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8/2/2019 GII GN NG PHNG TRNH PHI TUYN V PHNG TRNH VI PHN TRN MY TNH IN T
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I HC THI NGUYN
TRNG I HC SPHM
TRN TH HON
GII GN NG PHNG TRNH PHITUYN
V PHNG TRNH VI PHNTRN MY TNH IN T
LUN VN THC S TON HC
THI NGUYN - 200
8/2/2019 GII GN NG PHNG TRNH PHI TUYN V PHNG TRNH VI PHN TRN MY TNH IN T
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I HC THI NGUYN
TRNG I HC SPHM
TRN TH HON
GII GN NG PHNG TRNH PHI TUYNV PHNG TRNH VI PHN
TRN MY TNH IN T
Chuyn ngnh: Gii tch
M s: 60.46.01
LUN VN THC S TON HC
Ngi hng dn khoa hc:
TS T Duy Phng
THI NGUYN - 2007
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MC LC
Trang
Li ni u..............................................................................................2-3
Chng 1. Gii gn ng phng trnh phi tuyn trn my tnh in
t...............................................4
1. Gii gn ng phng trnh ( ) 0f x .......4
2. Cc phng php tm nghim gn ng ca phng trnh
( ) 0f x ......10
3. Tm nghim gn ng ca phng trnh ( ) 0f x trn my tnh in
t.....24
Chng 2. Gii gn ng nghim ca bi ton Cauchy cho phng trnh vi phn
thng trn my tnh in t..................48
1. Phng php gii gn ng bi ton Cauchy cho phng trnh vi phn
thng......482. Phng php Euler .........52
3. Phng php Runge-Kutta ......57
4. Gii bi ton Cauchy cho phng trnh vi phn trn my tnh in t
.........64
Kt lun..................................................................................................82
Ti liu tham kho...............................................................................83
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LI NI U
Cc bi ton thc t (trong thin vn, o c rung t,) dn n vic cn
phi gii cc phng trnh phi tuyn (phng trnh i s hoc phng trnh vi
phn), tuy nhin, cc phng trnh ny thng phc tp, do ni chung kh c th
gii c (a c v cc phng trnh c bn) bng cc bin i i s. Hn na,
v cc cng thc nghim (ca phng trnh phi tuyn hoc phng trnh vi phn)
thng phc tp, cng knh, nn cho d c cng thc nghim, vic kho st cc
tnh cht nghim qua cng thc cng vn gp phi rt nhiu kh khn. V vy, ngay
t thi Archimedes, cc phng php gii gn ng c xy dng. Nhiu
phng php (phng php Newton-Raphson gii gn ng phng trnh phi tuyn,phng php Euler v phng php Runge-Kutta gii phng trnh vi phn) tr
thnh kinh in v c s dng rng ri trong thc t.
Vi s pht trin ca cng c tin hc, cc phng php gii gn ng li
cng c ngha thc t ln. gii mt phng trnh bng tay trn giy, c khi
phi mt hng ngy vi nhng sai st d xy ra, th vi my tnh in t, thm ch
vi my tnh in t b ti, ch cn vi pht. Tuy nhin, vic thc hin cc tnh ton
ton hc trn my mt cch d dng cng i hi ngi s dng c hiu bit su sc
hn v l thuyt ton hc. Mt khc, nhiu vn l thuyt (s hi t, tc hi t,
chnh xc, phc tp tnh ton,) s c soi sng hn trong thc hnh tnh
ton c th. V vy, vic s dng thnh tho cng c tnh ton l cn thit cho mi
hc sinh, sinh vin. Cng c tnh ton s h tr c lc cho vic tip thu cc kin
thc l thuyt, ging dy l thuyt gn vi thc hnh tnh ton, s gip hc sinh, sinh
vin khng ch tip thu tt hn cc kin thc khoa hc, m cn tip cn tt hn vi
cc phng php v cng c tnh ton hin i.
Ni chung, trong cc trng ph thng v i hc hin nay, vic gn ging
dy l thuyt vi tnh ton thc hnh cn cha c y mnh. iu ny hon ton
khng phi v thiu cng c tnh ton, m c l l v vic ph bin cch s dng cc
cng c tnh ton cn t c quan tm.
Vi mc ch minh ha kh nng s dng my tnh in t trong dy v hc
mn Gii tch s, chng ti chn ti lun vn Gii gn ng phng trnh phi
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tuyn v phng trnh vi phn trn my tnh in t. Lun vn gm hai chng:
Chng 1 trnh by ngn gn cc phng php gii gn ng phng trnh phi
tuyn v c bit, minh ha v so snh cc phng php gii gn ng phng trnh
thng qua cc thao tc thc hnh c th trn my tnh in t khoa hc Casio fx-570ES. Chng 2 trnh by phng php Euler, phng php Euler ci tin v phng
php Runge-Kutta gii phng trnh vi phn thng. Cc phng php ny c so
snh v minh ha qua thc hnh tnh ton trn my tnh Casio fx-570 ESv trn
chng trnhMaple.
C th coi cc qui trnh v chng trnh trong lun vn l cc chng trnh
mu gii bt k phng trnh phi tuyn hoc phng trnh vi phn no (ch cn
khai bo li phng trnh cn gii). iu ny c chng ti thc hin trn rt
nhiu phng trnh c th.
Tc gi xin chn thnh cm n TS. T Duy Phng (Vin Ton hc), ngi
Thy hng dn tc gi hon thnh lun vn ny. Xin c cm n Trng i
hc S phm (i hc Thi Nguyn), ni tc gi hon thnh chng trnh cao
hc di s ging dy nhit tnh ca cc Thy. Xin c cm n Phng Gio dc
Ph Yn (Thi Nguyn), nitc gi cng tc, to mi iu kin thun li tc
gi hon thnh kha hc v lun vn. Cui cng, xin c cm n Gia nh ng
vin, gip v chia x nhng kh khn vi tc gi trong thi gain hc tp.
Thi Nguyn, 20.9.2007
Trn Th Hon
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CHNG I
GII GN NG PHNG TRNH
PHI TUYN TRN MY TNH IN T
1. GII GN NG PHNG TRNH ( ) 0f x
Phng trnh ( ) 0f x thng gp nhiu trong thc t. Tuy nhin, ngoi
mt s lp phng trnh n gin nh phng trnh bc nht, phng trnh bc hai,
phng trnh bc ba v bc bn l cc phng trnh c cng thc nghim biu din
qua cc h s, v mt vi lp phng trnh c gii nh cc k thut ca i s(phn tch ra tha s, t n ph,) a v cc phng trnh bc nht hoc bc
hai, hu ht cc phng trnh phi tuyn l khng gii c chnh xc (khng c
cng thc biu din nghim qua cc h s ca phng trnh), v vy ngi ta
thng tm cch tm nghim gn ng ca phng trnh. V ngay c khi bit cng
thc nghim, do tnh phc tp ca cng thc, gi tr s dng ca cng thc nhiu
khi cng khng cao. Th d, ngay c vi lp phng trnh n gin l phng trnh
a thc bc ba 3 2 0 ax bx cx d , mc d c cng thc Cardano gii,
nhng v cng thc ny cha nhiu cn thc kh cng knh (xem, th d:
Eric W. Weisstein: CRS Concise Encyclopedia of Mathematics, CRS Press, New
York, 1999, mc Cubic Equation, trang 362-365),
nn thc cht chng ta cng ch c th tm c nghim gn ng. Hn na, a s
cc phng trnh, thm ch nhng phng trnh rt n gin v mt hnh thc
nhng li xut pht t cc bi ton thc t, th d, phng trnh cosx x khng ccng thc biu din nghim thng qua cc php ton c bn (cng, tr, nhn, chia,
khai cn, ly tha), ni cch khc, khng gii c hoc rt kh gii bng cc php
bin i i s, nhng c th gii gn ng n chnh xc bt k rt d dng nh
php lp 1 cos n nx x , nht l trn my tnh in t b ti (ch cn bm lin tip
mt phm ).
Nhng phng trnh xut hin trong cc bi ton thc t (th d, khi o
c,) ni chung c thng tin u vo (th hin trn cc h s, trong cng thc) ch
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l gn ng (sai s trong o c, nh gi, tnh ton s b,...). V vy vic tm
nghim chnh xc cng khng c ngha thc t ln, trong khi vi cc phng
php gii gn ng phng trnh, ta thng c cng thc nh gi chnh xc ca
nghim gn ng v c th tm nghim n chnh xc bt k cho trc, nnphng phpgii gn ng phng trnh c ngha rt quan trng trong gii quyt
cc bi ton thc t.
Cc phng php gii chnh xc phng trnh ch mang tnh n l (cho tng
lp phng trnh), cn cc phng php gii gn ng phng trnh mang tnh ph
dng: mt phng php c th dng gii cho nhng lp phng trnh rt rng,
th d, ch i hi hm s l lin tc chng hn, v vy kh nng ng dng ca gii
gn ng l rt cao.
Gii gn ng phng trnh lin quan n nhiu vn quan trng khc ca
ton hc. Th d, theo iu kin cn cc tr (nh l Fermat), im 0x l im cc
tr (a phng) ca hm s ( )y F x th n phi l im dng, tc l
0 0'( ) '( ) 0 y x F x . Nh vy, tm im cc tr, trc tin ta phi gii phng
trnh ' '( ) : ( ) 0 y F x f x tm im dng (im c nghi ng l im cc
tr). Trong thc t tm nghim ti u, ta thng i tm cc im dng (nghi ng
l cc tr) nh gii gn ng phng trnh ' '( ) : ( ) 0 y F x f x .
Bi v mt trong nhng th mnh ca my tnh in t l kh nng lp li
mt cng vic vi tc cao, m gii gn ng phng trnh thc cht l vic thc
hin mt dy cc bc lp, nn nh my tnh m vic gii gn ng phng trnh
tr nn n gin, nhanh chng v thun tin. Khng nhng th, my tnh cn cho
php, thng qua lp trnh, m phng qu trnh thc hin bc lp gii phng trnh,
bi vy n l cng c tt trgip hc sinh v sinh vin tip thu cc kin thc ton
hc ni chung, cc phng php gii gn ng phng trnh ni ring. Do thc
hnh gii gn ng trn my tnh in t c mt ngha nht nh trong ging dy
v hc tp b mn ton trong cc trng ph thng v i hc.
Trong chng ny, gii gn ng phng trnh, chng ta lun gi thit
rng, ( )f x l mt hm xc nh v lin tc trn mt on no ca ng thng
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thc. Nhiu khi iu kin ny l xy dng phng php gii gn ng.
Trong mt s phng php, ta s gi thit rng ( )f x kh vi n cp cn thit (c
o hm cp mt hoc c o hm cp hai).
Nu ( ) 0f x th im x c gi l nghim hoc khng im ca
phng trnh ( ) 0f x . Ta cng gi thit rng cc nghim l c lp, tc l tn ti
mt ln cn ca im x khng cha cc nghim khc ca phng trnh. Khong
ln cn (cha x ) ny c gi l khong cch lica nghim x .
Cc bc gii gn ng phng trnh
Gii gn ng phng trnh ( ) 0f x c tin hnh theo hai bc:
Bc 1. Tm khong cha nghim
Mt phng trnh ni chung c nhiu nghim. Ta cn tm khong cha
nghim, tc l khong ( , )a b trong phng trnh c nghim (c duy nht
nghim), bng mt trong cc tiu chun sau.
nh l 1 (Bolzano-Cauchy)Nu hm ( )f x lin tc trn on ,a b v tha mn
iu kin ( ) ( ) 0f a f b th phng trnh ( ) 0f x c t nht mt nghim trong
khong ( , )a b .
ngha hnh hc ca nh l ny kh r rng: th ca mt hm s lin tc
l mt ng cong lin tc (lin nt), khi chuyn t im ( , ( ))A a f a sang im
( , ( )) B b f b nm hai pha khc nhau ca trc honh, ng cong ny phi ct trc
honh ti t nhtmt im (c th ti nhiu im).
Th d, hm s 3( ) 3 1 y f x x x c ( 2) 3 f ; ( 1) 1 f ; (0) 1 f v
(2) 1f nn phng trnh 3 3 1 0 x x c ba nghim phn bit trong cc
khong ( 3, 1) ; ( 1,0) v (0,2) .
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nh l 2 (H qu ca nh l 1) Gi s ( )f x l mt hm lin tc v n iu
cht trn on ,a b . Khi y nu ( ) ( ) 0 f a f b th phng trnh ( ) 0f x c duy
nht mt nghim trong khong ( , )a b .
ngha hnh hc ca nh l ny l: th ca mt hm s lin tc tng
cht (gim cht) l mt ng cong lin tc (lin nt) lun i ln (i xung). Khi di
chuyn t im ( , ( ))A a f a sang im ( , ( )) B b f b nm hai pha khc nhau ca
trc honh th th phi ct v ch ct trc honh mt ln (Hnh v).
Hai nh l trn ch i hi tnh lin tc m khng i hi tnh kh vi (tn ti
o hm) ca ( )f x . Nu ( )f x c o hm th c th dng tiu chun di y.
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nh l 3 (H qu ca nh l 2) Gi s hm s ( )f x c o hm ( )f x v o
hm ( )f x ca n khng i du (lun dng hoc lun m) trn on ,a b .
Khi y nu ( ) ( ) 0f a f b th phng trnh ( ) 0f x c duy nht mt nghim
trong khong ( , )a b .
T ba nh l trn, ta i n hai phng php tm khong cch li nghim ca
phng trnh ( ) 0f x (khong cha duy nht mt nghim): phng php hnh hc
v phng php gii tch.
Phng php gii tch
Gi s ta phi tm nghim ca phng trnh ( ) 0f x trong khong ( , )a b .
Ta i tnh gi tr ( )f a , ( )f b v cc gi tr ( )if x ca hm s ti mt s im
( , )i x a b , 1,2,...,i n . Nu hm ( )f x n iu cht trn khong 1, i ix x v
iu kin 1( ) ( ) 0 i if x f x c tha mn th 1, i ix x l mt khong cch li
nghim ca phng trnh ( ) 0f x . Nu thng tin v hm ( )f x qu t th ta
thng dng quy trnh chia on thng (chia khong ( , )a b thnh 2, 4, 8,phn) v
th iu kin 1( ) ( ) 0 i if x f x tm khong cch li nghim.
Mt a thc bc n c khng qu n nghim. V vy phng trnh a thc c
khng qu n khong cch li nghim.
Khi hm ( )f x tt (c o hm, c dng c th,...), ta c th kho st
th chia trc s thnh cc khong i du ca o hm (khong ng bin v
nghch bin ca hm s) v xc nh khong cch li nghim.
Phng php hnh hc
Trong trng hp th hm s tng i d v, ta c th v phc th
tm khong cch li nghim hoc gi tr th ca nghim nh l giao im (gn ng)
ca th vi trc honh. Cng c th dng cc my tnh ha (my tnh c kh
nng v hnh nh Casio Algebra fx-2.0 Plushoc Sharp EL-9650) hoc cc phn
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mm tnh ton (Maple, Matlab,) v th. Sau , nh tnh ton, ta tinh
chnh i n khong cch li nghim chnh xc hn.
Bc 2. Gii gn ng phng trnh
C bn phng php c bn gii gn ng phng trnh: phng php chia
i, phng php lp, phng php dy cung v phng php tip tuyn (phng
php Newton-Raphson). Nhm lm c s l thuyt cho cc tnh ton trong 3,
trong 2 chngti s vn tt trnh by ni dung ca cc phng php ny, ch yu
l da vo cc gio trnh Gii tch s [1] - [6].
2. CC PHNG PHP TM NGHIM GN NG CA
PHNG TRNH ( ) 0f x
1. Phng php chia i
Ni dung ca phng php chia i rt n gin: Gi s ( )f x l mt hm
lin tc trn on ,a b v ( ) ( ) 0f a f b . Khi y theo nh l Bolzano-Cauchy,
phng trnh ( ) 0f x c t nht mt nghim trong khong ( , )a b .
Chia i on ,a b v tnh ( )2
a bf .
Nu ( ) 02
a bf th 2
a bx l mt nghim ca phng trnh ( ) 0f x .
Nu ( ) 02
a bf th ( ) ( ) 02
a b f a f hoc ( ) ( ) 0
2
a b f f b nn phng
trnh c t nht mt nghim trong khong ( , )2
a ba hoc ( , )2
a bb .
Gi khong mi (khong nh) cha nghim l 1 1( , )a b .
Li chia i khong 1 1( , )a b v tnh gi tr ti im gia1 1
2
a bx .
Tip tc mi qu trnh ny ta i n:
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Hoc ti bc th n no ta c ( ) 02
n n
a bf , tc l
2
n n
a bx l
nghim, hoc ta c mt dy cc on thng lng nhau [ , ]n na b c cc tnh cht:
1 2 1... ... ... ...n na a a a b b b ,
( ) ( ) 0n nf a f b v 2
n n n
b ab a .
S hi t ca phng php chia i
Dy na l dy n iu tng, b chn trn bi b , dy nb l n iu
gim v b chn di bi a nn c hai dy u c gii hn.
Do2
n n n
b ab a nn lim 0
n n
nb a hay lim lim
n n
n na b x .
Do tnh lin tc ca hm s ( ) y f x ,ly gii hn trong biu thc
( ) ( ) 0n nf a f b ta c2 ( ) lim ( ). ( ) 0
n n
n f x f a f b .
Suy ra ( ) 0f x hayx l mt nghim ca phng trnh ( ) 0f x trong khong
( , )a b .
nh gi sai s
Ti bc th n ta c n na x b v2
n n n
b ab a .
Nu chn nghim gn ng l nx a th2
n n n
b a x x b a ;
Nu chn nghim gn ng l nx b th2
n n n
b a x x b a ;
Nu chn nghim gn ng l2
n n
a bx th ta c nh gi:
12 2
n nn
b a b ax x .
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Nh vy, sau bc th n , nn chn nghim gn ng l2
n nn
a bx c , ta s
c nghim chnh xc hn.
Nu chn2
n nn a bx th 12 2
n nn nb a b ax x . Do vi mi 0
cho trc ( chnh xc 0 cho trc) ta c nx x vi mi
2log
b an
.
Nu ti mi bc n ta u chn2
n nn
a bx
th ta cng c
1 1 2 1( ) ( )
2 2 2n n n n n n n
b a b a b a x x x x x x
.
Do khi tnh ton (trn my tnh b ti vi mn hnh hin th c 10 ch s
chng hn), ta c th dng tnh ton khi 1 1 .... n n n x x x ng n s thp
phn cn thit (th d, ta c th dng tnh ton khi c nghim chnh xc n 10
ch s, tc l 1010 ).
2. Phng php lp
Gi s ( , )a b l khong cch li nghim ca phng trnh ( ) 0f x . Gii
phng trnh ( ) 0f x bng phng php lp gm cc bc sau:
Bc 1. a phng trnh ( ) 0f x v phng trnh tng ng ( ) x g x .
Bc 2. Chn 0 ( , ) x a b lm nghim gn ng u tin.
Bc 3. Thay 0x x vo v phi ca phng trnh ( ) x g x ta c nghim gn
ng th nht 1 0( ) x g x . Li thay 1 0( ) x g x vo v phi ca phng trnh
( ) x g x ta c nghim gn ng th hai 2 1( ) x g x . Lp li qu trnh trn, ta
nhn c dy cc nghim gn ng
1 0( ) x g x , 2 1( ) x g x , 3 2( ) x g x , 4 3( ) x g x ,..., 1( )n n x g x , ...
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Nu dy cc nghim gn ng nx , 1,2,...n hi t, ngha l tn ti
lim
nn
x x th (vi gi thit hm ( )g x l lin tc trn on ,a b ) ta c:
1 1lim lim ( ) (lim ) ( )
n n nn n n
x x g x g x g x .
Chng t x l nghim ng ca phng trnh ( ) x g x (im bt ng ca nh
x g) hay x l nghim ng ca phng trnh ( ) 0f x .
Tnh hi t
C nhiu phng trnh dng ( ) x g x tng ng vi phng trnh
( ) 0
f x . Phi chn hm s ( )g x sao cho dy nx xy dng theo phng phplp l dy hi t v hi t nhanh ti nghim. Ta c tiu chun sau.
nh l 4. Gi s x l nghim ca phng trnh ( ) 0f x v phng trnh
( ) x g x tng ng vi phng trnh ( ) 0f x trn on ,a b . Nu ( )g x v
'( )g x l nhng hm s lin tc trn ,a b sao cho ( ) 1 , g x q x a b th
t mi v tr ban u 0 ( , ) x a b dy nx xy dng theo phng php lp
1( )n n x g x s hi t ti nghim duy nht x trong khong ( , )a b ca phng
trnh ( ) 0f x .
Chng minh.
Gi s 0 ( , ) x a b bt k. V x l nghim ca phng trnh ( ) 0f x trong
khong ( , )a b nn ta c ( ) x g x . Mt khc v 1 0( ) x g x nn
1 0( ) ( ) x x g x g x .
Theo nh l Lagrange tn ti mt im 0,c x x sao cho
1 0 0( ) ( ) '( )( ) x x g x g x g c x x .
Suy ra
1 0 0 0'( )( ) x x g c x x q x x x x .
Chng t 1 ( , ) x a b .
Tng t ta c:
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2 1 x x q x x ; 3 2 x x q x x ;...; 1 n nx x q x x ;...
T cc bt ng thc trn ta suy ra nu 0 ( , ) x a b th ( , )n x a b vi mi n v
21 2 0...
nn n nx x q x x q x x q x x .
Do 1q nn khi n v phi tin ti 0 . Chng t dy nx hi t ti x .
nh gi sai s
nh gi sai s ca nghim gn ng nx (nhn c bng phng php
lp) v nghim chnh xc x ca phng trnh ( ) 0f x ti bc th n ta xt hiu
nx x .
T chng minh trn ta c:
1 1 1 n n n n n n n nx x q x x q x x x x q x x q x x
Vy
1(1 ) n n nq x x q x x
hay
11
n n n
q x x x x
q
Mt khc, p dng cng thc s gia hu hn (cng thc Lagrange) ta c:
1 1 2 1 2( ) ( ) '( )( ) n n n n n n nx x g x g x g c x x
trong
1 2( , ) n n nc x x
Suy ra
1 1 2 1 2'( ) n n n n n n nx x g c x x q x x
T bt ng thc trn, cho n=2,3,4,... ta c:
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2 1 1 0 x x q x x
23 2 2 1 1 0 x x q x x q x x
...
11 1 0
nn n x x q x x .
Thay vo bt ng thc
11
n n n
q x x x x
q
ta c:
1
1 1 0 1 01 1 1
nn
n n n
q q q x x x x q x x x x
q q q
Cng thc trn cho thy phng php lp hi t cng nhanh nu q cng b.
T cng thc trn ta cng suy ra rng, t c xp x (nghim gn ng
sai khc nghim ng khng qu , nx x ), ta phi lm ( )N bc, trong
1 0
(1 )lg
( )lg
q
x xN
q
.
T cng thc 11 1 0
n
n n x x q x x ta c kt lun: nu dy nx hi t
th khi n ln hai nghim gn ng nx v 1nx xp x bng nhau. V vy khi sdng my tnh ta thng dng qu trnh lp khi cc kt qu lin tip 1nx , nx ,
1nx ,... t xp x yu cu (trng nhau ti s ch s thp phn sau du phy cn
thit).
Nhn xt. V ta coi ( , )a b l khong cch li nghim (cha nghim x ) ca
phng trnh ( ) 0f x nn trong nh l 4 ta gi thit s tn ti nghim x . Hn
na, ta i hi ( )g x phi l mt hm kh vi. Di y l mt phin bn ca
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nh l 4 (khng i hi trc tn ti nghim ca phng trnh ( ) 0f x v ch i
hi ( )g x l mt hm lin tc Lipschitz).
nh l 5. Gi s ( )g x l hm s xc nh trn khong ;a b sao cho:
i) ( ) ( ) , ; g x g y q x y x y a b ( ( )g x l Lipschitz trn ;a b ).
ii) Tn ti mt s ; a b sao cho ( ) (1 )( ) g q b a .
Khi y vi mi 0 ; x a b , dy nx xy dng theo phng php lp 1( )n n x g x
s hi t ti im bt ng (tc l ( ) x g x ) duy nht x trong khong ( , )a b ca
nh x g.
3. Phng php dy cung
Gi s ( , )a b l khong cch li nghim. Ta thay cung ca ng cong
( ) y f x trn on [ , ]a b bng dy trng cung y v coi giao im ca dy cung
(ng thng) vi trc honh l nghim xp x ca phng trnh ( ) 0f x .
xy dng dy xp x nx , ta xt hai trng hp:
Trng hp 1. '( ). ''( ) 0 f x f x .
xc nh, ta coi ( ) 0, ( ) 0, f a f b v
'( ) 0, ''( ) 0 f x f x (Hnh 1).
Dy cung AB l ng thng ni hai im Hnh 1
( , ( ))A a f a v ( , ( )) B b f b c phng trnh( )
( ) ( )
y f a x a
f b f a b a.
Honh giao im 1x ca ng thng AB vi trc honh chnh l nghim ca
phng trnh trn khi cho 0y .
Suy ra 1( )
( ) ( )
f a x a
f b f a b ahay 1
( )( )
( ) ( )
f a b ax a
f b f a.
Nghim x by gi nm trong khong 1( , )x b (xem Hnh 1).
a
b
f(a)
f(b)
x1x
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Thay khong ( , )a b bng khong 1( , )x b , ta i n nghim1 1
2 1
1
( )( )
( ) ( )
f x b xx x
f b f x.
Tip tc qu trnh trn, ta i n dy nghim xp x:
1
( )( )
( ) ( )
n n
n n
n
f x b xx x
f b f x.
Cng thc trn vn ng trong trng hp
( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x .
Trng hp 2. '( ). ''( ) 0 f x f x .
xc nh, coi
( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x (Hnh 2). Hnh 2
Dy cung AB l ng thng ni hai im ( , ( ))A a f a v
( , ( )) B b f b c phng trnh( )
( ) ( )
y f b x b
f b f a b a.
Honh giao im 1x ca ng thng AB vi trc honh chnh l nghim caphng trnh trn khi cho 0y .
Suy ra 1( )
( ) ( )
f b x b
f b f a b ahay 1
( )( )
( ) ( )
f b b ax b
f b f a.
Nghim x by gi nm trong khong 1( , )a x .
Thay ( , )a b bng khong 1( , )a x , ta i n nghim xp x
1 12 1
1
( )( )
( ) ( )
f x x ax x
f x f a.
Tip tc qu trnh trn, ta i n dy nghim xp x
1
( )( )
( ) ( )
n n
n n
n
f x x ax x
f x f a.
Cng thc trn vn ng trong trng hp
( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x .
a
(b)
X1 b
f(a)
x
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Ta c th tng kt thnh mt cng thc nh sau:
1
( )( )
( ) ( )
n n
n n
n
f x x d x x
f x f d ,
trong d b nu ( ). ''( ) 0f b f x , 0 x a ;
d a nu ( ). ''( ) 0 f a f x , 0 x b .
Tnh hi t
Dy xp x lin tip l mt dy tng, b chn trn (trng hp 1):
0 1 1... ... n na x x x x x b
hoc l dy gim , b chn di (trng hp 2):
0 1 1... .... n nb x x x x x a
nn hi t. Hn na, chuyn qua gii hn trong cng thc
1
( )( )
( ) ( )
n n
n n
n
f x x d x x
f x f d ta c
( )( )
( ) ( )
f x x d x x
f x f d .
Suy ra ( ) 0f x hay x l nghim ca phng trnh ( ) 0f x trong khong
( , )a b .
nh gi sai s
Gi s '( )f x khng i du trn ( , )a b v 0 '( ) m f x M vi mi
( , ) x a b .
Ta c cc cng thc nh gi sai s sau y:
( ) nn
f xx x
m
; 1
n n nM m
x x x x
m
.
Chng minh.
p dng nh l gi tr trung bnh Lagrange (cng thc s gia hu hn), ta c
( ) ( ) '( )( ) n nf x f x f c x x
vi ( , ) ( , ) nc x x a b .
V ( ) 0f x v 0 '( ) m f x nn
( ) ( ) '( )( ) n n nf x f x f c x x m x x .
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Suy ra( )
nnf x
x xm
.
Nh vy, nh gi chnh xc ca nghim nhn c bng phng php dy
cung, ta c th s dng cng thc max ( ) , [ , ](
nnf x x a bf x
x xm m
.
Ngoi ra, nu bit hai gi tr gn ng lin tip, ta c th nh gi sai s nh sau.
T trn (chng minh s hi t ca phng php dy cung) ta c:
1 11
1
( )( )
( ) ( )
n n
n n
n
f x x d x x
f x f d .
Suy ra 11 11
( ) ( )( ) ( )
n
n n n
n
f x f d f x x x
x d.
V x l nghim ng ca phng trnh ( ) 0f x nn ta c th vit:
11 1
1
( ) ( )( ) ( ) ( )
n
n n n
n
f x f d f x f x x x
x d.
p dng nh l gi tr trung bnh Lagrange, ta c
1 1 1'( )( ) ( ) ( ) n nf c x x f x f x
v
2 1 1'( )( ) ( ) ( ) n nf c x d f x f d ,
trong 1c nm gia x v 1nx , 2c nm gia 1nx v d.
Suy ra
11 1 1 1
1
2 11 2 1
1
( ) ( )'( )( ) ( ) ( ) ( )
'( )( )( ) '( )( ).
( )
nn n n n
n
nn n n n
n
f x f d f c x x f x f x x x
x d f c x d
x x f c x xx d
Vy
1 1 2 1'( )( ) '( )( ) n n n n nf c x x x x f c x x
hay
1 2 1 1
'( )( ) [ '( ) '( )]( ) n n n
f c x x f c f c x x
v
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2 11
1
'( ) '( )
'( )
n n n
f c f c x x x x
f c.
Theo gi thit ta c:
2 1'( ) '( ) f c f c M m .
Do
1
n n n
M m x x x x
m.
Nh vy, ta c hai cng thc nh gi sai s
max ( ) , [ , ]( ) nn
f x x a bf xx x
m m
v 1
n n nM m
x x x x
m
.
4. Phng php tip tuyn (Phng php Newton-Raphson)
Gi s ( , )a b l khong cch li nghim. Ta thay cung ca ng cong
( ) y f x trn on [ , ]a b bng tip tuyn ti im ( , ( ))A a f a hoc im
( , ( )) B b f b v coi giao im ca tip tuyn vi trc honh l nghim xp x ca
phng trnh ( ) 0f x
. xy dng dy xp x
nx , ta xt hai trng hp:
Trng hp 1. '( ). ''( ) 0 f x f x .
xc nh, ta coi
( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x (Hnh 3). Hnh 3
Phng trnh tip tuyn vi ng cong ( ) y f x ti im ( , ( ))B b f b c dng:
( ) '( )( )y f b f b x b . Honh giao im 1x ca tip tuyn vi trc honhchnh l nghim ca phng trnh trn khi cho 0y .
Suy ra 0 ( ) '( )( )f b f b x b hay 1( )
'( )
f bx b
f b . Nghim x by gi nm trong
khong 1( , )a x (xem Hnh 3). Thay khong ( , )a b bng khong 1( , )a x , ta i n
nghim xp x 12 1( )
'( )
f xx x
f b
.
Tip tc qu trnh trn, ta i n dy
(a)
a1b
f(b)
a
f(b)
x x1 b
f(a)
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nghim xp x: 1( )
'( )
nn n
n
f xx x
f x .
Cng thc trn vn ng trong trng hp Hnh 4
( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x (Hnh 4).
Trng hp 2. '( ). ''( ) 0 f x f x . xc nh, coi
( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x (Hnh 5).
Phng trnh tip tuyn ti im ( , ( ))A a f a c dng:
( ) '( )( )y f a f a x a .
Honh giao im 1x ca tip tuyn
vi trc honh chnh l nghim ca
phng trnh trn khi cho 0y . Hnh 5
Suy ra 0 ( ) '( )( )f a f a x a hay 1( )
'( )
f ax a
f a .
Nghim x by gi nm trong khong 1( , )x b (xem hnh 5).
Thay khong ( , )a b bng khong1
( , )x b , ta i n nghim 12 1
1
( )
'( )
f xx x
f x .
Tip tc qu trnh trn, ta i n dy nghim xp x:
1
( )
'( )
nn n
n
f xx x
f x .
Cng thc trn vn ng trong trng hp
( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x .
Tnh hi tDy cc xp x lin tip l mt dy n iu gim v b chn di (trng
hp 1) hoc n iu tng v b chn trn (trng hp 2) nn tn ti gii hn
lim
nn
x x . D thy rng x l nghim ca phng trnh ( ) 0f x .
f(a)
X1ax b
f(b)
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Tht vy, chuyn qua gii hn trong biu thc 1( )
'( )
nn n
n
f xx x
f x ta c:
( )
'( )
f xx x
f x
. Suy ra ( ) 0f x . Do ( , )a b l khongcch li nghim nn x x
chnh l nghim ban u.
nh gi sai s
Gi s 10 '( )m f x v 1''( ) f x M .
Khi y ta c nh gi sai s:
1 1
max ( ) , [ , ]( )nn
f x x a bf x
x x m m
v
21
112
n n n
M
x x x xm .
Chng minh. p dng nh l gi tr trung bnh Lagrange (cng thc s gia hu
hn), ta c
( ) ( ) '( )( ) n nf x f x f c x x
vi ( , ) ( , ) nc x x a b .
V ( ) 0f x v 0 '( )m f x nn
( ) ( ) '( )( ) n n nf x f x f c x x m x x .
Suy ra(
nnf x
x xm
.
Dng khai trin Taylor ca ( )f x ti 1nx :
21 1 1 1
1( ) ( ) '( )( ) ''( )( )
2n n n n n n nf x f x f x x x f c x x ,
trong c nm gia nx v 1nx .
Do 111
( )
'( )
nn n
n
f xx x
f x
nn 1 1 1( ) '( )( ) 0n n n n f x f x x x .
Thay vo ng thc trn ta c:
2
1
1( ) ''( )( )
2n n n f x f c x x .
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T cng thc trn v cng thc( )
nnf x
x xm
ta suy ra
21
(( )
2
nn n n
f x M x x x x
m m
.
Nh vy, tc hi t ca phng php tip tuyn l bc hai.
3. TM NGHIM GN NG CA PHNG TRNH
( ) 0f x TRN MY TNH IN T
Nh ta thy trong 2, c bn phng php gii gn ng phng trnh
( ) 0f x trong khong cch li nghim ,a b u dn ti thc hin mt dy lp
1 ( ) n nx x v dy lp ny hi t ti nghim (duy nht ) x ca phng trnh trong
khong ,a b .
Do c thit k sn cho php thc hin cc thao tc tnh ton lin tip,
vic thc hin dy lp trn cc my tnh in t khoa hc (Casio fx-500 MS, Casio
fx-570 MS, Sharp 506 WM, Casio fx-500 ES, Casio fx-570 ES) l kh n gin v
thun tin. Trong ny, chng ti trnh by cch s dng cc loi my ny cho
mc ch gii gn ng phng trnh theo cc phng php trnh by mc trn.
Thc hnh gii gn ng phng trnh trn my tnh in t khoa hc cho php cm
nhn r hn cc vn ca gii tch s (s hi t, tc hi t,) ca tng phng
php.
tin trnh by, chng ti chn my tnh in t khoa hc Casio 570 ES, l
loi my c nhiu u im trong cc tnh nng gii ton v c s dng tng iph bin hin nay trong cc trng ph thng v i hc.
My tnh in t khoa hc Casio 570 ESc mt s phm rt tin dng trong
tnh ton. N c thit k c th tnh ton i s (tnh ton theo cng thc) kt
hp vi nhng nh (9 nh). Thit k ny c bit thch hp cho thc hin dy
lp, do c bit thun tin cho vic gii gn ng phng trnh.
Bi 1. Gii phng trnh i s bc cao 9 10 0 x x .
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y tuy ch l mt phng trnh a thc, tuy nhin bc ca n kh cao nn
kh c th gii c bng cc k thut ca i s (t n ph, nhm s hng,)
a v phng trnh bc thp hn.
t 9( ) 10 y f x x x .
Do 8' 9 1 0 y x vi mi x nn hm s ng bin trn ton trc s. Ta d dng
tnh c (1) 8 f v (2) 504 0 f . Do , phng trnh 9 10 0 x x c
duy nht mt nghim trong khong (1;2) . so snh, ta s tm nghim gn ng
ca phng trnh ny theo c bn phng php trnh by trong 2.
Phng php chia i khong cha nghim
a gi tr 1x vo nh A : 1 SHIFT STO A
a gi tr 2x vo nh B : 2 SHIFT STO B
Ch : T nay v sau, cho tin, cc phm s c vit nh l cc s, cn cc
phm ch trn mn hnh c trong cc vung. Th du, phm s 2 ta vn vit l
s 2, cn phm A ta vit l A .
Tnh1 2 3
2 2 2
a bc v a vo nh C :
ALPHA A ALPHA B 2 SHIFT STO C
Khai bo cng thc 9( ) 10 f x x x :
ALPHA X ^x 9 ALPHA X 10 (3.1)
tnh gi tr ca biu thc ti mt im no ta bm phm CALC (Calculate-tnh): CALC
My hi (hin trn mn hnh): ?X
Khai bo3
2c (ang trong C ): ALPHA C ( 15331
512)
Ch : T nay v sau, tin trong trnh by, ta ghi ngay p s ca kt qu tnh
ton hin th trn mn hnh sau phm v trong ngoc. Th d, sau khi khai
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bo cng thc (3.1) v bm phm CALC , my hi ?X , ta khai bo ALPHA C
(tc l gi tr ca i s x cn tm gi tr ca hm s 9( ) 10 f x x x chnh
bng gi tr trong C ). Sau khi bm phm , my hin ngay trn mn hnh p
s15331
512, tc l
3 15331( )
2 512f v ta vit (
15331
512).
Nh vy, ta c (1) 8 0 f v3 15331
( ) 02 512
f . Chng t nghim ca phng
trnh nm trong khong 1 13
( , ) ( , ) (1; )2
a b a c . B c gii phng (gi tr c
2x trong nh B khng cn dng na).
Ch : V ti mi bc i ta ch cn nh ba gi tr ,i ia b hoc2
i ii
a bc nn ta
cng ch cn s dng ba nh A , B , C l .
Li chia i khong 1 13
( , ) ( , ) (1; )2
a b a c v gi 1 115
1,252 4
a bc vo
B (va c gii phng):
ALPHA A ALPHA C 2 SHIFT STO B
S dng phm i v dng cng thc (3.1) v dng phm CALC tnh gi
tr ca hm s ti 1 1,25c :
Bm phm CALC (Calculate-tnh): CALC
My hi: ?X
Khai bo 1 115
1,252 4
a bc ( c tnh v lu trong B ):
ALPHA B ( 1.299419403 )
Nh vy, ta c 15
( ) ( ) 1, 2994194034
f c f v 13 15331
( ) ( ) 02 512
f b f .
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Ch : tin trnh by, ta thng vit du bng thay cho du gn ng , th
d5
( ) 1,2994194034
f , mc d chnh xc hn, phi vit l:
5( ) 1,2994194034
f (chnh xc n 10 ch s).
Chng t nghim ca phng trnh nm trong khong
2 2 1 1
5 3( , ) ( , ) ( ; )
4 2 a b c b . nh A c gii phng (gi tr c 1a trong nh
A khng cn dng na).
Li chia i khong 2 2 1 1 5 3( , ) ( , ) ( ; )4 2 a b c b , tnh v gi
2 22
5 3114 2 1,375
2 2 8
a bc vo A :
ALPHA B ALPHA C 2 SHIFT STO A
S dng phm i v dng cng thc (3.1) v dng phm CALC tnh gi
tr ca hm s ti 211
1,3758
c :
Bm phm CALC (Calculate-tnh): CALC
My hi: ?X
Khai bo 211
1,3758
c ( c tnh v lu trong A ):
ALPHA A (8.943079315)
Nh vy, ta c 211
( ) ( ) 8,943079315 08
f c f v
2 1
5( ) ( ) ( ) 1, 299419403 0
4 f a f c f . Chng t nghim ca phng trnh
nm trong khong 3 3 2 25 11
( , ) ( , ) ( ; )4 8
a b a c . nh B c gii phng (gi tr
c 2 32
b trong nh B khng cn dng na).
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Li chia i khong 3 3 2 25 11
( , ) ( , ) ( ; )4 8
a b a c v gi
3 33
5 1121
4 8 1,31252 2 16
a bc vo B :
ALPHA A ALPHA C 2 SHIFT STO B
S dng phm i v dng cng thc (3.1) v dng phm CALC tnh gi
tr ca hm s ti 321
1,312516
c :
Bm phm CALC (Calculate-tnh): CALC
My hi: ?X
Khai bo 321
1,312516
c ( c tnh v lu trong B ):
ALPHA B (2.870795905)
Nh vy, ta c 321
( ) ( ) 2,870795905 016
f c f v
2 15( ) ( ) ( ) 1, 299419403 04
f a f c f . Chng t nghim ca phng trnh
nm trong khong 4 4 1 35 21
( , ) ( , ) ( ; )4 16
a b c c . nh C c gii phng (gi tr
c11
8c trong nh C khng cndng na).
Li chia i khong4 4 1 3
5 21( , ) ( , ) ( ; )
4 16 a b c c v gi
4 44
5 21414 16 1,28125
2 2 32
a b
c vo C :
ALPHA A ALPHA B 2 SHIFT STO C
S dng phm i v dng cng thc (3.1) v dng phm CALC tnh gi
tr ca hm s ti 4 41 1,2812532
c :
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Bm phm CALC (Calculate-tnh): CALC
My hi: ?X
Khai bo 441
1,28125
32
c ( c tnh v lu trong C ):
ALPHA C (0.5860042121)
Nh vy, ta c 441
( ) ( ) 0,586004212 032
f c f v
2 1
5( ) ( ) ( ) 1, 299419403 0
4 f a f c f . Chng t nghim ca phng trnh
nm trong khong 5 5 1 4 5 41( , ) ( , ) ( ; )4 32 a b c c . nh C c gii phng (gi tr
c3
21
16c trong nh C khng cn dng na).
Tip tc qu trnh ny, ta i n bng sau, trong mi dng cho ta cc gi tr nc ,
( )nf c v gi tr ( )nf a hoc ( )nf b tnh c dng trn, tri du vi ( )nf c .
V cc gi tr ( )n
f a hoc ( )n
f b c tnh n 10 ch s dng trn, v ch
cn du ca chng, nn tit kim ch, ta ch vit gi tr khc 0 u tin ca n
sau du phy. Mt khc, nhn vo mi dng, ta cng bit no (khng c) c gii
phng, nhng ch a gi tr mi ca nc vo.
nc trong ( )nf c ( )nf a ( )nf b Khong
a=1; b=2 -8 A 504 B ( ; )a b
1,5c
C 29,94335937 -8 A ( ; )a c
1 1, 25c B -1,299419403 29,9 C 1( ; )c c
2 1,375c A 8,943079315 -1,2 B 1 2( ; )c c
3 1,3125c C 2,870795905 -1,2 B 1 3( ; )c c
4 1, 28125c A 0,5860042121 -1,2 B 1 4( ; )c c
5 1, 265625c C -0,402449001 0,5 A 5 4( ; )c c
6 1, 2734375c
B 0,079843683 -0,4 C 5 6( ; )c c
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7 1, 26953125c A 0,164222375 0,07 B 7 6( ; )c c
8 1, 271484375c C 0,042927154 0,07 B 8 6( ; )c c
9 1, 272460938c A 0,018272819 0,04 C 8 9( ; )c c
10 1, 271972656c B 0,012373404 0,01 A 10 9( ; )c c
11 1, 272216797c C 0,0029381332 0,01 B 10 11( ; )c c
12 1, 272094127c A 0,00472052697 0,002 C 12 11( ; )c c
13 1, 272155762c B 0,00089191976 0,002 C 13 11( ; )c c
14 1, 272186279c A 0,0010229262 0,0008 B 13 14( ; )c c
15 1, 272171021c C 0,0000654581 0,0008 B 13 15( ; )c c
16 1, 272163391c A 0,00041324182 0,00006 C 16 15( ; )c c
17 1, 272167206c B 0,00017389435 0,00006 C 17 15( ; )c c
18 1, 272169113c A 0,00005421851 0,00006 C 18 15( ; )c c
19 1, 272170067c B 0,0000056196 0,00005 A 18 19( ; )c c
20 1, 27216959c C 0,00000024847 0,000005 B 20 19( ; )c c
21 1, 272169828c A 0,00000933952 0,000005 B 21 19( ; )c c
22 1, 272169948c C 0,00000185969 0,000005 B 22 19( ; )c c
23 1, 272170007c A 0,00000188 0,000001 C 22 23( ; )c c
24 1, 272169977c B 0,0000000101 0,000001 C 22 24( ; )c c
25 1, 272169963c A 0,00000092473 0,00000001 B 25 24( ; )c c
26 1, 27216997c C 0, 0000004573 0,00000001 B 26 24( ; )c c
27 1, 272169974c A 0,00000022325 0,00000001 B 27 24( ; )c c
28 1, 272169976c C 0,00000010655 0,00000001 B 28 24( ; )c c
29 1, 272169977c A 0,00000004821 0,00000001 B 29 25( ; )c c
Nghim gn ng l 29 1,272169977 x c v ( ) 0,00000004821 f x Li
bnh: Ta thy phng php chia i tuy n gin nhng ni chung hi t chm,
thao tc trn my kh phctp.
Phng php lp
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Phng trnh 9 10 0 x x tng ng vi 9 10 ( ) x x x .
V
89
1( )
10
xx
nn9 98 8
1 1( ) 1
9 8 x vi mi 1;2x . Do dy
lp 91 10 n nx x hi t ti nghim 1;2x .
Chn gi tr ban u 0 1,5x : 1.5
Ch : My c thit k mt cch thng minh l gi tr 0 1,5x va c khai
bo, hoc cc kt qu tnh ton hin trn mn hnh, cng s c chuyn ngay vo
Ans (kt qu), s dng vo cc tnh ton tip theo cho tin.
Khai bo 0( )x : ( 10 Ans )^x 1 9
Tnh gi tr ca 0( )x : Bm phm
Tnh cc gi tr tip theo: Bm phm
Sau 6 ln bm phm, ta i n dy nghim xp x:
1,5; 1,268436614; 1,27223043; 1,272168998; 1,272169993; 1,272169977;
1,272169977;
Kt qu nghim gn ng l: 1,272169977 (chnh xc n 10 ch s thp phn).
Phng php dy cung
Xt 9( ) 10 f x x x . Ta c 8'( ) 9 1 0 f x x v 7( ) 72 0 f x x
vi mi x trong khong (1;2) . Theo cng thc dy cung
1
( )( )
( ) ( )
n n
n n
n
f x x d x x
f x f d vi 2d ta c dy lp
9
1 9
( 10)( 2)
10 504
n
n
n n
n n
n
x x xx x
x x, 0 1x .
Khai bo gi tr u 0 1x : 1
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Khai bo cng thc9
1 9
( 10)( 2)
10 504
n
n
n n
n n
n
x x xx x
x x:
Ans ( Ans ^x 9 Ans 10 ) ( Ans 2 ) ( Ans ^x 9 Ans
514 )
Tnh cc gi tr 1nx bng cch bm lin tip phm . Ta c dy cc gi tr:
65
64; 1,03069279;
Phi sau khong 300 ln bm phm ta mi ra c kt qu 1,272169976.
Nh vy, phng php dy cung trong bi ny hi t rt chm.
Phng php tip tuyn
Theo cng thc tip tuyn 1( )
'( )
nn n
n
f xx x
f x, 0 2x ta c
9
1 8
10
9 1
n
n
n
n n
x xx x
x
.
Khai bo gi tr ban u: 2
Khai bo cng thc9
1 8
10
9 1
n
n
n
n n
x xx x
x:
Ans ( Ans ^x 9 Ans 10 ) ( 9 Ans ^x 8 1 )
Tnh cc gi tr 1nx bng cch bm lin tip phm . Ta c dy cc gi tr xp
x:
1,781344902; 1,592631378; 1,438653785; 1,331291548; 1,281547657;
1,272435766; 1,272170196; 1,27216977; 1,27216977.
Sau chn ln lp ta i n p s.
Kt lun: C bn phng php (chia i on cha nghim, lp, dy cung v tip
tuyn) u cho nghim gn ng ti 10 ch s thp phn. tm c nghim gn
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ng n 10 ch s thp phn theo phng php chia i cn 30 bc lp, v thao
tc phc tp trong mi bc. tm nghim gn ng n 10 ch s thp phn theo
phng php lp ch cn 6 bc. tm nghim gn ng n 10 ch s thp phn
theo phng php dy cung phi cn n 300 bc lp. tm nghim gn ngn 10 ch s thp phn theo phng php tip tuyn ch cn 9 bc lp. Thao tc
lp theo ba phng php sau rt n gin: sau khikhai bo gi tr ban u v cng
thc lp, ta ch cn lin tip bm phm cho n khi c gi tr khng i (im
bt ng), l nghim gn ng n 10 ch s thp phn.
Bi 2. Gii phng trnh cha cn thc 83 2 5 0 x x .
t 8( ) 3 2 5 y f x x x . Khng cn my ta cng c th d dng tnh
c (0) 5 f v 8 8(5) 3.5 2. 5 5 10 2 5 0 f . Do , phng trnh
83 2 5 0x x c t nht mt nghim trong khong (0;5) . xem dng iu
hnh hc ca th, ta c th nhMaple v th trong khong (0;5) nh lnh
plot (v th):
[> plot(3*x-2*x^(1/8)-5,x=0..5);
Nhn vo th hm s 8( ) 3 2 5 y f x x x ta thy, th ct trc honh ti
mt im trong khong (2;3) . Tuy nhin, chnh xc hn ta cn tnh gi tr ca
hm s ti cc im 2x v 3x .
Ta c th tnh cc gi tr ca hm s cho ti im 2x
nh lnh eval (evaluate -
tnh gi tr ca hm s ti 2x ) v lnh evalf(%) (tnh gi tr ca biu thc trn):
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[> eval(3*x-2*x^(1/8)-5,x=2);
1 2 2( )/1 8
[> evalf(%);
-1.181015466
Cng c th kt hp hai lnh trn vo mt lnh:
> evalf(eval(3*x-2*x^(1/8)-5,x=2));
-1.181015466
Tng t, ta c th tnh cc gi tr ca hm s cho ti im 3x nh lnh eval
v lnh evalf(%):[> eval(3*x-2*x^(1/8)-5,x=3);
4 2 3( )/1 8
[> evalf(%);
1.705594620
Tuy nhin, ta cn c th dng lnh subs (thay th) thay 3x vo biu thc
8( ) 3 2 5 f x x x c gi tr (3)f :
[> subs(x=3,3*x-2*x^(1/8)-5);
4 2 3( )/1 8
[> evalf(%);
1.705594620
C th kt hp hai lnh trn vo mt lnh: [> evalf(subs(x=3,3*x-2*x^(1/8)-5));
1.705594620
Tt nhin, ta cng c th dng my tnh khoa hc tnh cc gi tr trn. Th d, s
dng Casio fx-570 ES tnh gi tr ca hm s 83 2 5 x x ti 2x nh sau:
Khai bo hm s 83 2 5 x x :
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3 ALPHA X 2 ALPHA X x 1 8 5 (3.2)
Tnh gi tr hm s:
Bm phm CALC (Calculate-tnh)My hi: ?X
Khai bo 2x : 2
My tr li (p s hin trn mn hnh), mn hnh c dng:
1
83 2 5
1,181015465
X X .
tnh gi tr ca hm s 83 2 5 x x ti 3x ta khng cn khai bo li cng
thc hm s m ch cn s dng phm CALC :
Bm phm CALC (Calculate-tnh)
My hi: ?X
Khai bo 3x : 3
My tr li (p s hin trn mn hnh):1
83 2 5
1, 705594619
X X .
Vy phng trnh 83 2 5 0 x x c mt nghim duy nht trong khong
( ; ) (2;3)a b .
Phng php chia i khong cha nghim
a gi tr 2x vo nh A : 2 SHIFT STO A a gi tr 3x vo nh B : 3 SHIFT STO B
Tnh2 3 5
2 2 2
a bc v a vo nh C :
ALPHA A ALPHA B 2 SHIFT STO C
S dng phm i v dng cng thc (3.2) v dng phm CALC tnh gi
tr ca hm s ti 52c :
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Bm phm CALC (Calculate-tnh)
My hi: ?X
Khai bo5
2
c (ang trong C ):
ALPHA C (0.2572932161)
Nh vy, ta c (2) 1,181015465 0 f v5
( ) 0.2572932161 02
f . Chng
t nghim ca phng trnh nm trong khong 1 1( ; ) ( , ) (2;3) a b a c . B c
gii phng (gi tr c 3x trong nh B khng cn dng na).
Li chia i khong 1 1( ; ) ( , ) (2;3) a b a c v gi1 1
1 9 2,252 4 a bc vo
B :
ALPHA A ALPHA C 2 SHIFT STO B
S dng phm i v dng cng thc (3.2) v dng phm CALC tnh gi
tr ca hm s ti 19
2,254
c :
Bm phm CALC (Calculate-tnh):
My hi: ?X
Khai bo 19
4c (ang trong B ): CALC
ALPHA B ( 0.4633638394 )
Nh vy, ta c1
9( ) ( ) 0,4633638394 0
4 f c f v
1
5( ) ( ) 0, 2572932161 0
2 f b f . Chng t nghim ca phng trnh nm trong
khong 2 2 1 19 5
( ; ) ( , ) ( ; )4 2
a b a c . nh A c gii phng (gi tr c 2a
trong nh A khng cn dng na).
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Li chia i khong 2 2 1 19 5
( ; ) ( , ) ( ; )4 2
a b a c v gi
2 22
9 5194 2
2,3752 2 8
a b
c vo A :
ALPHA B ALPHA C 2 SHIFT STO A
S dng phm i v dng cng thc (3.2) v dng phm CALC tnh gi
tr ca hm s ti 219
2,3758
c :
Bm phm CALC (Calculate-tnh): CALC
My hi: ?X
Khai bo 219
2,3758
c (ang trong A ):
ALPHA A ( 0.103373306 )
Nh vy, ta c 219
( ) ( ) (2,375) 0.103373306 08
f c f f v
1
5( ) ( ) 0.2572932161 02 f b f . Chng t nghim ca phng trnh nm trong
khong 3 3 2 15
( ; ) ( ; ) (2,375; )2
a b c b . nh B c gii phng (gi tr c 19
4c
trong nh B khng cn dng na).
Li chia i khong 3 3 2 15
( ; ) ( ; ) (2,375; )2
a b c b v gi
2 23
19 5 398 2 2,43752 2 16
c bc vo B :
ALPHA A ALPHA C 2 SHIFT STO B
S dng phm i v dng cng thc (3.2) v dng phm CALC tnh gi
tr ca hm s ti 339
2,437516
c :
Bm phm CALC (Calculate-tnh): CALC
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My hi: ?X
Khai bo 339
2,437516
c (ang trong B ):
ALPHA B ( 0,076879549)
Nh vy, ta c 339
( ) ( ) 0, 076879549 016
f c f v
2
19( ) ( ) (2,375) 0.103373306 0
8 f c f f . Chng t nghim ca phng
trnh nm trong khong 4 4 2 319 39
( ; ) ( ; ) ( ; )8 16
a b c c . nh C c gii phng (gi
tr c 52
c trong nh C khng cn dng na).
Li chia i khong 4 4 2 319 39
( ; ) ( ; ) ( ; )8 16
a b c c v gi
4 44
19 39778 16 2,40625
2 2 32
a b
c vo C :
ALPHA A ALPHA B 2 SHIFT STO C
S dng phm i v dng cng thc (3.2) v dng phm CALC tnh gi
tr ca hm s ti 477
2,4062532
c :
Bm phm CALC (Calculate-tnh): CALC
My hi: ?X
Khai bo 477
2,4062532
c (ang trong C
):
ALPHA C ( 0,013267467 )
Nh vy, ta c 477
( ) ( ) 0,013267467 032
f c f v
2
19( ) ( ) (2,375) 0.103373306 0
8 f c f f . Chng t nghim ca phng
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trnh nm trong khong4 4 2 3
19 39( ; ) ( ; ) ( ; )
8 16a b c c . nh C c gii phng (gi
tr c5
2c trong nh C khng cn dng na).
Ta c bng kt qu sau:
nc trong ( )nf c ( )nf a ( )nf b Khong
a=2; b=3 -1,1 A 1,7 B ( ; )a b
2,5c C 0.2572932161 -1,1 A ( ; )a c
1 2,25c B -0.4633638394 0,2 C 1( ; )c c
2 2,2375c A -0,103373306 0,2 C 2( ; )c c
3 2,4375c B 0,076879549 -0,1 A 2 3( ; )c c
4 2,40625c C -0,013267467 0,07 B 3 4( ; )c c
5 2,421875c A 0,031800956 0,01 C 4 5( ; )c c
6 2,42140625c B 0,00926546534 0,01 C 4 6( ; )c c
7 2,41015625c A 0,0020013218 0,009 B 7 6( ; )c c
8 2,412109375c C 0,00363199172 0,002 A 7 8( ; )c c
9 2, 411132813c B 0,00081531494 0,002 A 7 9( ; )c c
10 2, 410644531c
C 0,00059300843 0,0008 B 10 9( ; )c c
11 2,410888672c
A 0,00011115202 0,0005 C 10 11( ; )c c
12 2,410766602c B 0,00024092851 0,0001 A 12 11( ; )c c
13 2,410827637c C 0,000006488831 0,0001 A 13 11( ; )c c
14 2,410858154c B 0,00002313185 0,000006 C 13 14( ; )c c
15 2,410842896c A 0,0002087824 0,00002 B 15 14( ; )c c
16 2,410850525c C 0,00000112682 0,0002 A 15 16( ; )c c
17 2, 41084671c B 0,0000098757 0,000001 C 17 16( ; )c c
18 2,410848618c A 0,00000437445 0,000001 C 18 17( ; )c c
19 2,410849571c B 0,00000162379 0,000001 C 19 17( ; )c c
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20 2,410850048c A 0,00000024847 0,000001 C 20 17( ; )c c
21 2,410850287c B 0,00000043919 0,0000002 A 20 21( ; )c c
22 2, 410850167c C 0,00000009536 0,0000002 A 20 22( ; )c c
23 2,410850108c B 0,00000007658 0,00000009 C 23 22( ; )c c
24 2,410850138c A 0,0000000094 0,00000007 B 23 24( ; )c c
25 2,410850123c C 0,00000003358 0,000000009 A 25 24( ; )c c
26 2,41085013c B 0,00000001209 0,000000009 A 26 24( ; )c c
27 2, 410850134c C 0,00000000132 0,000000009 A 27 24( ; )c c
28 2,410850136c B
0,00000000404 0,000000001 C 27 28( ; )c c
29 2,410850135c A 0,00000000135 0,000000001 C 27 29( ; )c c
30 2,4108501345c B 0,00000000000 0,00000000132 C 27 30( ; )c c
Nghim gn ng l30 2,4108501345 x c v ( ) 0,00000000000f x (chnh
xc n 10 ch s).
Phng php lp
Phng trnh 83 2 5 0 x x tng ng vi82 5
( )3
xx x .
Dy lp c dng8
1
2 5
3
nn
xx .
Chn gi tr ban u 0 2,5x : 2.5
Khai bo 0( )x : ( 2 Ans x 1 8 5 ) 3
Tnh gi tr ca 0( )x : Bm phm
Tnh cc gi tr tip theo: Bm phm
Ta c dy xp x:
2.5; 2.414235595; 2.410980682; 2.410855171; 2.410850329; 2.410850142;
2.410850135; 2.410850134; 2.410850134;
Sau 9 bc ta i n nghim gn ng n 10 ch s.
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Phng php dy cung
Theo cng thc dy cung 1( )( 3)
( ) (3)
n n
n n
n
f x xx x
f x f , 0 2x ta c
8
18
(3 2 5)( 3)
3 2 5 (3)
n n n
n n
n n
x x xx x
x x f , 0 2x .
Khai bo 0 2x : 2
Khai bo cng thc 8( ) 3 2 5 f x x x :
3 ALPHA X 2 ALPHA X x 1 8 5
Tnh gi tr ca (3)f : Bm phm CALC
My hi: X?
Khai bo: 3 (1.705594619)
Gi vo A : SHIFT STO A
Khai bo gi tr ban u: 2
Khai bo cng thc8
18
(3 2 5)( 3)
3 2 5 (3)
n n n
n n
n n
x x xx x x x f
:
Ans ( ( 3 Ans 2 Ans ^x 1 8 5 ) ( Ans 3 ) ( 3 Ans
2 Ans ^x 1 8 5 ALPHA A ) )
Tnh cc gi tr 1nx bng cch bm lin tip phm .Ta c dy cc gi tr:
2,409135779; 2,410843757; 2,410850111; 2,410850134; 2,410850134.
Phng php tip tuyn
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Theo cng thc lp trong phng php tip tuyn 1( )
'( )
nn n
n
f xx x
f x, 0 2x ta c
8
1
78
(3 2 5)
134
n n
n n
n
x xx x
x
.
Khai bo gi tr ban u 0 2x : 2
Khai bo cng thc8
1
78
(3 2 5)
13
4
n nn n
n
x xx x
x
:
Ans ( 3 Ans 2 Ans ^x 1 8 5 ) ( 3 1 ( 4 Ans ^x
7 8 ) )
Tnh cc gi tr 1nx bng cch bm lin tip phm . Ta c dy cc gi tr:
2,412410874; 2,410850152; 2,410850134; 2,410850134.
Sau ba ln lp ta in p s.
Kt lun: tm c nghim gn ng n 10 ch s thp phn theo phng
php chia i cn 30 bc lp, v thao tc phc tp trong mi bc. tm nghim
gn ng n 10 ch s thp phn theo phng php lp ch cn 9 bc. tm
nghim gn ng n 10 ch s thp phn theo phng php dy cung cn 4 bc
lp. tm nghim gn ng n 10 ch s thp phn theo phng php tip tuyn
ch cn 3 bc lp. Thao tc lp theo ba phng php sau rt n gin: sau khi khai
bo gi tr ban u v cng thc lp, ta ch cn lin tip bm phm cho n khic gi tr khng i (im bt ng), l nghim gn ng n 10 ch s thp
phn.
Bi 3. Gii phng trnh hn hp i s - lng gic 3 cos 0 x x .
t 3( ) cos y f x x x. Ta d dng tnh c (0) 1 f v
(1) 0,4596976941 0 f . Do , phng trnh 3 cos 0 x x c t nht mt
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nghim trong khong (0;1) . Nghim ca phng trnh 3 cos 0 x x c th c
coi nh l giao ca hai th hm s 3y x v cosy x . Ta c th nhMaple
v hai th trong khong (0;1) nh lnh plot (v th) trn cng mt h ta :
[> plot([x^3,cos(x)],x=0..1);
Nhn vo thta thy, nghim ca phng trnh nm trong khong (0,8;0,9).
s dng Casio fx-570 EStnh ton vi radian, trc tin ta phi vo MODE s
dng radian bng cch bm phm: SHIFT MODE 4 .
Phng php chia i
Khai bo hm 3( ) cos y f x x x: ALPHA X ^x 3 cos ALPHA X
Thc hin cc thao tc hon ton tng t nh trong Bi 1 v Bi 2, ta i n bng
kt qu di y.
nc trong ( )nf c ( )nf a ( )nf b Khong
a=0; b=1 -1 A 0,45969 B ( ; )a b
0,5c C -0,752582561 0,4 B ( ; )c b
10,75c A -0,3098138689 0,4 B 1( ; )c b
2 0,875c C 0,02892501684 -0,3 A 1 2( ; )c c
3 0,8125c B -0,1513086091 0,02 C 3 2( ; )c c
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4 0,84375c A -0,06398823736 0,02 C 4 2( ; )c c
5 0,859375c B -0.01824073474 0,02 C 5 2( ; )c c
6 0,8671875c A 0,05163610306 -0,01 B 6 2( ; )c c
7 0,86328125c C -0,006583038793 0,05 A 7 6( ; )c c
8 0,86523437c B -0,000720852907 0,05 A 8 6( ; )c c
9 0,866210937c C 0,002218591595 -0,0007 B 8 9( ; )c c
10 0,865722656c
A 0,000748172873 -0,0007 B 8 10( ; )c c
11 0,865478515c
C 0,000013485903 -0,0007 B 8 11( ; )c c
12 0,86535644c A -0,00035372701 0,00001 C 12 11( ; )c c
13 0,86541748c B -0,000170132846 0,00001 C 13 11( ; )c c
14 0,86544799c A -0,000078327132 0,00001 C 14 11( ; )c c
15 0,86546325c B -0,000032422234 0,00001 C 15 11( ; )c c
16 0,86547088c A -0,000009469276 0,00001 C 16 11( ; )c c
17 0,8654747c B 0,000002007338 -0,000009 A 16 17( ; )c c
18 0,86547279c C -0,00000373098 0,000002 B 18 17( ; )c c
19 0,86547374c A -0,000000861816
0,000002B
19 17( ; )c c
20 0,86547422c C 0,00000057276 -0,0000008 A 19 20( ; )c c
21 0,86547398c B -0,000000144521 0,0000005 C 20 21( ; )c c
22 0,86547410c A 0,000000214128 -0,0000001 B 21 22( ; )c c
23 0,86547404c C 0,000000034804 -0,0000001 B 22 23( ; )c c
24 0,86547401c A -0,000000054851 0,00000003 C 24 23( ; )c c
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25 0,86547402c B -0,000000010024 0,00000003 C 25 24( ; )c c
26 0,86547403c A 0,00000001239 -0,00000001 B 25 26( ; )c c
27 0,86547403c C 0,00000001183 -0,00000001 B 25 27( ; )c c
28 0,865474031c
A -0,000000004412 0,00000001 C 28 27( ; )c c
29 0,86547403c B -0,000000001615 0,00000001 C 29 27( ; )c c
30 0,86547403c A -0,000000000216 0,00000001 C 30 27( ; )c c
Nghim gn ng l30 0,865474033 x c radian v ( ) 0,000000000216 f x
Phng php lp
Phng trnh 3 cos 0 x x tng ng vi 3 cos ( ) x x x .
Dy lp c dng 31
cos
n n
x x .
Trc tin ta phi vo MODE s dng radian bng cch bm phm:
SHIFT MODE 4
Chn gi tr ban u 04
x
: SHIFT 4
Khai bo ( )nx : SHIFT3 cos Ans )
Tnh gi tr ca 0( )x : Bm phm
Tnh cc gi tr tip theo: Bm phm CALC
Ta c dy xp x:
0.8908987181; 0.8566779192; 0.8684331147; 0.8644689718; 0.8658143009;
0.8653587072; 0.8655131056; 0.8654607937; 0.865478519; 0.8654725131;
0.8654745481; 0.8654738586; 0.8654740922; 0.8654740131; 0.8654740399;
0.8654740308; 0.8654740339; 0.8654740328; 0.8654740332; 0.8654740331.
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Nh vy, sau 20 ln bm phm CALC , ta i n nghim xp x 0,8654740331x .
Phng php dy cung
Hm s3
( ) cos y f x x x c2
( ) 3 sin 0 y f x x x v
( ) 6 cos 0 y f x x x vi mi ;14
x
. Theo cng thc dy cung
1
( )( )
( ) ( )n n
n n
n
f x x bx x
f x f b
, 0
4x
ta c
3
1 3
( cos )( 1)
cos (1)
n n nn n
n n
x x xx x
x x f
, 0
4x
.
Khai bo 04
x
: SHIFT 4
Khai bo cng thc
3
1 3
( cos )( 1)
cos (1)n n n
n n
n n
x x xx x
x x f
:
Ans ( Ans SHIFT 3x cos Ans ) ) ( Ans 1 ) ( Ans SHIFT 3x
cos Ans ) ALPHA A )
Tnh cc gi tr 1nx bng cch bm lin tip phm . Ta c dy cc gi tr:
0.8554102741; 0.8642643557; 0.8653292731; 0.865456721; 0.8654719629;
0.86547376855; 0.8654740035; 0.8654740296; 0.8654740327; 0.8655740331;
0.8655740331; 0.8654740331.
Sau 10 ln bm phm , ta i n nghim xp x 0,8654740331x .
Phng php tip tuyn
Theo cng thc tip tuyn 1( )
'( )n
n n
n
f xx x
f x ta c
3
1 2
cos
3 sin
n nn n
n n
x xx x
x x
.
Ta chn gi tr ban u 04
x
.
Khai bo gi tr ban u: SHIFT 4
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Khai bo cng thc3
1 2
cos
3 sin
n nn n
n n
x xx x
x x
:
Ans ( Ans SHIFT 3x cos Ans ) ) ( 3 Ans 2x sin Ans )
Tnh cc gi tr 1nx bng cch bm lin tip phm . Ta c dy cc gi tr:
0.8724441024; 0.8655207565; 0.8654740352; 0.8654740331; 0.8654740331;
0.8654740331; 0.8654740331.
Sau bn ln bm phm ta i n p s 0,8654740331x .
Kt lun: C bn phng php u cho p s 0,8654740331x .
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CHNG II
GII GN NG NGHIM CA BI TON
CAUCHY CHO PHNG TRNH VI PHN THNG
TRN MY TNH IN T
1. PHNG PHP GII GN NG BI TON CAUCHY
CA PHNG TRNH VI PHN THNG
S lng phng trnh vi phn gii c bng cu phng (nghim c
biu din thng qua cc php tnh v cc hm c bn) l rt t. Thm ch khi c cng
thc nghim ca phng trnh vi phn, do s phc tp ca cng thc, cha chc ta
c th kho st c cc tnh cht ca nghim. V vy, nghin cu phng
trnh vi phn, ngi ta thng khng gii trc tip phng trnh, m s dng hai
phng php: phng php nh tnh - nghin cu cc tnh cht ca nghim (tn ti
v duy nht, tnh b chn, nghim tun hon, tnh n nh,...) thng qua v phi ca
phng trnh v phng php gii gn ng - tm nghim di dng xp x. Hai
phng php ny h tr v b sung nhau.
gii gn ng phng trnh vi phn, ngi ta thng dng hai phng
php: phng php gii tch - tm nghim xp x di dng mt dy cc hm s
lin tc hi t u ti nghim (l mt hm kh vi tha mn phng trnh vi phn v
iu kin u) trn mt khong ,a b no v phng php s - tm nghim xp
x di dng cc gi tr s ca nghim ti mt s im trn on ,a b v kt qu
c cho di dng bng, nh phng php ng gp khc Euler, phng php
Runge-Kutta, phng php a bc,...
Trong chng ny, chng ti trnh by phng php ng gp khc Euler
v phng php Runge-Kutta gii bi ton Cauchy cho phng trnh vi phn
thng. Nhm minh ha cho kh nng s dng my tnh in t gii phng
trnh vi phn, tr gip cho hc tpca sinh vin khi hc mn hc ny, chng ti th
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hin phng php Euler v phng php Runge-Kutta trn my tnh in t khoa
hc Casio fx-570 ESv trn chng trnhMaple.
1.1. Bi ton Cauchy ca phng trnh vi phn cp mt
Bi ton: Tm nghim ( ) y y x ca phng trnh
( , )y f x y (1.1)
tha mn iu kin ban u
0 0( ) y x y (1.2)
(tc l tm mt hm kh vi ( ) y y x sao cho 0 0( ) y x y v ( ) ( , ( ))y x f x y x
trong mt ln cn no ca 0x ).D dng chng minh c rng, phng trnh vi phn (1.1) tng ng vi
phng trnh tch phn
0
0( ) ( , ( )) x
x
y x y f s y s ds , (1.3)
theo ngha, mi nghim ca phng trnh (1.1) l nghim lin tc ca phng trnh
(1.3) v ngc li (xem, th d, [7], [8]).
Ta c nh l sau y v s tn ti v duy nht nghim ca phng trnh vi
phn (1.1) tha mn iu kin ban u (1.2).
nh l 1 (Picard-Lindelof) Gi s:
1. Hm s ( , ) f x y l lin tc theo c hai bin trong min ng, gii ni D :0 0
0 0
x a x x a
y b y y b
(Do D l mt min ng, gii ni nn t gi thit ny suy ra tn ti mt s dng
M sao cho ( , ) f x y M vi mi ( , ) x y D).
2. Hm hai bin ( , ) f x y tha mn iu kin Lipschitz theo bin y trongD u theo x, tc l tn ti mt hng s dngL sao cho
1 2 1 2( , ) ( , ) f x y f x y L y y
vi mi 1 2( , ) , ( , ) x y D x y D .
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Khi y tn ti duy nht mt nghim ( ) y y x ca phng trnh (1.1) trong khong
0 0 x H x x H , trong min( , )b
H aM
, tha mn iu kin ban u
(1.2).
Chng minh nh l ny c th xem trong [7], [8].
Nhn xt 1. Nu hm ( , ) f x y v( , )
f x y
y l lin tc trn hnh ch nht D th
( , )
f x yL
y vi mi ( , ) x y D . Hn na, theo nh l gi tr trung bnh, vi
mi cp 1( , )x y v 2( , )x y tn ti im*
( , ) x y D sao cho*
1 2 1 2 1 2
( , )( , ) ( , )
f x yf x y f x y y y L y y
y
.
Chng t ( , ) f x y tha mn iu kin Lipschitz.
Nhn xt 2. nh l Picard-Lindelof c tnh cht a phng(tn ti nghim trong
mt ln cn ca 0x l khong 0 0, x H x H ).
Vi mt s lp hm ( , )f x y , ta c th chng minh s tn ti nghim ton ccca
phng trnh (1.1). Th d, s tn ti nghim ton cc ca phng trnh tuyn tnh
' ( ) ( ) y P x y Q x trn ton on ,a b l h qu ca nh l sau.
nh l 2 Gi s ( , ) f x y l hm lin tc theo hai bin v tha mn iu kin
Lipschitz
1 2 1 2( , ) ( , ) f x y f x y L y y
trong min ch nht v hn [ , ] ( , ) a b v 0 0( , )x y l mt im trong ca
min . Khi y bi ton gi tr ban u (1.1)-(1.2) c nghim duy nhttrn on
,a b .
1.2. Bi ton Cauchy ca h phng trnh vi phn cp mt
Cc nh l tn ti nghim ca bi ton Cauchy cho phng trnh vi phn
cp mt c th pht biu tng t cho h n phng trnh vi phn cp mt
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1 1 1 2
2 2 1 2
1 2
( ) ( , , ,... )
( ) ( , , ,... )
.....................
( ) ( , , ,... )
n
n
n n n
y x f x y y y
y x f x y y y
y x f x y y y
(1.4)
H (1.4) c th vit di dng ( , )y f x y , trong 1 2( , ,..., )T
n y y y y
v 1 2( , ,..., )T
n f f f f l nhng vect n chiu. Di y chng ta pht biu mt
dng tng t ca nh l 1 cho h phng trnh vi phn cp mt.
nh l 1 Cho h phng trnh vi phn (1.4), trong hm fxc nh v lin
tc trn tp m D R Rn v thomn iu kin Lipschitz theo y u theo x
trn D :
1 2 1 2( , ) ( , ) f x y f x y K y y vi mi 1 2( , ) , ( , ) x y D x y D .
Khi y vi mi 0 0( , ) x y D tm c mt s 0d sao cho trn khong
0 0, x d x d tn ti v duy nht nghim ca phng trnh vi phn (1.4) tho
mn iu kin 0 0( ) y x y .
Chng minh: Xem [7], [8].
2.PHNG PHP EULER
2.1. Phng php Euler
Khi ( , )f x y l lin tc nhng khng l Lipschitz, nghim ca bi tonCauchy vn tn ti, nhng c th khng duy nht.
Th d 2.1. Xt bi ton Cauchy 3dy
ydx
, (0) 0y .
Bi ton ny c v s nghim: ngoi nghim ( ) 0y x cn c mt h nghim
( ) 0y x khi x c v
3
22( )( )
3
x cy x
khi x c , trong 0c bt k.
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Th d 2.2. Xt bi ton Cauchy 3(sin2 )dy
x ydx
, (0) 0y .
Bi ton ny c ba nghim ( ) 0y x ; 32 2
( ) sin3 3
y x x.
Nhn xt rng cc hm s 3( , ) f x y y v 3( , ) sin 2 . f x y x y khng l
Lipschitz theo y trong ln cn im (0,0).
Cc th d trn cho thy, nu hm ( , )f x y khng l Lipschitz th nh l v tn ti
duy nht nghim (nh l Picard-Lindenlof) khng cn ng. Tuy nhin ta vn c
nh l tn ti nghim sau y:
nh l 1 (Cauchy-Peano) Gi s ( , ) f x y l hm lin tc theo hai bin trong min
ng, gii ni D :
0 0
0 0
x a x x a
y b y y b
(Do D l mt min ng, gii ni nn t gi thit ny suy ra tn ti mt s dng
M sao cho ( , ) f x y M vi mi ( , ) x y D).
Khi y tn ti t nht mt nghim ( )y y x ca phng trnh (1.1) trong khong
HxxHx 00 , trong min( , )b
H aM
, tha mn iu kin ban u (1.2).
Chng minh. Nhm mc ch th hin phng php tnh gn ng nghim, di
y chng ti trnh by chng minh nh l Cauchy-Peano bng phng php da
trn khi nim ng gp khc Euler.
T 0 0( , )x y k hai ng thng vi h s gc l M v M .
Cc ng thng ny c phng trnh l
0 0( )y y M x x
v
0 0( )y y M x x
Chng ct hai ng thng
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song song 0 y y b
v 0 y y bti cc im c honh l
0
bx
Mv
0
bx
M(xem hnh v).
Gi h l di bc (stepsize)ca bin c lp x ( h c th dng hoc m, khi h
dng th nghim c xy dng v bn phi ca im 0x v ngc li, khi h m
th nghim c xy dng v bn tri ca 0x . Di y ta coi 0h , trng hp
0h c th chng minh tng t).
Ta tm gi tr gn ng ca nghim ti cc nt gi tr ca ix vi 1,2,...,i n
( 1
Hn
h):
1 0
2 1 0
0
;
2 ;
.........................
.
n
x x h
x x h x h
x x nh
dc ca ung tip tuyn trong th ca hm s ( ) y f x ti cc gi tr cax c cho bi cng thc : ( ) ( , ( ))y x f x y x .
Chng hn, ti 0x x dc ca ng tip tuyn l
0 0 0 0 0'( ) ( , ( )) ( , ) y x f x y x f x y .
Khi , phng trnh ng tip tuyn vi ng cong y y( x ) i qua im
0 0( , )x y c dng:
0 0 0 0 0 0( )( ) ( , )( )y y y x x x f x y x x
hay
0 0 0 0( , )( )y f x y x x y . (2.1)
Dng ng ny tm gi tr xp x ca nghim y y( x )ti 1x x (k hiu l
1y ), ta c
1 0 0 1 0 0 0 0 0( , )( ) ( , )y f x y x x y hf x y y .
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Tip , dng im 1 1,x y tnh gi tr ca y ti 2x x nh phng trnh
ng tip tuyn
1 1 1 1( , )( )y y f x y x x
hay
1 1 1 1( , )( )y f x y x x y .
Tip tc qu trnh ny, ti nx x ta tnh c:
1 ( , )n n n n y hf x y y . (2.2)
Ni cc im ( , )i ix y vi 1,...,i n , ta c mt ng gp khc ( )hy x , c gi
l ng gp khc Euler (xem hnh v). Vi mi h chn trc, ng gp khcEuler cho ta mt xp x ca ng cong nghim, n trng vi tip tuyn ca cc
ng cong nghim (khc nhau) ti mi im ( , )i ix y . Cng thc hin ca ng
gp khc ny l
0 0( ) h y t x ;
1 1 1 1( ) ( ) ( , ( ))( ) h h k k h k k y t y t f t y t t t , 1 k kt t t , 1,2,...,k n .
Khi h thay i ta c mt h cc ng gp khc Euler. Cho h tin ti 0 ta cmt h cc hm s (cc ng gp khc) gii ni u v lin tc ng bc trn
khong 0 0 x x x H .
Tht vy, th ca mi ng gp khc u nm trong tam gic Tgii hn bi cc
ng thng 0 0( ) y y M x x v 0 x x H (xem hnh v) nn chng b chn
u.
Do
''
'
'
( '') ( ') ( ) '' ' x
h h h
x
y x y x y s ds M x x vi mi h v
0 0', '' x x x x H nn h hm ( )hy x lin tc ng bc.
Theo b Arzel, t ( )hy x c th trch ra mt dy cc hm hi t u ti mt
hm lin tc ( ) y y x trn khong . Hm ( )y y x chnh l nghim ca phng
trnh vi phn (1.1) tha mn iu kin ban u (1.2).
nh l chng minh xong.
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Nhn xt. Phng php ng gp khc Euler ni chung khng cho ta tt c cc
nghim ca phng trnh (21)-(2.2). Th d, mi h ng gp khc Euler p dng
cho bi ton Cauchy 3dy
y
dx
, (0) 0y u hi t ti nghim 0y , v vy ta
khng tm c tt c cc nghim ca phng trnh ny.
Di y chng ta pht biu nh l Cauchy-Peano cho h phng trnh vi
phn (phng trnh vi phn vect).
nh l 1. Gi s
1.Hm : n f D R lin tc trn tp D,
0 0 0( , ) : ,D t x t t t a x x b .
2. M v l nhng s sao cho
( , ) , , min( , )b
f t x M t x D aM
.
Khi y phng trnh vi phn (1.1) c t nht mt nghim tho mn iu kin ban
u (1.2) trn on 0 0,t t .
Phng php Euler tuy cho xp x nghim ca bi ton Cauchy tng i
th, nhng n gin, d tnh ton, v khi h cng nh th xp x nghim cng tt,
iu ny c th hin r rng qua cc tnh ton c th trong cc v d trnh by
trong 4. Vi tc ca my tnh hin nay, ta cng c th s dng phng php
Euler trong cc bi ton thc t, c bit l minh ha phng php s trong
ging dy mn phng trnh vi phn.
Nhn xt rng c nhiu cch i n cng thc (2.2), l c s ca phng php
ng gp khc Euler (xem [8]).
2.1. Phng php Euler ci tin
Phng php Euler c tc hi t chm, n c th ci tin nh dc
trung bnh trong mi khong nhm tng tc chnh xc nh sau.
Trc tin, ta s dng tip tuyn ca ng cong qua im 0 0( , )x y :
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0 0 0 0( , )( ) y f x y x x y
tm gi tr xp x ca nghim ( ) y y x ti 1x x ( v k hiu l*
1y ) . Khi (
vi 1 0h x x ):
*1 0 0 0( , ) . y hf x y y
Nh vy, vi phng trnh ( , ) , y f x y ta tm c dc xp x ca ng tip
tuyn ti 1x x l*
1 1 1'( ) ( , )y x f x y . Gi tr trung bnh ca hai dc 0 0( , ) f x y
v*
1 1( , ) f x y ca tip tuyn ti hai im 0 0( , )x y v*
1 1( , )x y l
*0 0 1 1( , ) ( , )
2
f x y f x yv phng trnh ca ng thng i qua
0 0( , )x y vi dc*
0 0 1 1( , ) ( , )
2
f x y f x yl
*0 0 1 1
0 0
( , ) ( , )( )
2
f x y f x y y x x y.
Khi , ti 1x x gi tr gn ng ca ( )y y x c tnh theo cng thc trn l:
* *
0 0 1 1 0 0 1 11 1 0 0 0( , ) ( , ) ( , ) ( , )( )2 2
f x y f x y f x y f x y y x x y h y .
Tip tc vi cch lm ny, gi tr xp x trong mi bc ca phng php Euler ci
tin ph thuc vo hai tnh ton:
^1 1 1
^1 1
1
. ( , )
( , ) ( , )
2
n
n
n n n
n n n
n n
y h f x y y
f x y f x y y h y
(2.6)
T cng thc trn ta suy ra:
1 1
1 1. ( , ) . ( , . ( , ))
2 2n n n n n n n ny y h f x y h f x y h f x y . (2.7)
S dng cng thc ny, ta tm c nghim xp x tt hn so vi phng php
Euler. iu ny s c thy r trong cc v d 4.
3.PHNG PHP RUNGE - KUTTA
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Mt trong nhng phng php s gii phng trnh vi phn hiu qu, thng
dng v ph bin nht trong cc bi ton k thut l phng php Runge-Kutta. C
th suy ra cc cng thc trong phng php Runge-Kutta t cng thc xp x tch
phn. Trong mc ny, khc vi cc ti liu [1][6], chng ti trnh by phng phpRunge-Kutta xut pht t qui tc cu phng theo [7]. Cch lm ny cho php hiu
cc phng php s gii phng trnh vi phn mt cch nht qun hn (xem [7]).
lm c iu ny, trc tin chng ta nhc li quy tc cu phng c bn
(basic quadrature rules).
3.1. Quy tc cu phng c bn
Ni dung c bn ca quy tc cu phng l: tnh tch phn ( )b
a
f t dt ta
thay ( )f t bi mt a thc ni suy (interpolating polynomial). Tch phn ca hm
( )f t trn on ,a b c xp x bi tch phn ca hm a thc (tnh c chnh
xc).
Gi s ta c s im ni suy khc nhau 1 2, ,..., sc c c trong khong ,a b . a
thc ni suy Lagrange bc nh hn s c dng (xem [1]):
1
( ) ( ) ( )
s
j j
j
t f c L t ,
trong
1,
( )( )
( )
si
j
i i j j i
t cL t
c c.
Khi y
1
( ) ( )
b s
j j
ja
f t dt f c .
Cc trng s j c tnh theo cng thc
( ) . b
j j
a
L t dt
Nu 1s th a thc ni suy 1( ) ( )t f c v ta c:
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1( ) ( ) ( ). b
a
f t dt b a f c
Ta ni chnh xc(precision) ca quy tc cu phng l p nu quy tc ny chnh
xc cho mi a thc bc nh hn p , tc l vi mi a thc ( )kP t bc nh hn p
ta c:
1
( ) ( ).
b s
k j j
ja
P t dt f c
Nu 0( ) b a h th sai s trong quy tc cu phng ca chnh xc p l
10( ).ph
Ta xt mt s trng hp c bit.
Nu chn 1s v 1 c a th ta c
cng thc xp x tch phn bi
din tch hnh ch nht ABCD:
( ) ( ) ( ).
b
af t dt b a f a (3.1)
Nu ( )y x l nghim ca phng trnh vi phn (1.1) - (1.2) th:
( ) ( ) ( , ( )) .
x h
x
y x h y x f s x s ds (3.2)
Kt hp vi cng thc (3.1) ta i n cng thc: ( ) ( ) . ( , ( )). y x h y x h f x y x
T y ta li c cng thc Euler tin bit trong 2:
1 . ( , ).n n n ny y h f x y
Nu chn s=1 v c=b th ta c cng thc xp x tch phn bi din tch hnh ch
nht ABEF:
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( ) ( ) ( ). b
a
f t dt b a f b
T y ta c:
( ) ( ) . ( , ( )) y x h y x h f x h y x h
Suy ra cng thc Euler li:
1 1 1. ( , ). n n n ny y h f x y
Hai phng php Euler tin v li l nhng phng php Runge-Kutta bc nht (c
xp x bc nht).
Nu chn 2s v 1 2, c a c b th 1( ) ( )
t b
L t a b v 2( ) .( )
t a
L t b a
Suy ra
2
1 1
1 ( )( )
( ) ( ) 2 2
bb b
a a a
t b t b b a L t dt dt
a b a b
v
2
2 2
1 ( )
( ) .( ) ( ) 2 2
ab b
a a b
t a t a b a
L t dt dt b a b a
Chng t
( ) [ ( ) ( )]2
b
a
b a f t dt f a f b .
Nh vy nu xp x tch phn ( , ( ))
x h
x
f t y t dt bi cng thc trn (bi din tch
hnh thang ABED)th ta c:
( , ( )) [ ( , ( )) ( , ( ))].2
x h
x
hf t y t dt f x h y x h f x y x
T y ta c cng thc hnh thang:
1 1 1[ ( , ) ( , )].2
n n n n n nh
y y f x y f x y
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Nu chn 1s v 12
a bc th ta c cng thc xp x tch phn bi din tch
hnh ch nht ABMN:
( , ( )) . ( , ( )).2 2
x h
x
h hf t y t dt h f x y x
T y ta c cng thc tnh gi tr
xp x nghim ca phng trnh vi phn:
( ) ( ) . ( , ( ))2 2
h h
y x h y x h f x y x .
T cng thc trn ta c
1 [ ( , ( )].2 2
n n n nh h
y y h f x y x
Cng thc ny c gi lphng php im gia (midpoind method).
Phng php im gia v phng php hnh thang l hai phng php n, chng
c chnh xc 2p .
Nu chn 3s v 1 2 3, ,2
a bc a c c b th, t h b a , ta c:
1 2
( )( )22( ) ( )( ),
2( )( )
2
a bt t b
a b L t t t b
a b ha a b
2 2
( )( ) 4( ) ( )( ),
( )( )2 2
t a t b L t t a t b
a b a b ha b
3 2
( )( )22( ) ( )( ).
2( )( )
2
a bt a ta b
L t t a t a b h
b a b
Suy ra:
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1 1 2 2
3 22
2 2
3
2
2 2( ) ( )( ) ( )( )
2 2
2 2 ( ) ( )[( ) ( )( )] [ ( ) ]
2 3 2 2
2 ( ).
12 6
b b b
a a a
bb
a a
a b a b L t dt t t b dt t b t b dt
h h
a b t b a b t bt b t b dt
h h
b a h
h
v
2 2 2 2
3 2
2
4 4( ) ( )( ) ( )( ( ))
4 ( ) ( ) 4[ ( )] .3 2 6
b b b
a a a
b
a
L t dt t a t b dt t a t a a b dt h h
t a t a ha bh
Do tnh cht i xng (hoc tnh trc tip), ta c
3 16
h
.
T cc tnh ton trn ta i n cng thc Simpson:
( ) [ ( ) 4 ( ) ( )]6 2
b
a
h a b f t dt f a f f b .
Suy ra cng thc xp x nghim ca phng trnh vi phn
( ) ( ) [ ( , ( )) 4 ( , ( )) ( , ( ))]6 2 2
h h h
y x h y x f x y x f x y x f x h y x h
v cng thc sai phn
1 1 1[ ( , ) 4 ( , ( )) ( , )]6 2 2
n n n n n n n nh h hy y f x y f x y x f x y .
y l cng thc n ca phng php Runge- Kutta kinh in cp bn (classical
fourth-order Runge-Kutta method).
2. Dn ti Phng php Runge - Kutta
Tnh 1ny theo cng thc n i hi ti mi bc phi gii xp x mt
phng trnh phi tuyn, iu ny khng n gin v c kh nng lm tng sai s,nn ta c gng xy dng cc cng thc Runge-Kutta hin t cng thc hnh thang
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n, cng thc im gia n v cng thc Runge -Kutta kinh in cp bn n tng
ng nh sau.
Trong cng thc hnh thang n:
1 1 1[ ( , ) ( , )]2
n n n n n nh
y y f x y f x y
ta thay gi tr 1ny v phi bng cng thc Euler tin:
1 ( , ) n n n n y y hf x y .
Khi y ta c cng thc:
1 1 1[ ( , ) ( , )].
2
n n n n n nh
y y f x y f x y
Cng thc ny c gi l phng php hnh thang hin (explicit trapezoidal
method).
Bng cch s dng xp x bc nht ca ( )2
nh
y x theo phng php Euler tin:
1
2
( , )2
n n nn
h y y f x y
v thay vo cng thc ca phng php im gia n
1 [ ( , ( )].2 2
n n n nh h
y y h f x y x
ta nhn cphng php im gia hin (explicit midpoint method):
1 1
2
. ( , ).2
n n nn
h y y h f x y
T phng php Runge-Kutta n cp bn kinh in
1 1 1[ ( , ) 4 ( , ( )) ( , )]6 2 2
n n n n n n n nh h h
y y f x y f x y x f x y
ta c cng thc Runge-Kutta hin bc bn kinh in sau:
1 1 2 3 4( 2 2 ), 0,1,2,...6
n nh
y y k k k k n (3.3)
trong :
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1
12
23
4 3
( , );
( , );2 2
( , );2 2
( , ).
n n
n n
n n
n n
k f x y
h hkk f x y
h hkk f x y
k f x y hk
(3.4)
Nh vy, tnh c 1ny theo cng thc Runge-Kutta hin, ta ch cn tnh cc h
s ik , 1,2,3,4i theo gi tr ca hm s ( , ) f x y ti cc im trc . Vic ny
c th d dng thc hin c trn chng trnhMaple qua cc th d trong 4 di
y.
4. GII BI TON CAUCHY CHO PHNG TRNH VIPHN TRN MY TNH IN T
Do c ci t cc nh, my tnh khoa hc Casio fx-570 ESrt thun tin
cho vic thc hin cc thao tc qu trnh lp. Cng thc tnh xp x nghim theo
phng php Euler (2.2), phng php Euler ci tin (2.7) v v phng php
Runge-Kutta (3.3) cho thy, vic gii gn ng phng trnh vi phn (1.1)-(1.2)
thc cht l thc hin mt qu trnh lp, v vy c th d dng thc hin tnh ton
trn my tnh khoa hc Casio fx-570 EShoc lp trnh trnMaple.
Di y chng ti s trnh by cch gii bi ton Cauchy cho mt phng
trnh vi phn bng phng php Euler, Euler ci tin v phng php Runge-Kutta
vi cc bc ni suy khc nhau trn my tnh khoa hc Casio FX-570 ESv trn
Maple.
Bi 1. S dng phng php Euler, phng php Euler ci tin v phng phpRunge-Kutta vi di bc 0,1h v 0.05h tm xp x nghim ca
phng trnh 2 2dy
x ydx
tha mn iu kin ban u (0) 0y trn on 0 1; .
Phi tm nghim ca phng trnh 2 2 dy
x ydx
vi iu kin ban u
0 00, 0x y .
Vi 0,1h ta c:
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2 21 ( , ) 0,1( )n nn n n n n y hf x y y x y y .
Ta c:
0 0
2 21 00,1( ) (0,1).(0).(0) 0 0 y x y y .
Vi 1 0 0,1 x x h :
1 1
2 2 2 22 10,1 (0,1).(0,1 0 ) 0 0,001 y x y y .
Tip tc nh trn, ta tnh c cc gi tr ny theo cng thc:
2 21 ( , ) 0,1( )n n n n n n n y hf x y y x y y . (4.1)
Thc hin php lp (4.1) trn Casio fx-570 ES:
Khai bo cng thc 2 21 0,1( )n n n n y x y y :
0.1 ( ALPHA X2x + ALPHA Y 2y ) + ALPHA Y
Trong qui trnh ny, ta dng nh X cha gi tr nx v cc nh Y
cha gi tr ca ny .
Dng CALC tnh gi tr ca ny : CALC
My hi: X?
Khai bo0
0x : Bm phm 0 =
My hi: Y?
Khai bo0
0y : Bm phm 0 = (0)
Kt qu trn mn hnh l 0, tc l
2 2 2 21 0 0 00,1( ) 0,1(0 0 ) 0 0 y x y y .
a kt qu 1 0y vo nh Y : SHIFT STO Y
Tr v cng thc (4.1): Bm phm
Quy trnh:
Tnh tip: CALC
My hi: X? Khai bo 1 0 1x . : Bm phm 0.1 =
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My hi: Y? Bm phm = ( 1 0y v c sn trong nh Y nn khng cn
khai bo li).
Kt qu hin trn mn hnh:1
1000
, tc l
2 2 2 2 32 1 1 10,1( ) 0,1(0,1 0 ) 0 0,1 y x y y .
a kt qu vo nh Y : SHIFT STO Y
Tr v cng thc ban u:
Lp li quy trnh vi thay i duy nht l khi my hi X? th ta khai bo cc gi tr
tip theo: 0.2; 0.3; 0.4;...;1.0 ta s c bng gi tr nh trong bng sau.
n 1nx ny n 1nx ny
1 0 0 6 0,5 0,05511234067
2 0,1 1
1000 7 0,6 0,09141607768
3 0,2 35 001 10, 8 0,7 0,1412517676
4 0,3 0,0140026001 9 0,8 0,2072469738
5 0,4 0,03002220738 10 0,9 0,2925421046
Thc hin php lp (4.1) trnMaple 7:
TrongMaple, tm cc gi tr iy theo cng thc lp ta c th s dng mc
nh (option) remember(nh). Mc nh ny caMaplecho php nh cc gi tr c
tnh ny , m khng cn tnh li gi tr 1ny .
Trc tin ta khi ng chng trnhMaplenh lnh restart:
[> restart;
Khai bo hm f:
[>f:=(x,y)->x^2+y^2;
:=f ( ),x y x2 y2
Khai bo bcni suy 0,1h :
[> h:=0.1;
:=h .1
Khai bo cch tnh cc gi tr ca 1n n x x h (vi 0 0x ):
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[> x:=n->n*h;:=x n n h
Khai bo gi tr ban u ca y :
[> y(0):=0;:=( )y 0 0
Khai bo th tc tnh ny theo mc nh remember (nh):
[> y:=proc(n) option remember;
[> y(n-1)+h*f(x(n-1),y(n-1));
[> end;
:=y proc ( ) end procn option ;remember ( )y n 1 h ( )f ,( )x n 1 ( )y n 1
Khai bo lnh seq(sp xp theo dy) sp xp cc gi tr 0 1 9 10, ,..., , y y y y :
[> seq(y(i),i=0..10);
0 0. .001 .0050001 .01400260010.03002220738.05511234067.09141607768, , , , , , , ,
.1412517
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