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Page 1: !# $ 576 8)9:.* 4,>8)(HG ......"! #$! %'&( *),+.-/ & 0 1! 2 35463798;: =@?1acb a>d6e5?gfihkj h lm=@an4 4o4p4o4 4o4 4o4o4 4o4

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� � ��� ��� � ���� �

bPBW?V< ~\=o< :@A>HVFI=o< <;��D6=@?1A>D��ZT1H$=o< E5H�FVBW?V< ~\=o< <;��D6=@?V�@=�< F$=ª~s¦ D6?$�5:�?$D\=�H$8�� ~hBv8>:o<[EY~\H�A;D\EY?ªF�¦ :oT1HVB�A;D\EY?V< E5H�XV~\HV<U�5:�?$:@8CBW~6=@]e=@?1AUFI=c<[�I<[A>_@]e=o<UF�¦ :�T1HVBWA>D6E5?V<9F$D§¡":@8;=@?1A>D6=@~6~6=�<9EYHB5HIdwFI:�8>D6�5:�=�<PXVBW8>A>D6=@~6~6=�<O=o<MA�A>8;:�< D\]eX(E58>A;BW?1A;=54/¥�=@X(=@?ZF$BW?1A �/��=@8>A;B5D\?V=�<�:�T1HVBWA>D6E5?V<�?V=X(=@H$�Y=@?1AeXVB5<��A;8>=G8;:�<>E5~6H$=�< =�dIX$~6Dh��D\A>=@]e=�?YAª�w~s¦ BWDhFI=vF�¦ =�dIX$8;=�<;<[D6E5?Z< �@E5]eXZEY8[ACBW?1A ~6=�<��EY?V� A;D\EY?V<9HV<>H$=�~\~6=�<�FI= ~ ¦ B5?VBW~6�I<[=Y4 � ~(=�<[A�FIE5?Z��?$:��@=�<;<>B5D\8;=�FI=c8>=o��E5HV8>D689� FI=o<U]e:�A �$EIFI=�<?�H$]e:@8;DhTYHV=�<^TYHVD�=�dIX$~6E5D\A>:@=o<e<[H$8�~ ¦ E58CFID\?ZB�A>=�H$8^�5E5?1A�FIE5?V?$=@8eH$?$=G8;:�<>E5~6HIA>D6E5? FI=v��=�<:oTYHZB�A>D6E5?Z<@4 � :�BW?V]^EYD\?Z< �9�@=�A[A;=v8>:o<[EY~\H$A>D6E5?t?$=v<[=�8;B T1Hy¦ BWX$X$8;EI���$:@=GF$B5?V<�~hBwXV~\H$XZBW8>AFI=o<��@BY<@4� EWA>8;= XV8>E5LM=�AP�@E5?V<>Dh<MA;=^��FI:��@8>D68>=^FI=^]eB5?$D6_@8;=^B5X$X$8;E����V:@= H$?V=e<>E5~6HIA;D\EY?KXVBW8>A>Dh��HV~\D6_@8;=F�¦ H$?$= :�T1HVB�A;D\EY?�FID§¡"=@8;=@?1A;D\=�~\~6=�~6D\?$:oBWD68>=�XZBW8UHV?$= ��EI� A;D\EY?ª<[X$~6D6?$= FI=�?VE1=�HVF$<U=�T1H$D68>:�XVBW8��A;D6<�4

�y=O8CBWXVXZEY8[A�X$8;:�<>=@?1A>=OFI=O]�B5?$D\_�8>=OFI:@A;B5D\~6~\:�=�~6B �'B���E5?GFIEY?YA�?VE5HV<�Bq�YE5?V<�8;:�<>E5~6Hn�@=X$8;E5¢V~\_�]^= =�Ac~6=�<P���$EYD§dvT1H$= ?$EYHV<cBq�YE5?V< �'BWD\A;<�4ZR9?wBW?$?$=@dI= <>E5?1A���E5HV8>?$Dh<�~6=�<P��EYH$8;¢Z=o<X(=@8;]e=�A[ACBW?1A�FI=O��D6<>HVB5~\Dh<[=�8�~\=o<�8>:o<[H$~\A;BWA;<�EY¢IA>=�?1HZ<�BWD6?V<[DyT1H$=O~6=OX$8;E5�Y8;B5]^]e=Y4

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� ��������� � �

� ����.� � ��� ��� � ��� �������t�

����� �����+���+� � ! ��� ������� ���� E5A>8;=�X$8;EWLM=@AU��E5?Z<[Dh<MA;= FIEY?V��� FI:o��8;D\8;=�FI=�]�B5?$D\_�8>=cBWX$X$8C���$:�=�H$?$= <>E5~6HIA>D6E5?�XZBW8>A>D��

�@H$~\D6_@8;=�F�¦ HV?$=O:�T1HVBWA>D6E5?�F$D§¡"=@8;=@?1A>D6=@~6~6=E(f) = 0

XVB58�H$?$=O<[XV~\D6?$=O��H$¢VD6T1H$= FI=�?$E�=�HVF$<=oTYHVD\8;:@XVB58[A;D6<

(xi)i=0,..,n

47 E5HV8c��=�~6B �VEY?K��B����$=�8;���$=�8�~hB�<>E5~6HIA>D6E5?K<>E5HV<���E58;]e= F�¦ H$?$=^��E5] ¢$D6?VBWDh<[EY?v~\D6?$:oBWD68>= F$=��EY?V� A;D\EY?V<�<[D6]eX$~6=�< �I~\=o<���<[X$~6D6?$=�< �I~\=o<�D\?Z��E5?V?1HV=�<�:�ACBW?1A�~6=�<

aiF$BW?Z<

σ(x) =

n+1∑

i=−1

aiB(x − α

h− i)

� ?ªX(=@H$A�:��qB5~\HV=@8�~s¦ :�d$B5��A>D\A>HVF$=cFI=P~6B <[EY~\H$A>D6E5?�B5X$X$8;E����V:@=��Y8��Y��= � FI=�<���EY]eXVBW8CBWDh<[EY?V<�Y8;B5X �$DhTYHV=�<�=@AnBWH���BW~h��H$~�FI=

D(h) = 1N

∗∑N

j=1(σ(xj) − f(xj))2 TYHVDcX(=@8;]e=�AnF$=�@E5]eXVBW8;=@8"~ ¦ :o�@B58[Ay=�?YA;8>= ~hB�<>E5~6HIA>D6E5? B5X$X$8;E����V:@= =�A ~hB�<>E5~6HIA>D6E5?�=�d$BY� A>= X(E5HV8yH$?O?$EY] ¢$8;=

FI=�?$E�=�HVF$<�T1H$D���BW8;D6=54

����� � "���� !#�����! ����" -��� �+�+��y=o<�:�T1HVBWA>D6E5?V<��^8;:�<>E5HVFI8;=P?1HV]^:�8>DhT1H$=@]e=�?YA�<>H$8

[α, β]<>E5?1A�~\=o<�<[HVD\��BW?1A;=�<�#

$ E(f)(x) = f ′′(x) + f ′(x)Bq�5=o����EY]e]^=v�@E5?VFID\A>D6E5?Z< ~6D6]^D\A>=o<

f(0) = 0=�A

f(1) = 1

$ E(f)(x) = f ′′(x)−4f ′(x)+3f(x)−10sinxBq�5=o�U��EY]e]^=���EY?VFID\A>D6E5?V<,~6D\]eD\A>=o<

f(0) = 0=@A

f(1) = −1

$ E(f)(x) = f ′′(x)− 4f ′(x)+4f(x)− 5coshx+4sinhxBq�5=o����EY]^]e= �@E5?VF$D �

A>D6E5?V<�~6D\]eD\A>=o<f(0) = 0

=�Af(1) = cosh1

� �Z?GF$=��@E5]eXVB58>=�8�~hBe<[EY~\H$A>D6E5?�B5X$X$8;EI���$:@=�Bq�5=o� ~hBe<[EY~\HIA;D\EY?�=@dIBY� A;= �ID6~�=�<[A�?$:��@=�< �<;BWD68>=�FI=O8;:�<>E5HVF$8>=����VB5�@H$?$=�FI= ��=o<�A>8;E5Dh<�:�T1HVBWA>D6E5?V<�B5H�X$8;:�B5~6B5¢$~6=54

Q

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��������� ��� ���������� �������������

E(f)(x) = f ′′(x) + f ′(x)

f(0) = 0

f(1) = 1

� Be<>E5~6HIA>D6E5?n�5:�?$:@8CBW~6=PF$=O~s¦ :�T1HVBWA>D6E5? �$E5]eE5�Y_@?$=f ′′(x) + f ′(x) = 0

=�<[A�#f(x) = Acosx + Bsinx

�y=o<���E5?ZFID§A;D\EY?V<�D6?$D§A;D6B5~\=o<�FIEY?$?$=@?1A #{

A = 0Acos1 + Bsin1 = 1

� ?�A;8>EYH$�5=OFIEY?V�P~6Be<>E5~6HIA>D6E5?nXVB58[A;D6�@H$~\D6_@8;=OFI=�~6BeXV8>=�]^D6_@8;=P:�T1HVB�A;D\EY? #f(x) =

sin1

sinx

��������� !"�$#���� �%&� �����������'�

E(f)(x) = f ′′(x) − 4f ′(x) + 3f(x) − 10sinx

f(0) = 0

f(1) = −1

� Be<>E5~6HIA>D6E5?n�5:�?$:@8CBW~6=PF$=O~s¦ :�T1HVBWA>D6E5? �$E5]eE5�Y_@?$=f ′′(x) − 4f ′(x) + 3f(x) = 0

=�<[A�#f(x) = Aex + Be3x

� Bc<[EY~\HIA;D\EY? XVB58[A;D6�@H$~6D\_�8>=�FI=�~ ¦ :�T1HVB�A;D\EY? ��EY]eX$~\_@A>=f ′′(x)−4f ′(x)+3f(x) = 10sinx=�<[A�#

f0(x) = 2cosx + sinx

�y=o<���E5?ZFID§A;E5?V<�D6?$D\A>DhBW~6=�<�FIEY?$?$=�?YA #{

A + B + 2 = 0Ae + Be3 + 2cos1 + sin1 = −1� ?�A;8>EYH$�5=OFIEY?V�P~6Be<>E5~6HIA>D6E5?nXVB58[A;D6�@H$~\D6_@8;=OFI=�~6B�F$=@HIdID6_@]e=�:�T1HVB�A;D\EY? #

f(x) = Aex + Be3x + 2cosx + sinxB = − 1+2cos1+sin1−2e

e3−e

A = 2 − B

a

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��������� � ���'��� ���"� �� �����������'�

E(f)(x) = f ′′(x) − 4f ′(x) + 4f(x) − 5chx + 4shx

f(0) = 0

f(1) = ch1�/B^<>E5~6HIA;D\EY?��Y:@?$:�8;B5~\=�FI=�~s¦ :�T1HVBWA>D6E5? �$E5]eEY�5_@?V=f ′′(x) − 4f ′(x) + 4f(x) = 0

=�<[A�#f(x) = (A + Bx)e2x

�/B <>E5~6HIA>D6E5? XVBW8>A>Dh��H$~6D6_@8;=�FI=�~ ¦ :oTYHZB�A>D6E5? ��E5]eX$~6_�A;=f ′′(x)−4f ′(x)+4f(x) = 5chx−

4shx=�<[A�#

f0(x) = chx� =�<���EY?VFID\A>D6E5?V<�D6?$D\A>DhBW~6=�<�FIEY?$?$=�?YA #{

A + 1 = 0(A + B)e2 + ch1 = ch1� ?�A>8;E5H$�Y=�F$E5?V�P~hB^<>E5~6HIA;D\EY?�XVB58[A;D6�@H$~6D\_�8>=OFI=O~hB A>8;E5Dh<[D6_@]e=�:�T1HVBWA>D6E5? #

f(x) = (−1 + x)e2x + chx

j

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� ��������� � �

� � ����� � ���� ��� � �� ��� � � � ��� �

� ������� ��� ���

� ��� � !#� ���+� ��#��� � ��� ��������3��� ���+���^�$�$�b�

� ?wF$: �V?$D\A � X(E5D6?1A;<�:oTYHVD\8;:@XVB58[A;D6<(Xj)i=1,..,N

<[H$8�?$E5A>8;= D6?1A>=�8>��BW~6~6= FI=^8>:o<[EY~\H�A;D\EY?

[α, β]� � :�ACBW?1Aª<[HVXZ:�8>D6=@H$8��YE5D68e~6B58>�Y=@]e=@?1Aª<[HVXZ:�8>D6=@H$8ª� ?.=�A�EY?.���V=@8C���$=�~6=�<

(ai)i=−1,..,n

TYHVD�]^D6?$D6]eD6<>=@?1A�~6= ��8;D§A;_@8;= #

e(a) =

N∑

j=1

E(f)2(Xj)

<>E5HV<�~hB���EY?YA;8;B5D\?1A;=PFI=O<;B�A;D6< �'BWD68>=�~6=�<��@E5?VFID\A>D6E5?Z<�BWHId�~6D6]^D\A>=o<@47 E5HV8 ]eD\?VD\]eDh<[=�8

e(a)�/~6=�8CBWDh<>E5?$?$=�]e=@?1A =�<[A ~6=n]��]e=�T1H$=nXZEYH$8 ~6=�X��5~6��?$E5]e=nFI=�<

]eE5D6?VFI8;=�<���BW8;8>:o<�F$D6<;��8;=�AC<@4� D\?V<>D �

∂e(a)

∂aj

= 0 ∀j ε[1, n − 1]

=o<MAUHV?$=c�@E5?VFID\A>D6E5?�?V=���:o<><;BWD68;= �W=c=�A;B5?1AUH$?$= ��E5?V��A>D6E5?�FI=o<9? �C3���BW8;DhBW¢$~6=�<(ai)i=1,..,n−1=@Ac�W¦ =�<[AcH$?V= ��EY?VFID\A>D6E5?K<>HI©ª<>B5?1A>=54Z= =�<[APT1HVBYFI8CB�A>DhT1H$=O=@?w���VB5T1H$=

(aj)=�AP~\=o< ��E�=@©��

�@D\=�?YAC<�FI=�<(aj)

2 <>E5?1A�A>EYHV<�XZE1<[D\A>D �'< �$D\~/<@¦ BW�YD§A�FIE5?Z��¢$D\=�?vF�¦ HV?n]eD6?$D6] H$]G4b ¦ �5X$8;_�<P~\=o<OX$8;E5X$8;D6:�A>:o<�FI=�<OB5X$X$8;Eqd�D6]�B�A;D\EY?V< � � <>X$~\D6?$= ��E5? ��E5?V?VBWD\A

a0=@A

an�Y8��Y��=

B5HIdK��E5?ZFID§A;D\EY?V<�BWHIdK~6D6]^D\A>=o< �"=�?w=@¡(=@Aσ(x)

XVB5<;<>= XVB58c~6=�<P=@d1A;8>:�]eD§A;:�<�4"bc=eX$~6HV< �"EY?B�~\=o<�X$8;E5~6E5?$�Y=@]e=@?1A;<P<>H$D\��B5?YAC< #

a−1 = 2 ∗ a0 − a1=@A

an+1 = 2 ∗ an − an−14�7 B58

�@E5?V<>:�T1H$=�?YA���~\=OX$8;E5¢V~\_�]^=O<>=�8;B5]^_�?$=O�eFI:�A;=@8;]^D6?$=�8�~\=o<(ai)i=1,..,n−1

4� ?�:���8;D\A�F$E5?V�

e(a)=@?n8>=�]^XV~6B��@BW?1A

a−1=�A

an+1XVB58�~\=�H$8;<�=@dIX$8>=o<><>D6E5?V<�8>=o<[X(=���A>D6�5=o<

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=�A�EY?�8;=@�Y8>EYH$XZ=P~6=�<�A>=�8>]e=o<�=@? ��E5?V��A>D6E5?vFI=o<ai4

e(a) =

N∑

j=1

(

n∑

i=0

(ai ∗ coeffai) − SM(Xj))

Bq�5=o� �I=@?v?VEWA;B5?1AcJI|«~6B ��EY?V� A;D\EY?G8>=�X$8>:o<[=�?1A;BW?1A�~6= <[=o��E5?ZF�]e=�] ¢$8;=O=�Ac� ~hB ��E5?Z� A>D6E5?8>=�X$8;:�<>=@?1A;B5?YA�~hB XZBW8>A>D6=P~6D6?$:�B5D\8;=OFI=�~s¦ :�T1HVBWA>D6E5?GFID§¡":@8;=@?1A;D\=�~\~6= #

coeffai= L(Bi)(xj) ∀i ε[2, n− 2]

coeffa0= (2 ∗ L(B−1) + L(B0))(xj )

coeffa1= (−L(B−1) + L(B1))(xj)

coeffan−1= (−L(Bn+1) + L(Bn−1))(xj)

coeffan= (2 ∗ L(Bn+1) + L(Bn))(xj)

� ?ªFI:@8;D6�5=�BW~6E58C<e(a)

XVB5898CBWX$X(E58>AU�(ai0)i0=1,..,n−1

4YR :@A;BW?1AU~6D6?$:�B5D\8;=�F$BW?Z<9~6= �@BYFI8>=FI= ��=�X$8;EWLM=@A �IEY?�EY¢IA>D6=@?1A BW~6E58C<�H$?v<[�I<[A>_�]^=�~6D\?V:�BWD68;=P=�?

ai

4� ?n<>=O8;=�A>8;E5HV�5=cBq�Y=���FI=o<�:�T1HVB�A;D\EY?V<�FI=�~hB ��EY8>]e= #

2 ∗

N∑

j=1

coeffai0∗ (

n+1∑

i=−1

ai ∗ coeffai− SM(xj)) = 0

E58coeffai0

=�<[A�H$?$=v��EY] ¢$D6?VB5D6<>E5?�~6D6?$=�B5D\8;=nF$= ����<>X$~\D6?$=G=�AeE5?t~6=�] H$~§A;D\XV~\D6=KBq�5=o�∑n+1

i=−1 ai ∗ coeffai

��EY?YA;=@?VB5?1A9=@~6~\=�] �]^=PFI=�< ����<>X$~6D\?V=�<9�@= T1H$DVX(=@8;]e=�A�FI=c<[D6]eX$~\D���V=@8o4R9? =@¡(=@A ��<>D5EY? FI:@8;D6�5= XVB58 8CBWX$X(E58>A �

(ai0)�@E5? B�XZEYH$8

4 ≤ |i0−k| Bk(x)∗Bi0(x) = 0�

H$?$=OFI=o<�FI=@HId�<>X$~6D\?V=�<�:�ACBW?1A���E58C��:�]^=�?1A�?�H$~6~\=Y4 � ?�BeFIEY?V�cF$=�<�=�dIX$8;=�<;<>D\EY?V<�F$H�AM��X(= #2 ∗

N∑

j=1

i0+3∑

i=i0−3

(coeffai0∗ai∗coeffai

) −

N∑

j=1

coeffai0∗ SM(xj) = 0

⇐⇒

N∑

j=1

i0+3∑

i=i0−3

(coeffai0∗ ai ∗ coeffai

) =

N∑

j=1

coeffai0∗ SM(xj)

� ?nBeB5D\?V<>DyH$?G<>��<[A>_�]e=O~\D6?$:�B5D\8;=�F�¦ EY8;F$8>=�? �C354� ��� � �+� ���#�Z !#��� �$� �����v��3���

�y=�<�<>�I<MA;_@]e=�<�~6D6?$:�B5D\8;=�<�] HV~§A;D6FIDhBW�YE5?VB5HId�<>E5?1A�8>:o<[EY~\HV<�XVB58�~6B^]e:�A �$EIFI= FIHGX$D6�5EWAFI= ��BWHZ<><�T1H$D <>=OFI:@8;E5H$~6=�=@?vFI=�HId�X �VBY<[=o< #$ ~hB^A>8;D6B5?$�5HV~6BWA>D6E5?�F$HG<>�I<MA;_@]e=�~\D6?$:oBWD68>=$ ~hBe8>:o<[EY~\H$A>D6E5?�F$HG<>�I<MA;_@]e=PA>8;D6B5?$�5HV~6B5D\8;=

3��

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������� � � � �������'�����������'��� � ��� � � �"� �� ��������� ��7 E5H$8P<[= 8;B5]e=@?$=�8 �ªH$?w<[�I<[A>_@]e=OA;8>DhBW?V�5H$~hBWD68>= �$E5? �qBª]^=@A[A;8>=^FI=�<�@:�8>E1<�<[EYHV<�~6=�<

D6_@]e=�<�A>=�8>]e=�<PFIDhBW�YE5?VB5HIdnF$B5?V<�~hBª]�BWA>8;D6�@= FIHKX$8;=@]eD6=@8c]^=�] ¢$8;= �Z]�BWDh< EY?v�qB�BWHV<;<>D]eEIFID��V=@8�~6=�<�A;=@8;]^=o<�FI=O~hB ]�BWA>8;D6�@=�F$HG<>=��@E5?VF�]e=@] ¢$8>=�=�?nXV8>EI��:oF$BW?1A�FI=O~hB^<>E58>A>= #

M =

m1,1 m1,2 · · · · · · m1,p

0

4 4 4m... m...

444444 4 4 4 4 4 4 4 4 4 4444440 mi,i

4 4 4 444444 mi+1,i

4 4 4 4 4 4444 444 4 4 4 4440 · · · 0 mr,i · · · mr,p

=�AN =

n1

n2444ni

ni+1444nr

B5H D6_@]e=�A>=�8>]e=�FIDhBW�5EY?VBW~1E5?^=�¡"=���A>H$=�~6=�< ��BW~h��H$~h< <[HV8,~6=�A>=�8>]e=�FIDhBW�YE5?VB5~i+1

<>H$D6��BW?1A�#$ mi+1,i+1 = mi+1,i+1 −

mi+1,i∗mi,i+1

mi,i

$ ni+1 = ni+1 −mi+1,i∗ni

mi,i

mi,i

=o<MA B5X$X(=@~6:P~6=OX$D6�5E5A �I=�A�D6~�?$= FIEYD§A�XZB5<��A;8>=�?�H$~ 4��������� � � � ��� �$��� �'��� � � � � � � �%&� � �������'����������R9?nXVBW8>A;B5?1A�FIHGFI=�8>?$D6=@8�A;=@8;]^=�FIDhBW�YE5?VB5~(FI=�~hB ]�BWA>8;D6�@=PFIHnX$8;=@]eD6=@8�]e=@] ¢V8>= �1EY?

X(=@HIA�FI:@A>=�8>]eD6?$=@8�~6=PA>=@8;]e=PF$H��Y=�� A;=@H$8����$=�8;���V: ��X$H$Dh<�FI=�X$8;E����V=c=�?�X$8;EI���$=��@BW~h��HV~\=�8A;E5HV<�~6=�<�A;=@8;]^=o<�=@?G8;=@]eE5?1A;B5?1A�4

MA = N

⇐⇒

m1,1 m1,2 · · · · · · m1,p

0

4 4 4m... m...

444444 4 4 4 4 4 4 4 4 4 4444440 mi,i

4 4 4 444444 0

4 4 4 4 4 4444 444 4 4 4 4440 · · · 0 · · · 0 mr,p

a1

a2444ai+1444444ar

=

n1

n2444ni444444nr

R A FIE5?Z� BWHOD6_@]e= A>=�8>]e=UF$H �5=o� A>=�H$8 <>E5~6HIA>D6E5?�EY?�A>8;E5H$�Y= �@�@E5?$?VB5D6<;<;BW?1Ay~\=9A>=@8;]e=(i+1)

#ai = ni −

mi,i+1∗ai+1

mi,i

3Y3

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3 �

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� ��������� � �

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bcB5?V<9~6=�]eEIFIH$~6=�] ��<[�I<[A>_@]e= �5~6= <[EYHV< � X$8;E5�Y8;B5]e]^=�8>=�]eX$~\Dh<;<>B5�5=�X(=@8;]^=@A�FI=�8;=@]��X$~6D\8c~\=o<�]�B�A;8>Dh��=o< FIHKXV8>=�]^D6=@8 ]^=�] ¢$8;= =@APFIH <>=��@E5?VFn]e=@] ¢$8;= FIH <>�I<MA;=@]e= ~6D\?$:oBWD68>=FI:��@E5H$~hBW?1A�FI=�~hBe]^D6?$D6]eD6<;B�A;D\EY?nFI=

e(a)4� ?ªB X$8;:�<>=@?1A;: F$BW?Z<9~6=c���VB5X$D§A;8>= ���5XZBW8>A>D6= 3�~6=�<�]�B�A>8;Dh��=�<UXZ=�8>]e=@A[A;B5?1A�FI= F$D6<;��8;:�A;D6<>=@8

~\= <>�I<MA;_@]e=O�^8>:o<[EYHVFI8;=54� E5A>8;=�A>8CBq�qB5D\~/��E5?Z<[Dh<MACBWD\A�BW~6E58C<��eA>8CB5FIHVD\8;=OFI= ]�BW?$D6_@8;=OD6?���E58;]�B�A>DhT1H$=O?$E1<�:�T1HVBWA>D6E5?V<XZEYH$8�X(E5H$�YE5D68�~6=�<�8>:o<[EYHVFI8;=54

J�H$8n~\= X$~hBW?�D\?���EY8>]�BWA>DhTYHV= ��D\~�?y¦ =�<[AnXZB5< LMHVFIDh��D6=@HId FI= <[A>EI���5=�8�~hBr]�B�A;8>Dh��= | FIHX$8;=@]eD\=�8 ]e=@] ¢$8>=�FI=�FID\]e=�?V<[D6E5?

(n − 1, n − 1)� ��BW8 =�~\~6=�?y¦ BvT1H$=�<>=@XIA FIDhBW�YE5?VB5~\=o<

?$E5?n?�H$~6~\=o<@4

3@�

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M =

m m m m 0 · · · · · · · · · · · · 0m m m m m 0 · · · · · · · · · 0m m m m m m 0 · · · · · · 0m m m m m m m 0 · · · 0

0

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 444444 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 40

0 · · · 0 m m m m m m m0 · · · · · · 0 m m m m m m0 · · · · · · · · · 0 m m m m m0 · · · · · · · · · · · · 0 m m m m

� ?OHIA;D\~6Dh<[=�FIE5?V�9~6B ]eBWA>8;D6�@=M1

A>=�~\~6=UT1H$=M1(i, j) = M(i, i+j)

X(E5HV8i = 1, ..., n−1=@A

j = −3, ..., 34

M1 =

0 0 0 m m m m0 0 m m m m m0 m m m m m mm m m m m m m444 444 444 444 444 444 444m m m m m m m444 444 444 444 444 444 444m m m m m m mm m m m m m 0m m m m m 0 0m m m m 0 0 0

7 E5H$8/~\=�<[=o��EY?VF�]e=@] ¢V8>= ��EY?OHIA>D6~\Dh<>=U~6=U�5=o� A;=@H$8,| ����?�;39~6D6�5?$=o< �@A>=@~�T1H$=M1∗A =

M2��� :@A;BW?1A�H$?G�Y=�� A;=@H$8��@E5~6E5?$?$=O�@E5?1A>=�?VBW?1A�~\=o<

(ai)i=1,..,n−14

�P8��5�@= BWHK8;:�<>H$~\A;B�APFI= ∂e(a)∂ai0

=0

�ZEY?KX(=@HIAP8>=�]^XV~\D68 ~6=�<c]�B�A>8;Dh��=�<c| 3 =@AP| �ª�ª~hB�~\D6�5?V=i04�bc=^XV~\HV< ����EY]^]e= E5?KA;8;Bq��B5D\~6~\=^Bq�5=o� FI=�<�<>E5]e]e=�< �V?$D6=�<�<>H$8

i0=�A�L ��E5?KX(=@H$A�~6=�<

X(=@8;] HIA;=@8o4bc= �'B �@E5?n�5:@?V:@8CBW~6= ��E5?GB^XZEYH$8�~6Be]�BWA>8;D6�@=PFIHGX$8;=@]eD6=@8�]e=@] ¢$8;= #

M1(i0, l) =N

j=1

(coeffai0∗ coeffai0+l

) ∀i0 ε[1, n − 1]

7 E5HV8c~hB�]�BWA>8;D6�@=^FIH <[=o��EY?VFK]e=@] ¢$8;= �"D\~ ?$= �'B5HIAPXZB5<PE5H$¢V~\D6=@8OT1H$=^~s¦ E5? FIE5D\APXZB5<;<[=�8a0 ∗ coeffa0

=�Aan ∗ coeffan

FI= ~ ¦ BWHIA;8>= � �WA>: FIHK<[�I<[A>_�]^=O~6D6?$:�B5D\8;= �a0=@A

an

:�A;B5?1A

3qQ

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��EY?$?�HV<@4 � ?^8>=�]eX$~\D\A9FIE5?Z�U~hB ]�BWA>8;D6�@=�F$H^<>=��@E5?VF ]^=�] ¢$8;=�FIH^<>�I<MA;_@]e=�~\D6?$:oBWD68>=�BWD6?V<>D #M2(i0) =

N∑

j=1

(coeffai0∗ (SM(xj) − a0 ∗ coeffa0

))∀i0 ε[1, 3]

=�AM2(i0) =

N∑

j=1

(coeffai0∗ SM(xj))∀i0 ε[4, n− 4]

=�A

M2(i0) =

N∑

j=1

(coeffai0∗ (SM(xj) − an ∗ coeffan

))∀i0 ε[n − 3, n − 1]

��=@]�B58;T1H$= # � ~ �'BWHIA��5:�8>=�8�� XZBW8>A�~6=�<�:�T1HVBWA>D6E5?V<���EY?YA;=@?VB5?1Aa0�a1�an−1

=�Aan

�1~6=@HV8;<��E�=�©ª�@D\=�?YAC<�8>=o<[X(=���A>D �'<�:�ACBW?1A�XVBW8>A>Dh��HV~\D6=@8C<@4��������� � � � � ���$�����'� � � ��� � � � �%

�y=�< ��E5?ZFID§A;D\EY?V<�BWH$d�~\D6]eD§A;=�<�D6]eXZE1<[=�?1A�F$=�<��@E5?VFID\A>D6E5?Z<�<[H$8�F$=@HIdªA;=@8;]e=�<�FIHG�5=o� �A>=�H$8�<[EY~\H$A>D6E5?

A#

$ A(0) = f(0)

$ A(n) = f(1)

E�� �,=o<MA�~hB ��E5?V��A>D6E5?n=�d$B5��A>=OFI=O~ ¦ :oT1HVB�A;D\EY?nFID\¡":@8;=@?1A>D6=@~6~\=Y47 E5H$8�<;B�A;D6< �'BWD68>=P~6=�<��@E5?VFID\A>D6E5?Z<�D\]eX(EY<>:@=�<�XZBW8�~\=o<�� � <>X$~6D\?V=�<�E5?GBeBWHV<;<>D #$ A(−1) = 2A(0) − A(1)

$ A(n + 1) = 2A(n) − A(n − 1)

�y=O<>��<[A>_�]e=P�^8;:�<>E5HVFI8;= ��B^FIE5?Z�P<�¦ =�¡"=�� A;H$=@8�<>H$8�~hB ]�BWA>8;D6�@=AFI=o<�~\D6�5?$=o<

1�

n− 14

� BG]�B�A;8>Dh��=�FIH XV8>=�]^D6=@8O]e=�] ¢$8;=�:�ACBW?1A 8>=oFI8;=�<;<[:�= ��D6~ �'BWHIA X$8>=�?VFI8;=e=@? ��EY]^X$A>=�~6=�<���VBW?V�5=@]e=�?YAC< F�¦ D\?ZFID6�@=�< XZEYH$8 ~6=�< F$D§¡":@8;=@?1A>=o< X �ZB5<>=�< FI=�~6BG8;:�<>E5~6HIA;D\EY? F$Hr<[�I<[A>_�]^=~\D6?$:oBWD68>=Y4��/B F$D6B5�5E5?ZBW~6=cT1H$Dy��EY?1A>D6=@?1A�~6=�<�XVD\� �WA;<�X(E5H$8�~hB ]e:@A �$EIFI=OFI= ��BWHZ<><�FI=��1D6=@?1A~6B ��E5~6E5?V?$= F�¦ D\?ZFID6�@=

0F$BW?V<U~6B ]�B�A>8;Dh��= FIH�X$8>=�]eD\=�8U]e=@] ¢$8>=cFIE5?1A�~6=�<���EY~\EY?$?$=�<U�5EY?YA

FI=−3�

+34

� BPA;8>DhBW?$�YH$~hB�A>D6E5?�FIH�<[�I<[A>_@]e=�~6D\?V:�BWD68;=�FI=@��D6=@?1AU=@?ªA>=@?ZBW?1A���EY]eXIA>= F$=�<U���ZBW?$�Y=@]e=@?1A;<F�¦ D\?VF$D6�@=�<�#$ M(j, i+1−j : kfin−j) = M(j, i+1−j : kfin−j)− M(j,i−j)M(i,1:kfin−i)

M(i,0)Bq�Y=��kfin = min(i + Nsurdiag, size(N))

E��Nsurdiag

=�<[Ac~6= ?$EY] ¢$8;= FI=<>H$8�FID6B5�5EY?VBW~ F$= ~hBª]�B�A;8>Dh��= F$HKX$8;=@]eD\=�8 ]e=@] ¢$8;= =�A

size(N))~\= ?$E5] ¢$8>= FI=

~6D\�Y?$=�<�F$HG<>�I<MA;_@]e=54

3�a

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$ N(j) = N(j) − M(j,i−j)N(i)M(i,0)

� B^8>:o<[EY~\H$A>D6E5?GFIHn<>�I<MA;_@]e=�A>8;DhBW?$�YH$~6B5D\8;=cF$=@��D\=�?1A BW~6E58C< #$ A(i) = N(i) − M(i,1:min(Nsurdiag,size(N)−i)A(i+1:min(i+Nsurdiag,size(N)))

M(i,0)

� ��� � �+� �$!��-� ���+�^b�+� ����Z���-�+�7 �b� ��+�+��y=o< FID§¡":@8;=@?1A;=�< ��EYH$8;¢Z=o< A;8;BY��:�=�</XZ=�8>]e=�A>A>=�?YAUF$=���Dh<[HZBW~6D6<>=@8,~\=�A;8;Bq��BWD6~1=�¡"=�� A;H$:�XVB58

~6=�XV8>EY�58CBW]e]e=54 � ? X(=@HIAUFIEY?V��E5¢V<>=@8;�5=�8/~hBc<>E5~6HIA;D\EY? BWXVX$8>EI���$:�=U=@A ~hBP<[EY~\HIA;D\EY? =@d$B5� A;=<>H$8OH$? ] @]e=��58CBWX �$= �"X$H$Dh< �@E5]eXVBW8;=@8�~6=�<O8;:�<>H$~\A;BWA;<�E5¢$A>=@?�HV<�X(E5HV8 FI=�<O��B5~\=�H$8;<OF$=XVBY<��nF$D§¡":@8;=@?1A>=o<�<>E5D\A�X(E5HV8�FI=�<�?$EY] ¢$8;=�<�F$=O?$E�=@HVF$<�F$D§¡":@8;=@?1A;<�4�/B�8;:�<>E5~6HIA>D6E5?wFIH <>�I<MA;_@]e=^~\D6?$:�B5D\8;=^] H$~\A>DhFIDhBW�5EY?VBW~

(E5¢$A>=@?�HwXVBW8�]eD\?VD\]eDh<>BWA>D6E5? FIH

�@8>D\A>_�8>=e(a) =

∑N

j=1 E(f)2(Xj) )=@A FIE5?V�P~hBe��E5?V?VBWDh<><;BW?Z��=�FI=�<

aiF$B5?V<

σ(x) =

n+1∑

i=−1

aiB(x − α

h− i)

X(=@8;]e=�A�F$=c�@B5~6�@H$~6=@89~hB <>E5~6HIA>D6E5?ªB5X$X$8;EI���$:@=σ(x)

FI= ~ ¦ :oT1HVB�A;D\EY?ªFID§¡":@8;=@?1A;D\=�~\~6=c���$EYD6<>D6=X(E5H$8

NX(E5D6?1A;<�:�T1H$D68;:@XVB58[A;D6<�<[HV8

[α, β]4$� B^<[EY~\H$A>D6E5?�B5X$X$8;E����V:@=

σ(x)=�A�~hB ��EY?V� A;D\EY?

=@dIBY� A;=f(x)

<>E5?1A A>E5H$A>=�<^~6=�<eFI=�HIdr�@B5~6�@H$~\:�=�<^BWHId ] @]e=o< X(E5D6?1A;<eF$B5?V<^~\=n]eE�F$H$~\=] ���@B5~6�@H$~ ��< ���M4 �'�5�V4�/B ��EY]eXVBW8CBWDh<[EY?��58CBWX �VD6T1H$=P<[EYHV<U~hB ��E58;]^=�FI=

D(h) = 1N

∗∑N

j=1(σ(xj)− f(xj))2?$:o��=o<><>D§A;= ¢(=�BWHZ��E5HVXKX$~6HV<cF$= �@B5~6�@H$~h<@4ZR9? =@¡(=@A��ª���ZB5T1H$= ?$EYH$�5=�~\~6= ��B5~\=�H$8cFIH XVB5<�

<>E5D\AcFIHK?VE5] ¢V8>= F$= ?$E�=@HZF$<nD6~ �'B5HIAc8;:�<>E5HZFI8>=OHV?K?$E5HV�5=�B5HK<[�I<[A>_�]^= ~\D6?$:�B5D\8;= ] H$~ �

A;D6FIDhBW�YE5?VB5~yT1H$DyX(=@8;]^=@A =@?Z<[H$D\A>=O~6= ��BW~h��H$~yFI=σ(x)

FI:@X(=@?VFVBW?1A FI= �v<>E5D\A FIHv?$E5] ¢$8;=FI=O?VE1=�HVF$< �$=�A�=�? �V?GD6~y=�<[A�XZE1<><>D6¢$~\= F$= �@BW~h��HV~\=�8

D(h)4(¥�=�A;8;Bq��BWD6~�=�<[A�=�¡"=���A>H$: FVBW?V<

~6=�]^EIFIHV~\=O] ����EY]eXVBW8CBWDh<[EY?y4 �'�5�$4� =�A>8CB5��:�F$=���=�< A>8;E5Dh<9��EYH$8>¢(=�<9=�<[A9�5:@8;:�Bq�5=o���P?1HVX$~\E5AUF$BW?Z< ~\= ]^EIFIHV~\=�] ��A;8;BY��=54 �'�5�V4� =�<���EYH$8>¢(=�<OF$=^~hBG<[EY~\H$A>D6E5? BWX$X$8;E �V:@=^=�A FI=�~hB�<>E5~6HIA>D6E5? =�d$BY� A>=�<>E5?1APA>8CB5��:�=�<�Bq�5=o�H$?V= :����V=@~6~\= ?$E58;]�BW~6= BW~6E58C< T1H$= ~6Bn��EY]eXVBW8CBWDh<[EY?G�58CBWX �$DhT1H$= =o<MA�=�¡"=���A>H$:�=^Bq�Y=��OH$?V=:o���$=@~6~\=�~6E5�1BW8;D§A �$]eD6T1H$=�X(E5H$8�HV?$= X$~6HV<��Y8;B5?VFI=O��~hBW8>A>:�4(�y=�<�X(E5D6?1A;< =�A�~6=�<���BW~6=@HV8;<�=�?�@=�<�X(E5D6?YAC<�<[EY?1A�:o��8;D§AC<�F$BW?Z<�A>8;E5Dh< �Z���$D6=@8C<�FID§¡":@8;=@?1AC< #$ <[EY~ ��B5X$X$8;E����V:@=$ <[EY~ ��=@dIBY� A;=$ A>8CB5��: ��b �

�P?�H$X$~6EWAeHIA;D\~6D6<>=GFIE5?V��~6=�<eFIE5?V?$:@=o<^�@E5?1A>=�?�H$=�<^F$BW?Z<^�@=�< �Z���$D6=@8C<^XZEYH$8e�5:@?V:@8;=@8 ~6=�<�Y8;B5X �$=�<�4�� ���VB5T1H$=��E5Dh< �$E5?nX$8;E5X(EY<>=P�^~ ¦ HIA;D\~6Dh<>BWA>=@HV8 F�¦ D\]eX$8;D\]e=�8�~\=�A;8;BY��:PE5¢IA;=@?�Hy4

3�j

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3 �

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$ f ��:o<[H$~\A;BWA;</E5¢IA;=@?�HV</Bq�5=��9~s¦ :�T1HVBWA>D6E5? FID\¡(:�8>=�?1A>D6=@~6~\=�f�=�A,~\=o</F$E5?$?$:�=�</<[H$D6��BW?1A>=o< #$ E(f)(x) = f ′′(x)− 4f ′(x)+4f(x)− 5coshx+4sinhx

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f(0) = 0=�A

f(1) = cosh1� BO<>X$~\D6?$=�HIA;D\~6Dh<[:�=�BOf5�P?$E�=@HVFV< =�A9=o<MA���BW~h��H$~6:@=�<>H$8UfY�5�cXZEYD\?1A;<�45�U¦ D6?1A>=@8;��BW~6~\=�F$=8>=o<[EY~\HIA;D\EY?e=�<[A

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