!" *%+& *(,#-%!&#$ .-"/ 0(123#$ 412(5#$ 216 7(38,#.

12421

379

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

123

1

35 xx '&

) 9( :";+%< 03(12:

1x 2x 0 0

3

35

1241

32

1666667.36

191 &&x

619

3619

5277778.03619

2 &&x

) -( =;%< 03(12- .-(>:

1x 2x

0 0

3

32

619

8

11 1527777.3

72227

1 &&x

72227

216133

6157407.0216133

2 &&x

=;%<; =;%< :,3(12 *& ?+ *@ A8B - 2;")&#$ .8#$ C2D@ *$-1;< E

=;%< 03(12 *@;– :";+%< 03(12 *& %'"1%3, :F G15@; 7-@ .-(>.

(''

(' 14

01

130

31

10

01

14

31

13

1

13

4

13

3

13

1

13

1

13

4 10

0101

10

31'

('

) * 13131det,

131

134

133

131

1 '&'+&

!

"

##

$

%

'&( ' AA

%–&

(

'

'(

3

1

2

1

45

4

12

1

12

1

12

1

31

21

5

1

4

1

3

1

4

1

3

1

2

1

31

21 1

0

0

1

0

0

11

(2-3) s¹d9


380

30

6

1

100

110

1

6

1

00

110

131

21

6

1

180

1

31

21

'('

910181,36306,30123

&',&'&''&& xxx

!

"

###

$

%

'&

!

"

###

$

%

30

36

9

3

2

1

x

x

x

%–%

33.

50.

0.1

09.08.0.0

08.08.0.0

33.50.0.1

0.0

0.0

0.1

20.25.33.

25.33.50.

33.50.0.1

'

'(

!"#$:

!

"##$

%

(&+

(&+-

11.1089.33.33.

17.165.33.5

17

3.6

0.1

0.10.00.0

0.10.10.0

33.50.0.1

17.

3.6

0.1

01.0.00.0

0.10.10.0

33.50.0.1

'('

!"#$:

!

"##$

%

(&+

(&.

50.504.3.608.

3.625.6850

173&x

233.23173.62

'('&''&x

4.76.5121

61.55.11

1&',&

//x

!

"

###

$

%

'&

!

"

###

$

%

17

23

4.7

3

2

1

x

x

x

(3-3) s¹d9

381

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

03"%5#$ 0#H5&#$ IJ%,/)2-3( E03(K-CB-%!&#$ 0D;&<& 2")#%" 738, L@ %&@ E

5&#$ IJ%,/M$;N#$ *D 4-(!" :'F 0(#%8#$ 0#H%'(F H2O#$ *@ L@ E -((3,P# Q#R; 1("+

-(-6#$(severe limitation) M(13, :F M5%8#$ SPD(rounding off) T,%"%58.

%–'

) @ ) (i (

!

"##$

%0

!

"##$

%0

!

"##$

%01

0

1y,x,

1

2

1

3

1

2

1

2

1

x

x

!

"##$

%&

!

"##$

% !

"##$

%(

0

11

2

1

3

1

2

1

2

1

x

x

)ii ( 03(12G-J )0(15+#$ 0U(N#$(:

('

('

(6

1

10

11

0

1

0

112

1

2

1

12

1

2

1

3

1

2

1

2

1

234

'&

&(

'(

6

4

6

4

10

01

2

1

x

x

)iii (03(12 G-J )*((;/!& *(&K1#(:

('

(5.0

0.1

08.0

50.0

0

1

0

1

33.050.0

50.01

('

(' 3.6

2.4

10

01

3.6

1

10

5.01

234

'&

&

3.6

2.4

2

1

x

x

:F .8#$)ii( :F .8#$ %&/(" 7(K-)iii ( .(P3#$ --!#$ M"5" 1("+ H2O T(F

)=2( 3,#$ V%2O@ M"5"; %'" A%W,8B%" M5%8P# X;&5&#$ 0(;/!&#$ Y%K1Z$ *& 1 M(

)round off errors.(

2.00.42.41 &'&5x

3.02 &5x

%505.042.0.

1

11 &&&5& x

xrerrorrel

(4-3) s¹d9

!"#$%&':


382

%505.063.0.

2

22 &&&5& x

xrerrorrel

) M) (i(

(&'

'&'1 0

11

3

1

2

1

2

1

66

6

0657414.0,2675918.16

134 &7&6

)ii( 0F;WN&#$ *@ L@ E0"<;& *(,(,$R#$ *(,&(3#$ *& ?+ *Z $1A/A

-(-8,#%" 0"<;&03(12 *[F ESG !"#$%&# '!( )* +,-.%%/.

)iii( 2121 2

11121 xxxx !"!#

1221 230

31

21 xxxx !"!#

68359375.5

7890625.3

4375.5

625.3

25.5

5.3

5:

5:

2

1

x

x

6

4

2

1

"

"

x

x

01&234# 564# 167 +,-.%% 8&-%%34# '!.4# 9: ;6<!.

2109375.07890625.30.41

! !$x

31640625.068359375.50.62 ! !$x

!"#$: =.6%3 ,!> !,0.4# ?,0!/4# 0,@ 9* A0& 01&234# 564# 167 +,-.%4#.

0.59

0.24

0.12

5.240.100

0.121.00

2.62.21

96.0

020.0

0.6

5.696.00.5

36.05.40.2

1.31.150.0

"

%&'():

%&

'()

*

" "

" "

0.5904.59,0.1004.10

0.2498.23,0.1204.12

2460

240

0.12

122000

1200.10

2.62.21

"

(5-3) s¹d9

383

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

%&'():

%&

'()

*

" !

" !

24602459592400

12205.12245.241200

02.2

240

0.12

100

1200.10

2.62.21

"

+ , !"#$:02.20163934.2 "02.2!-"

80.5804.5524.1228.512

0.122.62.2

40.224040.242240120

" ! !

"!##

! !" !

x

zyx

yzy

)+(

90.5

364.

192.

45.200.10

24.212.40

3.1192.01

0.6

02.0

96.

1.31.15.0

36.05.40.2

5.696.00.5

"

+ ,90.5904.5,00.1004.1,12.4116.4 """

00.2

088.

192.

100

544.0.10

3.1192.01

99.5

088.

192.

99.200

544.0.10

3.1192.01

" "

6.223.11192.192.

00.12544.088.

00.2

!. . !

!.# !

!"

x

y

z

)B( 01&234# 564#: x = -2.6, y = 1.0, z = 2.0

CD 564# .!,0 9: ): ) + (- E!6& FG$-834# 5!$&%& -H!D -73( C%4#1

9I3! -3 ,&I: )JI%,34# ,K78L4 .L034# 3!.4# 91I%– 01&234# 564# C08%

FG$-83L4.


384

M–N

) : ( 1121)3(&)1( ! ! #!" zyx

1133)3(&)2( !"!"!" zyy

1!!!/ zyx

)+ (

1102

102

102

1101100

1102100

10101

1

2

2

111

121

1110

13

13

13

1313

1313

131313

#.

.

.

##

# "

+!,.%4-&

0

2

102

000

110

10101

102

102

102

10100

10100

10101 131313

13

13

13

1313

1313

1313 .

"

.

.

.

0

2

00

2

1021010 131313

!

!

!

"01

02

3

!

!#"

.!##

x

zy

arbitraryz

zy

zyx

44 ! !!" zyx ,2,0

E!64 ),-!%O# F&-P ):.

) B(

3

2

2

030

110

121

1

2

2

1110

1110

12113

%&'()*+,

1

1

2

100

010

121

2

3

2

110

030

121

"

"

112222

1

1

!# !# !

!

!"

zyx

y

z

) $ ( CD 8&%34# .!,04#)B (01&234# 564# C08% CD): ( Q/R7 CD .!,04# -37!&

)+ ( "0-O F-&-ST C08%. FG$-834# 5!$&% !3U: V21! #WU1– E!6&

,&I: )JI%,34# ,K78L4 .L034# 3!.4# 91I% 9I3! -3– XLY 51K6L4

.!($ F-&-ST.

(6-3) s¹d9

385

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

Z$& !,0.4# ?,0!/4# 0,@ =!.6%4 9!%4$-834# 5.

2121 4.08.31925 xxxx #!"!

1221 25.05.164 xxxx !"!#

1x 2x

0 0

3.8 0.55

4.02 0.495

3.998 0.5005

5K67 9!%!%[# 9!%4$-834# 56&1 01&234# 564# XLY

5.0,4 21 !! xx

002.998.341 ! !5/

0005.05.05005.02 ! !5

M–\

4$-834# ]!,6%& ]4W1 ^ !,0.4# ?,0!/4# 0,@ =!.6%4 FG$-834# +!%,% $Y:

CL! -3I ^?,!O*# 4$-834# V&K%4 X41*#:

"01

02

3

!##

!##

! #

952

1310

2010

321

321

321

xxx

xxx

xxx

321 1.01.02 xxx # !

312 1.01.03.1 xxx !

213 4.02.08.1 xxx !

1x 2x 3x 16 26 36

0 0

986.1

2

0054.1

1.1

123

1.00064

0.96 .014 .0946 .04064

ii78" 1.06

(7-3) s¹d9

(8-3) s¹d9


386

5(: _-0O*# `!3S 9* #,;7 930.1 a!K4# ,!,I% b(17D ^. 91I! ]4W&1

564# ) X4T 5K% ($ $1$6 CD -6!6K0.1 (1U:

00064.1,0054.1,986.1 321 !!! xxx

] ;6<3: 1U 01&234# 564#: 1,1,2 321 !!! xxx[

M–c

d6 Z,# 93 5I =!.6%4 ]4W1 ^ 8&#,4# 4$-834# V&K%4 !7-P4# 4$-834# ] 9!0,@4

9!!4-%4#:

e- #,RK )1-/! G FG$-834# D1RK3 CD C/!",4# ,0.4# CD 53-83 ):.

iii

a 79 0

f- !,0.4# ?,0!/4# .

34

423

312

21

5.0

5.05.0

5.05.0

5.05.0

xx

xxx

xxx

xx

!

#!

#!

#!

1x 2x 3x 4x

0 0 0 0

0.5 0.25 0.125 0.0625

0.625 0.3750 0.21875 0.109375

) : (

"

100

010

0011

5

1

4

1

3

1

4

1

3

1

2

1

3

1

2

1

"

10

01

001

0

0

1

3

1

2

1

45

4

12

1

12

1

12

1

31

21

(9-3) s¹d9

(10-3) s¹d9

387

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

"

11

0126

064

00

110

01

6

1

180

1

61

18018030

18019236

30369

100

010

001

-- ! "#$%&' ()*+,-&' ". -/- 01:

!

"

###

$

%

&'

30

36

9

x

- 2 3,-&' "4 56-&'7! ":

!

"

###

$

%

&

&&

&

'(&

18018030

18019236

303691

!- "4 56-&' 8**1- A:

) * 21601

1801

1211det '++'(

)9(

!

"

###

$

%

&

&

'(','( &

50.40

12.42

59.7

zxzx 1

!

"

###

$

%

&'(','( &

180

180

30

uxux 1

!

"

###

$

%

&

&

'(','( &

180

192

36

xx 1--

yx '(


388

:–;;

; – )<(

!

"

###

$

%

&

&'

!

"

###

$

%

'

!

"

###

$

%

&'(

6

3

0

y,x,

1402

112

111

3

2

1

x

x

x

)9(

,

&

&

&&

&

,

&

&

&

&

3

5.1

5.1

310

5.15.00

5.05.01

6

0

3

402

111

112

,

&&

0

3

5.1

000

310

5.05.01

8*=5>- ". -/-&'

? – )<(

!

"

###

$

%

'

!

"

###

$

%

'

!

"

###

$

%

&

&

&&

'(

4

4

20

y,x,

211

422

224

3

2

1

x

x

x

)9 (

,

&

&

&

&

&&

,

&

&

&&

1

6

5

5.25.00

330

5.05.01

4

4

20

211

422

224

2

2

5

200

110

5.00.51

&

&&

&&

@A. 061> 75A%&' B# ,C&+D :

35.15.05

312

1

1

2

3

'&&'

&'&&'

&'

x

x

x

8*=5>- =#E ()*+,-&' ". -/- F< G<.

". -/-&' 01: ) *t133x &&'

"4 56-&' 8**1- ="#H3C=-&' =6+>,&' 9=I 06+1. F< G<:

) * 24234det '++'(

(11-3) s¹d9

389

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

:-;?

4.21.04.0

0.11.03.0

1.005.01.0

2

2

22

.&&'

..&'

.&'

xyyz

xzxy

zyx

) < (",D+CC-&' (+D#=JCA& "#=#=3C&' "J#=$&':

) <: 40.2,0.1,1.0 ''' zyx

+#>+K:

99.14.201.04.0

021.10.1024.0003.0

088.01.0288.01.0

'.&&'

'..&'

&'.&'

z

y

x

+K&+K:

9920084.14.20089848.04169764.0

9801648.01017512.000232320.0

0062391.01.0198005.01042441.0

'..&'

'.&&'

'.&'

z

y

x

7&+C&' 0 */&' 74 LM+C>&' NO! P#%AC F3-# : x y z

0 0 0

0.1 1.0 2.40

-0.088 1.021 1.99

0.0062391 0.9801648 1.9920084

$ DI-&' 01&' F#-%C F3-# Q>- : 2,1,0 ''' zyx

,-&' ()*+,-&' 74 B# ,C&+D 01&' 'O! "16 F- RJ1C&' F3-# 8+$.

) 9 ("J#=$ G-S:

) <:

9924264.14.200997.0397686.0

997.01003.0

1.0

'.&&'

'.&'

'

z

y

x

+#>+K:

9997636.14.20000912965.04001452.0

0001816.11000181868.0610000002499.0

0009128.01.01984881.009940092.0

'.&&'

'..&'

'.&'

z

y

x

(12-3) s¹d9


390

7&+C&' 0 */&' 74 LM+C>&' NO! P#%AC F3-# : x y z

0 0 0

0.1 0.997 1.9924264

0.0009128 1.0001816 1.9997636

$ DI-&' 01&' F#-%C F3-# Q>- :

2,1,0 ''' zyx

RD+2&' 01&' ! Q25> $SJ4 F#=S#=3C *S,D +S>! 0S1&' 'O! F#-%C F3-# Q>< )T U ) )*SD

"KVK F- ( "J#=$ FW X&O G-S +YD=+JC 74 Z=2<.

:-;:

) < (

xyyz

xzxy

zyx

1.04.00.1

1.03.02.0

05.01.005.1

2

2

22

&&'

.&'

&.'

x y z

0 0 0

1.05 -0.13075 1.006890525

1.003292 -0.0921313 1.0058481

1.00101813 -.0001256832 1.000012575

)9( F< [1V> \#1 $ DI-&' 01&' F#-%C F3-# ],>:

1,0,1 ,,, zyx

"JJ1C- +Y,#-/ F< */> ()*+,-&' 74 ]#J&' NOYD B# ,C&+D .

7! ^+$%W' F< */> 7>+K&' =#=3C&' LM+C>& "D2>&+D:

11

00

22

105.010058.0

105.01009.0

105.0103.0

&&

&&

+/+01

+/+01

+/+01

z

y

x

"-#_ F< G<x "1#16 F##=`. F#-_=&.

"-#_y a#16 **. 9=_W "1#16.

"-#_z *1' G=`. ]_=& "1#16.

(13-3) s¹d9

391

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

)9( 74 "D =I- @& W' "&*+,-&' "#>+K&' "&*+,-&' F- b=$'0.498.

) *) *) *t

t

t

x

x

r

95.1973.0

276.0139.0

00100.0000360.0

1 &'

&'

&'

2

@A. 061> c=%< 8=- F#21C&' "#A-. R#D$CD :

) *tx 99.1996.02 &'

)9 (

) * ) *) * ) * 37

32

1

''

'3'(

DpCp

pp

)d( "4 56-&'B "#D=+JC.

:–;e: ) * ) * ) * 7.18,01.100.16 0(4' 5x

:–;f:

) *tx 0000.20000.1&'

],>: "4 56-&'A $=`&' "A#A..

:–;g:

666

7

8

999

:

;

&

&

'666

7

8

999

:

;

'

400

5.75.40

112

,

115.1

015.1

001

UL ) <(

666

7

8

999

:

;

&

&

&

'666

7

8

999

:

;

&'

800

5.375.00

35.12

,

122

015.0

001

UL ) 9(

(14-3) s¹d9

(15-3) s¹d9

(16-3) s¹d9

(17-3) s¹d9

(18-3) s¹d9


392

:–;h:

666

7

8

999

:

;

&

&'

154699.18164972.00

0224743.17071069.0

00414213.1

L

:–?i:

)<( ) *tx 15.05.0'

)9( ) *tx 011 &'

:–?;:

ixi <'1

:–??:

31

31

131

//&

,/,/

a

ap

:–?::

67

89:

; &'6

7

89:

;'6

7

89:

;'

00

40,

01

00,

0

0ULD

==

66

7

8

99

:

;

&'67

89:

; &>6

7

89:

;&'?

&

&2

1

40

40

00

40

1

0

=

==

=SG

22

44,0 ==@ ',&' p

414

141 22

2A,A,/,/ ==

=p

:–?j:

"4 56-&'A "#=$J&' 8=$#2&' "#,$_.

(19-3) s¹d9

(20-3) s¹d9

(21-3) s¹d9

(22-3) s¹d9

(23-3) s¹d9

(24-3) s¹d9

393

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

:–?k:

)<(

apa

ajj 2,

02

20'6

7

89:

;'?

2

24,

40

20ap

a

aSGSG '6

7

89:

;'? &&

F#CJ#=$&' F- 03 9=+JCB//&21

21 a

)9(

ap

a

ajj

2,01

40'

667

8

99:

;'?

22

4,40

40

ap

a

aSGSG '

66

7

8

99

:

;'? &&

F#CJ#=$&' F- 03 9=+JCC D E,FA&/ 22 a"#a

)d(

2,

02

0abpb

ajj

!!"

#

$$%

& '

2,

20

0abpab

aSGSG

!!"

#

$$%

& ' ((

!"#!$%&' ( )* +$,#")* 2ab

–!":

-!$%#&' .$%!/&' -!0%1 -2345(&' 6 7 809. k = 16

–!#:

) :( 9 -234;5(&' )!<="& >*/&3?" -!(@$'3A BC%A .$3;5&' >;2tLL + >;*3 7

%$?&' B#="! : +D! E9: FD9 $5,90&' G!(D !!0" *(!:

(25-3) s¹d9

(26-3) s¹d9

(27-3) s¹d9


394

21

21 **( a

2232

2222

312111

1111

0,,1

lal

aaal

lall

**(,*,(

21

21

2121

21

2

211

331

2

**(

,*,*(

,(

(

a

aa

a

a

a

) +( 809. . -!0%#&' -!$%#&' .$%!/&' %$? B#="&)F!F="&,C -CD3( -2345(&' 6 3:(.

–!$

zx,yzyx

,yx

-

- tt

t

LLLL

LLAA

..

.

.

.

/

0

11111

2

3

((

66.112.100

078.1452.05.0

0066.15.0

0002

L

-

.

.

.

.

.

/

0

11111

2

3

(((-

6

4

8

7

66.112.100

078.1452.05.0

0066.15.0

0002

yzL

..

.

.

.

/

0

11111

2

3

(

82.3

307.

77.3

5.3

z

8!1 H%A&' 8,K9&':

L t x = z

.

.

.

.

.

/

0

11111

2

3

(

(

(

-

82.3

307.

77.3

5.3

66.1000

12.178.100

0452.66.10

05.5.2

(28-3) s¹d9

395

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

–!%

-#!$% +$,#"" 809 J -2345(&' 6 A -!$%#&' .$%!/&' -!0%1 .

4122

552

39

48

34

4213

312

321

)/x(x

)/xx(xx

)/xx(x

)/xx(x

4(

((4

44(

44(

k 1x 2x 3x 4x

0 0 0 0 0

1 2 3 -1 3

2 1 2 -1.2 3.5

3 1.2 34/15 -1.8 3.6

6)0()1( 105xx1

(56((77k

jT 7

8 9

.

.

.

.

.

.

/

0

111111

2

3

(

(

(

(

:4:(

02100

520

51

51

0310

31

041

410

ULT j

8 9 154

21,

54,

32,

21max *

;jT7

8 9 33,1,3,2maxxxx )1()0()1( (;

67

836.661053.

5154

6

-

<-5*- (..

/

0

11

2

3

k

k

k

–

2x1 – x2 = 0

- x1 + 1.25 x2 = 1.5

- x1 = x2 / 2

x2 = ( 1.5 + x1 ) / 1.25

(29-3) s¹d9

(33-3) s¹d9


396

k 1x 2x

0 0 0

1 0 1.2

2 .6 1.68

3 .84 1.872

UL)(DT SG1(

( 4(

../

0112

3 ( ..

/

0112

3

( ..

/

0112

3

00

10,

01

00,

25.10

02ULD

.

.

/

0

11

2

3 - (

520

210

SGT

5.0 ;(SGT7

../

0112

3 ..

/

0112

3(..

/

0112

3

21

0

0

0

21

0x-x 01

.. )()(

2.1xx )0()1( (-;

8 9 81052.15.01

5.0 (565(k

265.25 -<- kk

– "

42

46

42

23

312

21

)/x(x

)/xx(x

)/x(x

4

44

4

k 1x 2x 3x

0 0 0 0

1 .5 1.5 .5

2 .875 1.75 .875

3 .9375 1.9375 .9375

8 9...

/

0

111

2

3

(

(

(...

/

0

111

2

3

(

(( :4:(

000

25.00

025.0

025.0

0025.

000

ULT j

(36-3) s¹d9

397

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

...

/

0

111

2

3

025.0

25.025.

025.0

5.25.25. 4 ;jT7

5.1xx

5.

5.1

5.

xx )0()1()0()1( (-...

/

0

111

2

3

(

8 9 8 9 51055.1.51

5. (56(k

1687.15 -<- kk

– #

.

.

.

/

0

111

2

3

(

(( ....

/

0

1111

2

3

(

3900

64240

1894

,

1412

014

001

UL

):(

.

.

.

/

0

111

2

3

(

( - -

39

56

11

zyzyx LLU

.

.

.

/

0

111

2

3

(

(

-

131

1

xzxU

)+(

.

.

.

/

0

111

2

3

(

( -

39

40

11

zyzL

..

.

.

/

0

1111

2

3

( -

1

121

xzxU

(37-3) s¹d9


398

– %

-...

/

0

111

2

3

((

((

((

511

151

115

A

8 9 -*

.

.

.

.

/

0

1111

2

3

:4:(

;1

52

52,

52,

52max

051

51

510

51

51

510

)(

j

j

T

ULT

>L'F"C' ED"( M6 +$,#"" >C3*,D -#!$%.

5/)5(

5/)10(

5/)5(

213

312

321

xxx

xxx

xxx

44(

44

44

k 1x 2x 3x

0 0 0 0

1 1 2 -1

2 1.2 2 -.4

3 1.32 2.16 -.36

4 1.36 2.192 -.304

(39-3) s¹d9

399

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

xxy y! y2! y3!

1 1

.025

1.05 1.025 -.001

.024 0

1.10 1.049 -.001

.023

1.15 1.072

) (1 – !"# $%&'()*%+ ,-

015.105.0

025.003.0103.1 "#$%y

2 – ./&%&0# 12345" 6(2/7 .8/9 1/+-(+:

6.005.003.0

05.0103.1.

0 & '# phpx

px

( )

( ) ( )01512.1

001.06.036.021025.06.01

211

. 2

00

&"&#"#

#!"&#!# ypp

ypyy

( ) 015.10148891.103.103.1 ' yExact

)./3:; <%=3> .?@?"(

A-!"# 6> B> = 349)./3:; <%=3> .?@?" (C"# 6& $' ,D 6/("%)1( E)2(.

(1-4) s¹d9

lÐ«d�« qBH�« �UM¹d9 WÐuł√

) >(

x ( )


400

F–G 052.0cos52.0cos & ' xxyxx

H%I/J 25-&"# 6> B>x %KH7; 62'( ,("#y = 0

L5; $9C7 $%&'()@" M7#3I* .8/9 1/+-(+: ( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )07.10667.1

1.102.02.04.02.

02.004.0

102.02.04.02.

02.004.0

9.002.04.02.04.

02.002.0

'

"&&&&&&#

"#&#&#

"#&#& x

F–N

) > (

x f(x) ! 2! 3! 4!

0 0 1.5

4* 1.5 0.0

1.5 -1.0

2* 3.0 -1.0 2.0

0.5 1.0

43* 3.5 0.0

0.5

0),(32

. 003

02

00 #!++,

-../

0#!++

,

-../

0#!# xforwardy

py

pypyyp

( ) ( ) ( )0.1321

2105.10.0 &""

&&##"# pppp

( ) ( ) ( ) ( )24321

321"""

&&&# pppp

1 2 xxhxx

ppppp **4

4

00,8178121 234 & & ##&

32

6.4

6 ' *

** px

) (

( ) ( ) ( ) ( ) 9958847.0328

3217

328

32

121 234

345

678 ##& y

(2-4) s¹d9

(3-4) s¹d9

401

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

9166667.01211

41

32

41

64 # #" **exacty

0792180.0 Error

F–F ' &' 0sin1sin lxxxx

.&/= H%I/J 25-&"#x %KH7; ,("#y = 0 ( ) ( )

( ) ( ) 112.16.02.16.

12.002.0 "&&#&&# x

( ) ( )

( ) ( ) 1.112.02.16.02.

12.016.0 "&&#&&##

( ) ( )

( ) ( ) 2.102.12.16.12.

02.016.0 "##### = 1.11

F–O ( )

.)(.)()(.)(

)()).()(())()().((

)(.)()(.)()(.)(

LHSxgxfhxghxf

hxgxfhxfxghxgxfRHS

xgxfhxghxfxgxfLHS

&##

#&##&#

&## !%

F–P x y ! 2! 3! 4! 5!

0.0 1

0.32

0.1 1.32 0.04

0.36 0

0.2 1.68 0.04 0

0.40 0 0

0.3 2.08 0.04 0

0.44 0

0.4 2.52 0.04

0.48

0.5 3.00

( )

( )

2

20

02.03.01

211

04.032.01

211

.

pp

ppp

ypp

ypyy

##

"&"##

#!"&#!#

(4-4) s¹d9

(5-4) s¹d9

(6-4) s¹d9


402

0608.10008.006.01

231

101.00

02.0

2

0

## 9

##

' '# '#

xy

xxy

xppxphxx

F–Q x p(x) ! 2!-1 7

0

0 7 2

2

1 9 2

4

2 13 2

6

3 19 2

8

4 27 2

10

5 37 2

12

6 49 2

14

7 63

F–R 625.3 y

F–S

) >( 14891.1,13137.1,00499.1

) ( 1025.1 x

F–TU

( ) 2294555247.0 xf

F–TT

7718.27 y

(7-4) s¹d9

(8-4) s¹d9

(9-4) s¹d9

(10-4) s¹d9

(11-4) s¹d9

403

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

F-TG y = 2.7357

F–TN

)>(

x y ! 2! 3!00 %x 1

6

2 7 48

54 48

4 61 96

150

6 211

( ) ( ) 966480 22 : ! ff

( ) ( ) 482543 2 ff ;;

( ) 4833 f;

) ( ( ) ( ) ( )

( ) ( ) ( ).821

21812461

.321

21211

3

03

02

00

pp

pppppp

yppp

ypp

ypyy p

#&

&&#&##

!""&&#!"

&#!#

.1

220

3

0

#&

& &

xxy

xxh

xp

xp

)V( 1111)1( #& y

F–TF:

( ) 136482.063.3 f

F–TO :

3200/3 2max *<=

(12-4) s¹d9

(13-4) s¹d9

(14-4) s¹d9

(15-4) s¹d9


404

(1-5) s¹d9

) (

772.0718.2490.3

1

!

! " #$ % kkkk yyyy

220.0718.2490.32482.4

2 122

%&!

%! " %% kkkk yyyy

[ ' ( ' (718.2490.3490.3482.4 !!! ! ]

6865.0

)117.2490.3(21)()(

21 1

21

11

! $!$ ! !!% kkkk yyyyy)*

x f(x) * 2* 3* 4* 5* 6*

-3x10

0.25 1284

365

0.50 1649 103

468 30

0.75 2117 133 8

601 38 3

1.00 2718 171 11 -2

772 49 1

1.25 3490 220 12

992 61

1.50 4482 281

1273

1.75 5755

002.06 ! + ky*

) "(

' (, ' (' ( -

11301

61

21

44

12

12

110

!!%

%!!! . !!

yy

yyyyh

y

**

**

,- 7172666.210)812(

301

)133220(61)21173490(

25.021)1(

3 &!%

%!!!& .+

!

f

' ( ' (' ( 0002699.1

7172666.2718.2

11

1 . +ff/

f�U)« qBH�« �UM¹d9 WÐuł√

405

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

' ( ' (' (

0. xfxf

x/

' (22

2

21

f

ff

f

fff

f

ffffx

.

...!

.

..!.

.

...!.1. ./

, - yh

y .....901

1211 642

2*** %! ..

' (' (

-,720779.2

1090211

121171

25.0

11 3

2

&!&! ..+ !f

' ( ' ( ' (' ( ' (

001636.0001636.11

26662717

9779272027181

1

1111

22

!

1

1&! .

..1! +f

ff2

) #(

' (

' ( ' (' (

xe

exfxf

x

exf

x

x

x

3 .

0

1/

' ( 11 exact/

' ( ' ( 03 0 xdx

xdx 01

//

' ( ' ( 010 03 exactxx 22

' ( 2699000.012699000.11 ! $/

' ( 636001.00636001.01 ! $2

-!

$%&'( )*+ ,! -./0yy ... , 10+1.03 = x :

' ( 456

789 "!%" . 0

20 12

211 ypy

hy

' ( ' ( 498.0001.012.121025.0

05.01 45

6789 !&!%

, -' (

4.0)001.0(05.0

1021

22! !& .. %" y

hy

' ( - 5055297.0498.001512.1.03.103.1

& . yy/

' ( xx

xyyx 3 . 5.02

1../

(2-5) s¹d9


406

' ( 0055297.003.1 $/

0 dxd/2 2345( )(x/ 67)

' ( ' ( ' (22 498.04.001512.103.1 %!& .%.. yyynumerical

2

= 0.158044

' ( 158044.003.1 $2

' 89:!%&;< 2404'3;< =*( "3>3 9'3? @AB;< 6 C/D ) 9'E. -!1F;< (

(1 G4HB0<!2404'3;< IJK = .

) (

' ( ' (' ( ' ( 61,21

00,10

6,13

1

2

3

.. .

..! .

..! .

%!

yy

yy

xyxy

xxy

)"(

baabax

babax

byayx

! 0! 0 %%

0 %!

%.%..

263026:1

0:0

0

) (

x y " 2" 3" 4" 310!& 310!&

0 0

.541

0.1 .541 86

.627 6

0.2 1.168 92 0

.719 6

0.3 1.887 98 0

.817 6

(3-5) s¹d9

(4-5) s¹d9

407

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

0.4 704.2 104 0

.921 6

0.5 3.625 110 0

1.031 6

0.6 656.4 116 0

1.147 6

0.7 5.803 122 0

1.269 6

0.8 7.072 128 0

1.397 6

0.9 8.469 134

1.531

1.0 10.000

)"( =0?&& =F91 -( 67 =L5;45;< =LF91;< $LK 24L04'3;< IJK M& NH%% 1!1/ 89'5?;3" 2345.

)#( ' ( ' ( ' (

03

02

0 32121

211

. ypp

ypp

yyy pp "&&!:!%"&

!%":%

2.01.0

1.012.00 ! 0% pphxx p

' ( ' (

' ( ' (659328.0106

3218.18.02.0

092.021

8.02.0627.2.0541.012.

3 &&&&!!%

%&&!%&%

!

y

' ( ' ( %"%!%"!%"% 0323

022

00 2361

21 ypppyppyyy p

' ( 456

789 %"%!%"!%" .

;;

<

=

>>

?

@ 0263

6112

211 32

02

0 yppypyh

y

' ( ' ( ' (

0032.6

006.22.112.61092.014.0

21627.0

1.0112.

456

789 &%!%&!% .y

456

789

!!%!!!!! ;;

<

=

>>

?

@

;;

<

=

>>

?

@ 11

30111

61112

1 44420 yyyyyy

hDy ****

' ( ' ( 75.9104.116.61704.2656.4

2.015.0 45

6789 !!! .y

01211 42

20

2 yh

yD 456

789 ! **

' ( , - 11110.001.015.0 ..y


408

)1(

x y " 2" 3" 4" 5" 6"

310!& 310!& 310!& 310!& 310!&

0 0

.541

0.1 .541 86

.627 6

0.2 1.168 92 0

.719 6 -360

0.3 1.887 98 -360 2160

.817 -354 1800

0.4 2.704 -256 1440 -5400

.561 1086 -3600

0.5 3.265 830 -2160 7200

1.391 -1074 3600

0.6 4.656 -244 1440 -5400

1.147 366 -1800

0.7 5.803 122 -360 2160

1.269 6 360

0.8 7.072 128 0

1.397 6

0.9 8.469 144

1.541

1.0 10.000

N!9LLH;< 6 CLL/D0i" LLL; =LL&'( O7i 894LLPQ< 4LL&R<1 4LLS; 2LL04?LL>H04S . 4LL& 119LL% )LL;T U1 @LLAB;< V(9LL;< 6WLL: 6X<i" $;4LL%;43! Y =3;4LL>;< V'LLZ;<! =LL3F!&;< V'LLZ;< 6'LL3

V'LL( 6WLL:i" V'LLZ; 4Ai [<9HLL. 1LL\% VLL; . -!1LLF;< ]LL.%0& $;<!LL/ 6LL& ^1LL%3' @LLAB;4: 9 _*5& -?P $: [4\>%& 9P%0'! L0+ `L>1 6!L?% 6 )L;T =LRA4B;< =L'R<1%3,< =L&'Z;<6"

@AB 4S*? . !'!F% 10 V'( 6'3 9C4i" )_'/4Ai (=LRA4B;< =L&'Z;< V4L& ALB;< -!L/ Y 6'%9C4LL0%&;< 6'LL%&'Z;< 6 )LL0\&3– `LL%/%! ALLB;< N!LL:– =LL&'Z;< 4LL&S; LL>H04S 894LLPQ<!

4S>H0 204? 6Ti ='F!a )6,4( i =L&'Z;< 4&S;! YL>H04S 894LPQ<!L>?+4S 2L04? 6Ti ='19: )i = 5(.

' ( 061215.03 ..y

(5-5) s¹d9

409

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

) (

x y " 2" 3"1.08 .295

0

1.12 .295 -.005

-.005 0

1.16 .290 -.005

-.010

1.20 .280

' (0

30

200

3211

yp

ypp

ypyy ";;<

=>>?

@%"&

!%"%

= 0.295 – 0.0025 p (p – 1)

272504.008.10 ! !

! xx

h

xxp

p

' ( ' (595.14375.35625.1

282527250025.295.

2 !%!

!!!

xx

xxy

)"( )&C\;< ='4S0;< =AZ0 10+ 0 .y

' ( ' (125.35625.12

296.0,1.1,

295625.0595.178125.3890625.1

1.14375.35625.120

! &! ..

+

!%!

0%&!

y

yx

y

xx

)#( !"#$% &'()*$% &+,* -(*('. /0123 4(+#!$% &'56(78$% &$9(#!$% ::

! ! !29732004.18352.3

41.11.13296.0401.1125.31.1

2

222

"#"#"#

$%"%$%&$$

nnn

n

) ;<n =>1! ?'@A 992( .

(6-5) s¹d9


410

)B ( 1)=( f(x) = sin2x + 2 cos 3x

x f(x) ' ' 2 ' 3

0 2

-1.75

6( 0.25 0.25

-1.5 3.5

362 (( " - 1.25 3.75

2.25

263 (( " 1

C*%D>E ;1*(F G'.+*.1 :

2

236

236 %$$$

$$$"

))

*

+

,,

-

.

))

*

+

,,

-

.

))

*

+

,,

-

.

))

*

+

,,

-

.

))

*

+

,,

-

.

))

*

+

,,

-

.

(((

((( xxx

y

!25.0

26360

6

230

%$$$

$$$&

))

*

+

,,

-

.

))

*

+

,,

-

.

))

*

+

,,

-

.

))

*

+

,,

-

.

))

*

+

,,

-

.

(((((

(( xxx

! !25.1

23630

3

260

$%)*+,

-. $)

*+,

-. $)

*+,

-. $

)*+,

-. $)

*+,

-. $$

&(((((

(( xxx

! ! ! ! ! ! !1

32620

2

360

%$$$

$$$&

(((((

(( xxx

)H(

! ! !

! !

%38946.11%1687008.0

481200814

cos212

2sin12

3125.15.36

5.15.5.

25.02

5.05.075.15.02

12

03

02

.

50

6

012

12

3200

"/#"/

"&")*+,

-.

"%$$

&

%$

&$&")*+,

-.0

1))*

+,,-

.&1))

*

+,,-

.&1&"

"$

"$

"#"

rrorrror

. exacty

f

yp

yp

ypyy

.

h

xxp x

op

(

(7-5) s¹d9

411

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

234

567 )

*+,

-.

$$$$$" 1161

1121 22 yyyyh

D ''

! !8 9 7559859.025.06125.1

62

13

"$$$%

":0(

(f

xxxxxy 3sin62sin3sin6cossin2 $"$":

! 0.8660255 sin 6 3

2sin 3exacty "$": (((

%70627.12%1100395.0 "/#"/ rrorrror

234

567 $"

41211 2

2

2 ''h

D

8 9 9118905.025.0

6

16 2

"

)*+,

-.

")*+,

-.::

(

(f

xxy 3cos182cos2 $"::

! 1 0 3

2cos 6exacty "$":: ((

%81095.8%0881095.0 "/#"/ rrorrror

)9(

83

6

2169

p "" (

(( $

&;))*

+,,-

.&;))

*

+,,-

.&;&" 0

3200

30

2y

py

pypyy

! !75.3

285

83

25.2831

169 %

$<&%&"(f

! !

6264648.15.36

813

85

83

"%$<$<

&

x y ' ' 2 ' 3 4'

0 1

-0.08

0.2 0.92 0

-0.08 -0.02

0.4 0.84 -0.02 0.02

-0.1 0

0.6 0.74 -0.02

-0.12

0.8 0.62

(8-5) s¹d9


412

) B (9(>'I y 9*20.22 =x J*8D(0.2x0 "

1.02.0

2.022.0 "$"p

:!(!<% ;1*(,$% G'.+8.1:

! ! 9129.09.01.0202.01.008.092.0 "$$%$"y

)=(

234

567 )

*+,

-. $$$":

$$

1

2

1

2

11 61

21 yyyyh

y ''

! !8 9 441667.0002.06192.074.0

4.014.0 $"$$$$":y

234

567 $"

42

2

2

1211 ''

hD

! 8 9 541667.002.012102.0

04.014.0 $"%$$"::y

!"#$:

&$%9$% K19> 1L K%MN$% :3 +#!$% K19>$%xy sin" ()56(78 :$(8$(.1

xy cos": .3 O$P$1(:5' (!Q :L K%MN$% %PL :3 &.15+!$% R',5$ &+1.6!$% &.1>< :

) B( !"#$% &'()*$% &+,* :

!2

5707963.1 (="x

!2

sin000.1 (="y

) =(

! !5.1sin997495.05.1 $=$"::y

! !5.1cos0707372.05.1 $=$":::y

) H(

! !42.1cos1502255.042.1 =":y

! !42.1sin9886518.042.1 $=$"::y

&.15+!$% R',$% 9(>'I &'99#$% GD+$% $S :5' (!'3 D'T*1 :

(9-5) s¹d9

413

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

)B ( !"#$% &'()*$% &+,* 9*20":y :NQ#$% C*%D>E ;1*(F G'.+8.1–

UV!– 91!2 R'F 52y: &!'F 52 KA@* K19>$% :3x: ! ! !

! ! !

7.16.15.1

4.11288.17.0292.17.0707.17.

1288.00292.00707.0

%&%&%&

%&&$

&&$"

x

x y ' ' 2 4'

410$% 410$% 410$%

1.4 .9855

120

1.5 .9975 - 99

21 - 1

1.6 .9996 - 100

-79

1.7 .9917

99.001.1099)5.1(

4

2

2

$"%$""::$

h

yy

'

WDJB &,'D+. (L9(>'S ;Q!' 1B:

! 996.01.02

1700.00292.02

5.1 11 "%$$"

:=23

456

7 $":: $

yyh

yyy

! 06.001.1065.1

4

2

2

$"%$"

:=223

4

556

7"::

$

yyh

yy

'

43658.0$":y

U#$% ;!F &)15(:

! 234

567 1$&1":

0

2

0 211 ypy

hpy

! ! !8 9012010 2211 yyypyy

hphxy &$$&$"&:

1X61. p = 0:

(11-5) s¹d9

(10-5) s¹d9


414

! !8 90120100 2211 yyyyy

hxyy &$$$":=:

8 9210 4321 yyyh

$&$"

:

! !>32

3fh"/

-:

! tr

h

fffxf "#"#

#$%&&

2

*0

*1

*2

1

2

'()

*+, -

./0

12 $#-

./0

12 $$-

./0

12 $%" *

002*

11*

222

21 ffffffh

r

! kkr

hh

$$ 3%##334" 102121105.0122

412

2

564" ht

!hh

h

ktotal "%5#34" $

4

2

2 12102

4/1

4min

1024

--

.

/

00

1

2

53

%$k

h

-!":

h 0.1 0.2 0.3

!8.1f &&$ 0.9600 0.9675 0.9656

) !"#$ %&'"( )*"+:( 0.9738 .81 sin .8)1(fExact $%$%&&

7% 2.0h ,!-. /01( 23)- . 4+ 5*&678 796:78 ,!-. ;: <=-! 8>?@

0.2 ABC78 %!578 D"'( 2?(0.1, 0.2, 0.3) EFC:78 :!5F7:

2.21

410244/1

min 8--

.

/

00

1

2 $3%h

(12-5) s¹d9

(15-5) s¹d9

(16-5) s¹d9

415

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

sI

! 04719.41

5ln "#" xxxIexact

310143.1 #$"E

i) % & ! 99429.096593.25882.210212/ "''''"(

)t

% & 96593.70711.25882.410312/ ''''"( )

S

! 00003.1) .86603 (.5 2 "''

ii) 00003.,00571.,1 "*"*" stexactI

iii) % & % & 006.010512212

622+,$+-.//#"* # hhfhabt

)0

% & % &% & 15.01051802180

6444+,$+-.#"* # hhfhabS

)1

% & % &23,012

21 22 2",'##"// # xxexf x

% & % & ! % & 032202124 222

"'##,"#'#'## xxxxxe x

346.023,0 .,2", hx

1 = 0.18516 3 0.19

6931471.021

6932539.0322

74

5445.01

121

""(

"456

789 ''''"(

:;

:<=

:>

:?@

:;

:<=

:>

:?@

nex

(1-6) s¹d9

(2-6) s¹d9

(3-6) s¹d9

(4-6) s¹d9

”œU��« qBH�« �UM¹d9 WÐuł√

048333.4"


416

!5

4

4

1

!418041

1

100684.1

xth

act

"

#$$%&

'%&

((

)

*

++

,

-

#

4

4max1020832.5

11804

24 #'%''

%&

5

54min106276.1

21804

24 #'%''

%&

–!

2751253.10511802

54./'.'$.& # hh0

2751253.42

./%%#% xxrrab

h 000

1241877.11 %/1/ xx

00003.13,12

%/%%/ Irh 0

–

i) 5

)4( 24)(x

xf %

1391578.0105124

1801 54

./'.$$ # hh

J

EE

EKI

JIii

151

1516

16

)

21

2

1

#2%3/45

46

7

%

"%

&"%

iii) 8 9 6944444.03625

5.1145.01

35.0

5.0 %%'""%%hJ

!8 9 6932539.0322

74

5445.01

325.0

25.0 %'""""%%hK

6931745.06944444.01516932539.0

1516 %'#'%3

6931471.02ln %%3exact

E = 0.6931745 – 0.6931471 = 0.0000274

" #$ %$&'()* +,-($ &./(0 &1'12 &/,34 .56 &7(,869,/: )i( #$ ;1<7

=>.? #:@A(,)?'( &8'(>"&-A9 #B9: ,C@D7 E:>F9" +319 (/B$ +'- +"AF?@,/ h.

(5-6) s¹d9

(6-6) s¹d9

417

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

–G

i) ! 3

max

2105.0

121 #':;;$ <fh

! 2122

max3%%;;/%;; f

xxf

054772255.0105.0212

32:/':$ # hh

ii) x 1 1.5 2 I1 = 0.7083333

x1 1 0.6666667 0.5

x 1 1.25 1.5 1.75 2

x1 1 0.8 0.6666667 0.5714285 0.5

I2 = 0.6970237

I = I1 + E1 , I = I2 + E2 , E1 = 4E2

()*+

,- %%% !hE

hhh :,

2125.0,5.0 2

21 =

6932538.03

4 12 %#>3 II

/'%/%3%3 #3101067.06931471.02 Errornexact

I &'()* +,-($ &H<H9 I'12.

#'?9,19" =?3'?7 JK* #:@A(,)?'( &8'(> L'/>?/ M7$ %$25.0,5.0 %% hh

&-A9" JK* ,7K21@D7,C NFO/ P(1769" M/) EA*,8/ ,C'K* Q217 =?9" 0.05477 <h.

–R I = 3.1044

–S

? %$%3/%2/

0

sin104.3cos

cossin

0@

@@@@ dex

)'&8/,@9" &9O@69" Q6,B? I/2 @D7M( .

(7-6) s¹d9

(8-6) s¹d9

(9-6) s¹d9


418

–TU

i) 69564.00

%tI

ii) 002493.0693147.02 %/%3%3 Enex

iii) 69377.0%tI

iv) 6931466.0%RI &'()* +,-($ &?@9 I'12 :V: +.7-

v) 6931466.0%SI

P(1769" M/) EA*,- &3'?7 #6 ('HB/ QWX$ #:@/6@ EA*,- &3'?7 #$ IW":

=X)iii( Y'1 EA*,- Z":3 ,67'/ &'()* +,-($ &?@9 I'12 #:@/6@ EA*,8/ Z":39"

M/) >8X #''()* #'6-(9 I'12 P(1769" . JK* #:@A(,)?'( &8'(> L'/>?/ #B9:

P(1769" M/) EA*,8/ #'('A8?#:@/6@ EA*,- Z":3 JK* ,7K21 M@D7 .

–TT

?

?

%%

#$%3/%

1

0

2

2

0

1

22

310268.0sin

1sin1

duu

duu

uutu

–T[

4675.146746.1 /%I

–T\

i) ]=3: >,879" AA* #0 _#0 :$ _

5.24.012

2%#%#%

habr

#$ %$r ,1'12 "AA* %:,@? ] .

6863.4%tI

ii) 015526.0670774.4 %/%3 Eex

iii) 6707666.4%sI

410074.0104.7 46 %A/'%'% ##E

(10-6) s¹d9

(11-6) s¹d9

(12-6) s¹d9

(13-6) s¹d9

419

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

iv) /'. #4

max

2105.0

12xeh

00901.0105.012

422./'. # heh

":&6'- #$ IWth ('`2 &6'89 &/@79,/ "A3 Esh Y'1 1.0%sh.

–Ta

i) 00003.1%sI

ii) 400003.01 %/%/%3 NEex

iii) 0195441.0105.01122

42./'.$$ # hh0

26.012

B%::/ 0st hh

–T!

i) dtda C%

/% ?? dtdtddta

TT C00

! !00

CC #%? TdtaT

! ! ?%/% dtaT T

000 CC

! 8 9 .sec/5.3087.31302

10100

10mdta %"%% ?C

! D8

E9 .sec/5.17430.38

7.356.337.3127.40302

10500

50

m

dta

%"

"""""%% ?C

ii) ! !8 ""%% ? 5010210

50 CCC hdtS

!D ! !E94030202 CCC """

! 8 9 6357.3126.33302

1020 %'""%C

! D E8 9 5.9816.337.3127.35302

1030 %"""%C

! D E8 9 13507.356.337.31238302

1040 %""""%C

D E !.925.3939925

13505.98163525.17435.3082

10

kmm

S

""

####"$

(14-6) s¹d9

(15-6) s¹d9


420

!" #$)%&'()' *+,-.(:(

% & % & % & % &' ( % & !

% & !.850.3939850

5.9812135063545.17435.3083

10

3024020450103

kmm

hS

""

)####"

)####" *****

/–0/

i) a) +," dteey t2

0

5.025.0)5.0(

1 23:

% & 25.02 ,-" eetf t

25.02

2)( ,". eettf t

% & 25.022

122)( ,-#".. etetf t

% & % & 222

max105.0

12105.0 ,, )/..,0)/ 1fhabE

% & % &8

125.02212

05.022.025.02

max

heehE "#)))," ,

2.0105.08

22/0)/ , hh

b) 1.0"h -4!3

' (! 4252687.02

21.0)5.0(

16.009.004.001.0

25.0025.0

"####

##)" ,

eeee

eeey

c) 21008323.0 ,)"E

% & 0899738.15.02

222)

25.0

5.0

0

5.0

0

22

""

"20-"0" + +

ex

t

ye

dtedtttteItxii

/–05

1) !+ ##"" )()(4)(3

)( 2102

3xgxgxghdxxgI

&6789'(:; %&'()' *+,-< =:

(16-6) s¹d9

(17-6) s¹d9

421

‰uBH�« �UM¹d9 WÐuł√ ∫(2)r�— o×K*«

5.0223

12",")

," abh

!)3()5.2(4)2(35.0 ggg ##"2

% & % &+ #"32

2x

xdyyxxg

% & % & % & % &++ #3"#"16

2

16

2

4;42 yydyydyyg 44

% & ! % & % &! ! 1822013463716942

37 "#)#"##" 444

% & % &+ +)

#"#"

35.22

5.2

25.31

5.2

,25.625.6)5.2( dyydyyg

375.14275.28 ""h

% & % & ! 84375.664875.1625.645.3775.83375.145.2 "#)##"g

% & % & !+ ")##"#" 5.19125.37463123

5.25933

54dyyg

! 31253.7925.191284375.664418235.0 "#)#"2

2) % &%223.076253.155.79031253.792 "","E


422

(1) r�— —U³²šô« qŠ

1. a) !"#$%& '!( )#*#$&+, -!'./01,'2

2284631111319234

421112 !

!"!"!"!#!

!"!""!"!$y

2488423412324558

212123 !

!!!!#!

"!!"!!#

= 25020.5

b) $% 25432 – 25020.5 = 411.5

0161.025432

5.411 $$%r

104400

17.59

1043000

14.59003.

78.46

17.59

130.6291.5

14.59003.)2

""&

"

a.

' ( !"#$:176466.1763003./291.5 !

001.1104300104400 !!y

.003 x = 59.17 – 59.14 * 1.001 = -0.03

00.10003./03. "!"!# x

b) $$!%! 00.2000.1010xinError

001.0000.1001.1 !"!yinError

14.59

78.46

14.590

130.6291.5

17.59

78.46

14.59003.

130.6291.5) "

"c

& 'exacty 000.114.59/14.59 !!

5.291 x = 46.78 + 6.130 = 52.91 #

& 'exactx 00.10291.5/91.52 !!

!"#$%&'(#) *!$&+" ,+&-+ ./+&0(#) *1#) 234 %5361 78 9&(!:#) ;&5%< =&!1 >

<(! %( ?+<8 7@<"?(#) ?65'3# 9:3.(#) .

3 . a) A:1"( ?!B 9!?.:#) C?.!-#) .?D E ,?%:"#) (05 F b) 8x + 2y + 3z = 30

x – 9y + 2z = 1 #

2x + 3y + 6z = 31

x = (30 – 2 y – 3 z) / 8

y = (1 – x – 2 z) / – 9

423

(3) r�— o×K*«W�UŽ �«—U³²šô« Ã–U/ WÐuł√

z = (31 – 2 x – 3 y) / 6 i x y z

0 1 1 1

1 3.1250 0.4583 3.8959

2 2.1745 0.9963 3.9437

3 2.0220 0.9899 3.9977

Error = 0.0020 0.0101 0.0023

4) & ' & ' 2

max

2

105.0;312

"()**""! tEfabhtE

4/)(2xexf "!

2

2/1)( xxexf ""!*

)12(2/1)( 22

%"""!** xxexf

0max)( !!** xatxf

2/1)(max!** xf

#()!((!# "222

max105.0

24211

12hh

tE

3/13464.0 !#) hleth

x: 0 1/3 2/3 1

f(x): 0.25 0.223709829 0.160295097 0.09196986

& ' & ' & ' & '+ ,- .184996618.0

32

31210

2

!

%%%! ffffhI

5) a)

t P / 2/ 3/ 4/

50 77

18

60 95 14

32 -7

70 127 7 -5

39 -12

80 166 -5

34

90 200

5.0,5.01051975 !!! qpEverett


424

& '5321

)5.()5(.)5.1(1665.01975 "

(("

%(!0 Py

& '7321

)5.()5(.)5.1(1275.0

(("

%(%

& '554321

)5.1()5.()5(.)5.1()5.2("

((((""

%

= 147.191406

84,900,6.1061984 !!"!"! pxxpBackward

6.0109084,0 "!#(%!%! ppphxx p

& ' & ' & '521

6.16.0346.02001984 "(

""("!0 Py

& '126

)6.2()6.1()6.( """"%

& ' 32.180524

)6.3()6.2()6.1()6.( !"""""%

b) %13.0001302.147

1471914061471975

1!"2!r3

%46.3034612.785186

32.1807851861984

1!2"2!r3

c) 90,0? 0 !!! xpatdtdP

& ' & ' & ' %4((""%4(

"%4%! oyppp

ypp

ypyy 30

200 321

2121

1

& ' & ' & '

%4((("""

04

4321321

ypppp

& '- %4%"%4"%4!* 5678

9: 322 263

6112

211 ppp

hy

. 0)6.......4(241 43 yp 4"

- .)5(246)12(

62)5(

2134

101

19900"""%""!0*

! dt

dpy

p

= 3.375

6) ix

ixi

e

exixix "

"

%

""!%

11

x0 = 0.5

x1 = 0.566311

x2 = 0.5671432

x3 – x2 = 1.2 ( 10 - 7 = 0.012 ( 10 -5 < 0.5 (10 -5

8 78x3 9!?D4 G%H?8 9-(I# 91!16 56714.0!# x

& '

425

W�UF�« �«—U³²šô« Ã–U/ WÐuł√ ∫(3)r�— o×K*«

(2) r�— —U³²šô« qŠ

1) a) f(0) = -1 < 0 , f (1) = .45969 > 0 #

]1,0[;< r

& ' #;=>%!* ]1,0[0sin2 xxxxf

f (x): increasing 9!$!)@" in #]1,0[ .+0#%+ $1)/ ?JK

b) #"!*#!x

xxgxxg

cos2

sin)(cos)(

x

xxg

cos2

sin)( !*

C?"L#) MN]1,0[ 9#)$#)sin x !$!)@"& %(5!+ 9cos x 9!6H%5" O&N P#J#/ >)(xg*

$54 2(Q'#) %R"(!H 2#S *6" x = 1.

1572.01cos2

1sin)(

max)!!*! xgm

? +?%:"9!9!T)$"+) 9(!H 7U+ !T$"+( > .

c) & ' #@#(!@ " 05.1.572.105 2 nnom 33

64.5 !#A# nn

or: 73.1cos1,05.0011!!!@"

"xxx

mmn

3

& '

505.0173.43.

57.!#@"# n

n

d) i xi

0 1

1 0.7350526

2 0.8612755

3 0.8071371

4 0.8316064

5 0.8207859

6 0.8256188

2. a) & '

2

2

1

1

ee

xeexx

ix

iix

ii"

"""!%

x0 = 1, x1 = 1.5819767, x2 = 1.8055081

b) f(r) 0 f (2) = 0


426

& ' & ' 0222 !"!*#"!* eerfeexf x

20)()( 2 !#B!**#!** mtymultiplicierfexf x

c) & '& '

& '2

2

1

1.2

ee

xeeix

xf

xfmxx

ix

iix

i

iii

"

"""!

*"!%

d) 1639534.21

1.211 10 !"

"!#!e

xx

1639534.01 !"! rx3

3. a) y). ( e.Fy). ( e.F x.y

x.x 20sin20,20cos20 2020 "!!

3.0

02max1!

!!

yx

xx FF

12.0

22max1!

!!

yx

yFyF

& ' DyxFF yx ;=)!%@% ,142.012.03.0

*V(#%+/:

12.022max1!

!!

yxxx GG

3.0

02max1!

!!

yxyy GG

& ' DyxGG yx ;=)%# ,1

b) i xi yi

0 0 2

1 0.9210609 0.3894183

2 1.1986263 0.0935426

3 1.2706775 0.0237752

4 1.28933 0.0061308

4. a) A:1"( ?!B 9!?.:#) C?.!-#) .?D E ,?%:"#) (05 F

b) 9!?.:#) C?.!-#) .?D A!:1"# 9'+)?#)/ 9V#%V#) !"#$%'(#) *W$+5 ,?%:"#) %(0 M#%"#%+/.

x1 = – 0.5 x2

x2 = (4 – 2 x1 – 2 x3) / 5

x3 = (– 4 – 2 x2 – 2 x4) / 5

x4 = (1– 2 x3) / 5 i x1 x2 x3 x4


427

0 0 0 0 0

1 0 0.8 -1.12 0.648

2 -0.4 1.408 -1.6224 0.84896

3 -0.704 1.73056 -1.831808 0.9327232

5. a) 555

6

7

888

9

: "

!555

6

7

888

9

:

"

!

200

040

633

,

123

014

001

UL

b) zxyzyx = , U = L = LU #

555

6

7

888

9

:

!#555

6

7

888

9

:

!#

1

2

1

x,

2

8

3

z

c) tttt .Ay)(AAA """ !!#!#! y.xyxyx 11

yyxx 1""!# A.A. ttt

6141

1

2

1

121xxyy 1 !%%!555

6

7

888

9

:

!!"" ) ( . AA ttt

d) A = L U #!#!C yxyx (LU) .A tt

yxyx. !**D! ULLU tt

X@K#) MN %(< *1#) *(<8 GV )b( *)Y-#) (.

]9Q1Z( :9N/L6( ?!/$" L 9N/L6( M.'!U [!16 \<'#)/ > [.

6) a) x y / 2/ 3/0 1

6

2 7 48

54 48

4 61 96

150

6 211 b) 3821 ppy p %"!

22

00 xxh

xxp

p !"!"

!

13 %"!# xxy


428

c) xyxy 6,13 2 !**"!*

,0)0(,1)0(

,6)1(,2)1(,1)1(

!**"!*

!**!*!

yy

yyy

d) x = 0 # - a + b = 0 # a = b

x = 1 # 6 + 2 a + b = 0 # 3a = - 6 # a = - 2 = b

7) a) + , + ,- .)2(2)5.2()5.1(4)3()1(61 fffffI S %%%%!

= 6.9981904

b) & ' & '& 'max

44

180EfabhE "@

f (x) = x2 ln x, ,ln2)( xxxxf %!*

& '& ' ,2,ln23)( 3

xxfxxf !%!**

& '122max

)4(

2)()4( !!#"! xatf

xf x

& ' & ' 34

max10388.12*13

1805.0 "(!"@E

c) max

001388.00004313.0 EEactual 0)!

d) 1972.53ln23;)(12 maxmax

2!%!****"@ ffabhE

& ' & ' #()"# "32105.01972.513

12h

Shh 0)))) 5.0024.0


429

(3) r�— —U³²šô« qŠ

1) a) #"!*#"! xxgxg(x) 1)(ln3

- . convgtxxg x #;=)!* 3,5.111)(

b) 1641158223068528222 210 ., x.x,x !!!

2) y,exy)f(x, x ""! "3 yeyxy)g(x, "%"! 4

xx ef "%! 3 1!xg

1"!yf yy eg """! 4

),f (),( fk),( fh yx 000000 00 "!%

),g (),( gk),( gh yx 000000 00 "!%

195,

196

1

1400

00

00 !!#FGH

"!"

!"kh

skh

kh

3158.196

1960001 !!%!%! hxx

2632.195

1950001 !!%!%! kyy

3) a) 22 1 ee(x)f), (xeef(x) xx "!*""!

50818051976758111 210 ., x ., xx !!!

b) 02( 22 = ee)f(root)f "!*0*

2

02 2

!#

B**!**#**

mtymultiplici

) = e(f(root)f(x) = ef x

c) & '& '

& '& 'i

ii

i

iii xf

xfx

xfxf

mxx *"!*"!% 21

9534163.2,1 10 !! xx

4) a) 555

6

7

888

9

:

!555

6

7

888

9

:

!

100

110

222

,

120

011

001

UL

yxbybxbx !!#!#! , ULLU A

555

6

7

888

9

:"

!#555

6

7

888

9

:

!#

2

2

2

x

2

4

4

y


430

b) #! L.UA

#!#!#! bx.bx.)(bx tttt LULU.A

bx. 11 !UL

byyx 11 !#! LU

+/*1 ?!IE) G%Q5#) by1 !L G%Q5#) *1+ GVyx1 !U 5 /$K x.

5) yyyx 11 DC CD(BA !"# "

vyvy1 B Let B "$"

factorize vy BBBB ULULB "$"

Algorithm : # of arith. operations

compute u1 = D y n2

u2 = C u1 n2

factorize B = LB UB 3

3n

solve for v: LB UB v = y n2

compute u3 = v – u2 n

factorize A = LA UA 3

3n

solve for x: LA UA x = u3 n2

!"#$%#& '&!$()* ++%&*:

Total # of operations nnn %%& 23

32 4

6) ' # ( ) 19.05.0,1.0,9.0max *""aT+

' # ( ) 16.06.0,4.0,2.0max *""bT+

' # ( ) 12.12.1,8.0,4.0max ,""cT+

,$"-"./"01&* 2*+3.4* -a, b -5 1*+ 6/"01&* 2*+3.4* 789": ;b -5

)()( ab ++ * 6/"01&* <0!/. -:,"=b 6/"01&* <0!/. -$ >04?a.

7) a) ' # ' # -./

012 %%%%"

34

356

37

389

::;

<==>

?::;

<==>

?::;

<==>

?

43

21

41210

2fffffhI t

2788706.05.8

161

5.312

1111

24/1 "-.

/012 %%%%" ::

;

<==>

?

b) ' # ' # ' #@@ fEfhabE AABC$AA "1216

112

2

' # ' #32 101200

10110

1011)(

xf

xf

xxf

%"AA$

% "A$%"


431

0416.11216

2001216

10max

"B"AAB"

"xfE

239789530

1011ln)101(ln

101

1011

0

1

1

0

.

xdxx

Iexact

"

"%"%" D

max

039081.0 EIIE texactactual *" "


432

433

oKD*« QD)«qO−Fð

W�œwKFH�« QD)«sJðù WOKLŽW¹d³ł W�œUF�

WO�“—«ušwKOK%V¹dIð

wHK)« i¹uF²�«WOHK)« ‚ËdH�« q�UF�

s¹b(« �«– �ö�UF�nOBM²�« WI¹dÞ WO�“—«uš

nOBM²�« WI¹dÞœËb(« ◊Ëdý

d�JM� jšW¹e�d*« ‚ËdH�« q�UF�

…eOL*« W�œUF*«©wJ��uAð® WI¹dÞ

©wJ��uAð® WO�“—«ušlDI�«

X³¦*« b(«W²³¦*« WO³OFJ²�« W×¹dA�« WO�“—«uš

W²³¦*« W×¹dA�«WIKG� WGO�

©ÍbIŽ® V�d� œbŽ©ÍbIŽ® V�d� —cł

r−�M�W³�d*« q�UJ²�« mO�

Absolute error

Acceleration

Accuracy

Actual error

Aitken's process

Algebraic equation

Algorithm

Analytic

Approximation

Back substitution

Backward difference operator

Binomial coefficients

Bisection algorithm

Bisection method

Boundary conditions

Broken line

Central difference operator

Characteristic equation

Choleski's method

Chloleski's algorithm

Chopping

Clamped boundary (conditions)

Clamped cubic spline algorithm

Clamped spline

Closed formula

Complex number

Complex root

Compatible

Composite integration formulas

ÍeOK$ù« `KDB*« wÐdF�« `KDB*«

(4) r�— o×K*«Index W¹eOK$ù«Ë WOÐdF�« �U×KDB*« qO�œ

ÎU¹b???−Ð√ ÎU???³???Oðdð W???³ðd???� W¹e???OK$ù« W???GK�U?Ð �U??×?KDB?*UÐ W??L?zU??� w?K¹ U???L??O???�ÆWOÐdF�« WGK�« w� UN� WKÐUI*« �U×KDB*«Ë

434

ÍeOK$ù« `KDB*« wÐdF�« `KDB*«W³�d*« Êu�³LÝ WGO�

W³�d*« ·d×M*« t³ý WGO�…dFI� W�«œ

◊dA�« œbŽ—«dL²ÝUÐ q{UH²K� WKÐU� W�«œ

»—UIð»—UI²�« qO−Fð

W¹d¹dJ²�« ‚dD�« »—UIðWÐb×� W�«œ

©d�«d�® …bŽU�©�Ëd�® WI¹dÞWO³OFJð W�œUF�

WO�ULJ²Ý« WO³OFJð W×¹dýWO³OFJ²�« W×¹dA�UÐ ‰ULJ²Ýô«

©sÝËœ® q�UJðWHO¦� W�uHB�

WI²A�…œb×�

W¹dDI�« …dDO��«‚ËdH�« jD��

‚ËdH�« ‰Ëbł‚ËdH�«

WOK{UH²�«WOK{UH²�« W‡DÝu²*« WLOI�« W¹dE½

q{UH²�«q{UH²�« q�UF�

�U�uHBLK� dýU³*« qOK×²�«…dýU³� WI¹dÞ

WKB²� dOž W�«œbŽU³ð

©q²O�Ëœ® WI¹dÞnÓŽUC� —cł

WOð«– WLO�wð«– t−²�

dOOF²�« l� ·c(«dOOFð ÊËœ ·c(«

QD)«

Composite Simpson formula

Composite trapezoidal formula

Concave function

Condition number

Continuously differentiable function

Convergence

Convergence acceleration

Convergence of iterative methods

Convex function

Cramer's rule

Crout's method

Cubic equation

Cubic spline interpolant

Cubic spline interpolation

Dawson integral

Dense matrix

Derivative

Determinant

Diagonal dominance

Difference scheme

Difference table

Differences

Differentiability

Differential mean value theorem

Differentiation

Differentiation operator

Direct factorization of matrices

Direct method

Discontinuous function

Divergence

Doolittle's method

Double root

Eigenvalue

Eigenvector

Elimination with normalization

Elimination without normalization

Error


435

ÍeOK$ù« `KDB*« wÐdF�« `KDB*«QD)« bŠ

QD)« d¹bIðQD)« u/

QD)« —UA²½«ÍbOK�ù« —UOF*«

rOOIð©�dH¹≈® WGO�◊u³C*« œbF�«◊u³C*« q(«

œułËQD�K� wÝ√ u/

¡UHO²Ýô«W¹UNM�« v�≈ ¡UHO²Ýô«

W�uHB� ©‰ULŽ≈ Ë√® qOK%W¹dDI�« WOŁö¦�« �U�uHB*« qOK%

nz«e�« l{u�« WI¹dÞ…œËb;« ‚ËdH�« WI¹dÞ©WON²M�® …œËb×� …d²�

WON²M� WOKLŽW²ÐU¦�« WDIM�« WI¹dÞ

©ÁU&ô«® XÐU¦�« lÞUI�«©ÁU&ô«® XÐU¦�« ”UL*«

WLzUF�« WDIM�« WGO�»cÐcð

WO�U�_« ‚ËdH�« q�UF�w�U�_« i¹uF²�«

d(« b(«”ËU' ‚ËdH�« WGO�

”ËU' ·c(« WO�“—«uš”ËU' ·c(« WI¹dÞ

wze'« “UJð—ôUÐ ·c×K� ”ËUł WI¹dÞq�UJ²K� ”ËUł WGO�

Ê«œ—uł ≠ ”ËUł WI¹dÞ‰b¹“ ≠ ”ËUł WI¹dÞ

wÝbM¼ »—UIðf½U−²� ÂUE½

d½—u¼ WI¹dÞ

Error bound

Error estimate

Error growth

Error propagation

Euclidean norm

Evaluation

Everett formula

Exact number

Exact solution

Existence

Exponential growth of error

Extrapolation

Extrapolation to the limit

Factorization of a matrix

Factorization of tridiagonal matrices

False position method

Finite difference method

Finite interval

Finite process

Fixed point method

Fixed secant

Fixed tangent

Floating point form

Fluctuation

Forward difference operator

Forward substitution

Free boundary

Gauss difference formula

Gauss elimination algorithm

Gauss elimination method

Gauss elimination with partial pivoting

Gaussian quadrature formula

Gauss-Jordan method

Gauss-Seidal method

Geometric convergence

Homogeneous system

Horner scheme

W¹eOK$ù«Ë WOÐdF�« �U×KDB*« qO�œ ∫(4)r�— o×K*«

436

ÍeOK$ù« `KDB*« wÐdF�« `KDB*«◊dA�« WKOKŽ W�uHB�

◊dA�« WKOKŽ W�Q��◊dA�« qOKŽ ÂUE½

Àb×� —UOF�¡«dI²Ýô«

WOzUN½ ô WOKLŽ·UDF½« WDI½wz«b²Ð« QDš

—«dI²Ýô« ÂbŽWOK�UJ²�« WDÝu²*« WLOI�« W¹dE½

‰ULJ²Ýô«…d²�

w�JF�« ‰ULJ²Ýô«d¹dJ²�«

d¹dJ²�« W�uHB�Íd¹dJ²�« 5�×²�«

©W¹—«dJð Ë√® W¹d¹dJð WI¹dÞW¹d¹dJ²�« WOHB²�«

©wÐu�Uł® WI¹dÞ!«dłô W¹œËbŠÍuKŽ bŠ dG�√

W¹UN½QD�K� wDš u/

wD)« ‰ULJ²Ýô«wDš q�UF�

wD)« ©¡UCH�« Ë√® ⁄«dH�«wDš ÂUE½

—Ëc'« l�«u�wKH��« b(«

WOKH��« W¹UNM�«WOKHÝ WO¦K¦� W�uHB�W�uHB� ”uJF� œU−¹≈

W�uHB*« —UOF�w³�½ QDš d³�√

WDÝu²*« WLOI�« q�UF�WDÝu²*« WLOI�« W¹dE½WDÝu²*« WDIM�« …bŽU�

Ill-conditioned matrix

Ill-conditioned problem

Ill-conditioned system

Induced norm

Induction

Infinite process

Inflection point

Initial error

Instability

Integral Mean Value Theorem

Interpolation

Interval

Inverse interpolation

Iteration

Iteration matrix

Iterative improvement

Iterative method

Iterative refinement

Jacobi's method

Lagrange polynomial

Least upper bound

Limit

Linear growth of error

Linear interpolation

Linear operator

Linear space

Linear system

Localization of roots

Lower bound

Lower limit

Lower triangular matrix

Matrix inversion

Matrix norm

Maximum relative error

Mean value operator

Mean value theorem

Midpoint rule


437

ÍeOK$ù« `KDB*« wÐdF�« `KDB*«©—dJ�® nŽUC²�

©—dJ�® nŽUC²� —cł©Á—«dJð �«d� œbŽ® —cł nŽUCð

»—UC�«WOFO³D�« WO³OFJ²�« W×¹dA�« WO�“—«uš

wFO³Þ —UOF�WOFO³D�« W×¹dA�«

qš«b²*« »dC�«WOHK)« ‚ËdHK� sðuO½ WGO�

fðu� ≠ sðuO½ WGO�WO�U�_« ‚ËdHK� sðuO½ WGO�

sðuO½ WI¹dÞ�ôœUF� rEM� sðuO½ WI¹dÞ

Êu��«— ≠ sðuO½ WI¹dÞWODš dOž �ôœUF� ÂUE½

…œdHM� dOž �ôœUF�œdHM� dOž q�UJð

…œdHM� dOž W�uHB�©ÍdH� dOž® t�Uð dOž qŠ

W�uHB*« —UOF�t−²*« —UOF�

dOOF²�«d]OF�

ÍœbF�« ¡UG�ù«W¹œbF�« �UÐU�(«ÍœbF�« q{UH²�«ÍœbF�« q�UJ²�«ÍœbF�« q�UJ²�«WŠu²H� WGO�

WOÐU�(« �UOKLF�« VOðdðwÐcÐcð

wze'« q�UJ²�«wze'« “UJð—ô«

∆e&w¾¹e& wDš ‰ULJ²Ý«

�U¹œËb(UÐ w¾¹e'« V¹dI²�«w¾¹e& wFOÐdð ‰ULJ²Ý«

Multiple

Multiple root

Multiplicity of a root

Multiplier

Natural cubic spline algorithm

Natural norm

Natural spline

Nested multiplication

Newton backward difference formula

Newton-Cotes formulas

Newton forward difference formula

Newton method

Newton method for systems of equations

Newton - Raphson method

Nonlinear system of equations

Nonsingular equations

Nonsingular integral

Nonsingular matrix

Nontrivial solution

Norm of a matrix

Norm of a vector

Normalization

Normalized

Numerical cancellation

Numerical computations

Numerical differentiation

Numerical integration

Numerical quadrature

Open formula

Order of arithmetic operations

Oscillatory

Partial integration

Partial pivoting

Partition

Piecewise linear interpolation

Piecewise polynomial approximation

Piecewise quadratic interpolation


438

ÍeOK$ù« `KDB*« wÐdF�« `KDB*« ÍbðË Ë√ ÍeJðd� dBMŽ

“UJð—ô«©œËbŠ …dO¦�® W¹œËbŠ

b¹b×²�UÐ W³łu� W�uHB�¡UDš_« ©b�«uð Ë√® —UA²½«

wFOÐdð »—UIðwFOÐdð ‰ULJ²Ý«WOFOÐdð W¹œËbŠ

©Wł—b�«® WOŽUÐ— W�œUF�WNO³A�« sðuO½ WI¹dÞ

»—UI²�« ‰bF�rOI�« WOIOIŠ W�«œ

©W¹œuŽ® W¹œ«bð—« WGO�nz«e�« l{u�« WI¹dÞ

WO�UE½ W�«œw³�M�« QD)«

w³�M�« QD)« bŠ—cł W�«“≈

©w³Ý«d�« Ë√® wI³²*« t−²*«ÊuÝœ—UA²¹— ¡UHO²Ý«

W�œUF� —cłV¹dI²�«

V¹dI²�« QDšWO½U¦�« ‚ËdH�«

WK�K�²*« „uJH�©WK�KÝ® WO�U²²�

WŠ«“ù« q�UF�W¹uMF� ÂU�—√

Êu�³LÝ …bŽU�Êu�³L�� 3/8 …bŽU�

WODš WO½¬ �ôœUF�WODš dOž WO½¬ �ôœUF�

œdHM� q�UJðœ^dHð

…œdHM� W�uHB��ôœUF*« qŠ

WA¼ Ë√ …dŁUM²� W�uHB�

Pivot element

Pivoting

Polynomial

Positive definite matrix

Propagation of errors

Quadratic convergence

Quadratic interpolation

Quadratic polynomial

Quartic equation

Quasi-Newton method

Rate of convergence

Real valued function

Recursive formula

Regula Falsi method

Regular function

Relative error

Relative error bound

Removal of a root

Residual vector

Richardson extrapolation

Root of an equation

Rounding

Rounding error (or Round-off error)

Second differences

Series expansion

Sequence

Shifting operator

Significant digits

Simpson rule

Simpson three-eights rule

Simultaneous linear equations

Simultaneous nonlinear equations

Singular integral

Singularity

Singular matrix

Solution of equations

Sparse matrix


439

ÍeOK$ù« `KDB*« wÐdF�« `KDB*«W�uHB* wHOD�« dDI�« nB½

WO�ULJ²Ý« W×¹dýW�uHB*« —UDA½«

—«dI²Ýô«dI²��

¢s�MH²Ý ¢ WI¹dÞ…uD)« r−Š

W¹dDI�« …dDO��« WOFD� W�uHB�i¹uF²�«

WFÐU²²*« �U³¹dI²�« WI¹dÞWKŁUL²� W�uHB�WO³O�d²�« WL�I�«

©WODš® �ôœUF� ÂUE½WODš dOž �ôœUF� ÂUE½

—uK¹Uð WK�K�²�`�U�ð ≠ �ËUHð

WO�U�²� W�œUF�·d×M*« t³ý …bŽU�

w¦K¦*« ©o¹dH²�« Ë√® qOK×²�«w¦K¦� ÂUE½

W¹dDI�« WOŁöŁ W�uHB�W¹dDI�« wŁöŁ ÂUE½

WFD²I� Èu� WK�K�²�5ðd� q{UH²K� WKÐU�

bOŠË qŠdI²�� dOž

ÍuKF�« b(«UOKF�« W¹UNM�«

WO�u� WO¦K¦� W�uHB�w�u� w¦K¦� ÂUE½

©ÁU&ô«® dOG²*« lÞUI�«©ÁU&ô«® dOG²*« ”UL*«

t−²*« —UOF�„uK��« ÈuÓÝ

◊dA�« WM�Š W�uHB�◊dA�« s�Š ÂUE½W�œUF� Ë√ W�«œ dH�

Spectral radius of a matrix

Spline interpolant

Splitting of a matrix

Stability

Stable

Steffensen algorithm

Step-size

Strictly diagonally dominant matrix

Substitution

Successive approximations method

Symmetric matrix

Synthetic division

System of (linear) equations

System of nonlinear equations

Taylor series

Tolerance

Transcendental equation

Trapezoidal rule

Triangular decomposition

Triangular system

Tridiagonal matrix

Tridiagonal system

Truncated power series

Twice differentiable

Unique solution

Unstable

Upper bound

Upper limit

Upper triangular matrix

Upper triangular system

Variable secant

Variable tangent

Vector norm

Well-behaved

Well-conditioned matrix

Well-conditioned system

Zero of a function (or of an equation)


n�RLK� V²�»uÝU(« rKŽË �UO{U¹d�« w�

Æ 1992 X¹uJ�« ÆÆ rKI�« —«œ ¨4 ◊ ¨Ê«dð—uH�« WGKÐ VÝU(« W−�dÐ ≠ 1

Æ 1993 X¹uJ�« ÆÆ rKI�« —«œ ¨2 ◊ ¨�«dHA�« WžUO�Ë �U�uKF*« W¹dE½ w� W�bI� ≠ 2

Æ 1986 X¹uJ�« ÆÆ rKI�« —«œ ¨WOL�d�« �UJ³A�« ≠ 3

Æ 1988 X¹uJ�« ÆÆ rKI�« —«œ ¨ÍœbF�« qOK×²�« ≠ 4

Æ 1995 X¹uJ�« ÆÆ rKI�« —«œ ¨2 ◊ ¨ wD)« d³'« ≠ 5

Æ 1999 X¹uJ�« ÆÆ rKI�« —«œ ¨ 2◊ ¨‰UJÝU³�« WGKÐ VÝU(« W−�dÐ ≠ 6

Æ1994 X¹uJ�« ÆÆ rKI�« —«œ ¨Ê«uý— …eLŠ Æœ l� ¨‰UJÝU³�« WGKÐ W�bI²*« W−�d³�« ≠ 7

Æ1993 X¹uJ�« ÆÆ rKI�« —«œ ¨ ©WLłdð® WOL�d�« WK�UJ²*« dz«Ëb�« ≠ 8

1997 X¹uJ�« ÆÆ rKI�« —«œ ¨ ‰UJÝU³�« WGKÐ W−�d³�«Ë �UO�“—«u)« ≠ 9

Æ1998 X¹uJ�« ÆÆ rKI�« —«œ ¨ Ê«uý— …eLŠ Æœ l� ¨ �UODF*« vMÐ ≠10

Æ 2000 X¹uJ�« ÆÆ rKI�« —«œ ¨ W¹œUF�« WOK{UH²�« �ôœUFLK� W¹œbF�« ‰uK(« ≠11

Æ 2001 X¹uJ�« ÆÆ rKI�« —«œ ¨ WOze'« WOK{UH²�« �ôœUFLK� W¹œbF�« ‰uK(« ≠12

Æ 2002 X¹uJ�« ÆÆ rKI�« —«œ ¨C++ WGKÐ VÝU(« W−�dÐ ≠13

Æ 2003 X¹uJ�« ÆÆ rKI�« —«œ ¨C++ WGKÐ W�bI²*« W−�d³�« ≠ 14

Æ 2005 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ »uÝU(« rKŽ w� WFDI²*« �UO{U¹d�« ≠ 15

2006 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ Ê«uý— …eLŠ Æœ l� ¨C++ WGKÐ �U½UO³�« q�UO¼ ≠ 16

Æ 2006 X¹uJ�« ÆÆ √d�« —«œ ¨C WGKÐ W�bI²*« W−�d³�« ≠ 17

Æ 2007 X¹uJ�« ÆÆ √d�« —«œ ¨C WGKÐ »uÝU(« W−�dÐ ≠ 18

Æ 2010 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ »uÝU(« rE½ ≠ 19

Æ 2010 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ WOKJA�« �UGK�«Ë WOðU�uðË_«Ë W³Ýu(« W¹dE½ ≠ 20

Æ 2012 ÆÆ X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ WOKJA�« �UGK�«Ë WOðU�uðË_«Ë W³Ýu(« w� W�bI²*« W¹dEM�« ≠ 21

Æ 2012 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�« ≠ 22

ÔrOJÓÚ(« ÔrOKÓFÚ�« ÓX½Ó√ Óp]½≈ UÓMÓ²ÚL]KÓŽ UÓ� ]ô≈ UÓMÓ� ÓrÚKŽ Óô ÓpÓ½UÓ×Ú³ÔÝ˚©33∫ …dI³�« …—uÝ®

NUMERICAL METHODSand

NUMERICAL ANALYSIS

Dr. Abu-Bakr Ahmad El-SayedDepartment of Computer Science

University of Kuwait

Documents

ÍœbF « qOK×² «Ë W¹œbF « ‚dDmathcsbooks.com/assets/files/الطرق العددية والتحليل العددي... · ÍœbF « qOK×² «Ë W¹œbF « ‚dD « bO «