Introduction à l’analyse numérique

Introduction

à l’analyse numérique

Introduction

à l’analyse numérique

Jacques Rappaz Marco Picasso

Presses polytechniques et universitaires romandes

Les auteurs et l’éditeur remercient l’Ecole polytechnique fédérale de Lausanne dont le soutien financier a rendu possible la publication de cet ouvrage.

L A COLLECTION «E NSEIGNEMENT DES MATHÉMATIQUES » EST ÉDITÉE SOUS LA DIRECTION DU PROFESSEUR R OBERT C. D ALANG

Recherche opérationnelle pour ingénieurs I

Dominique de Werra, Thomas M. Liebling, Jean-François Hêche Recherche opérationnelle pour ingénieurs II

Jean-François Hêche, Thomas M. Liebling, Dominique de Werra Calcul différentiel et intégral

Jacques Douchet et Bruno Zwahlen 1 Fonctions réelles d’une variable réelle 2 Fonctions réelles de plusieurs variables réelles

3 Fonctions réelles d’une variable réelle – Exercices résolus 4 Fonctions réelles de plusieurs variables réelles – Exercices résolus Algèbre linéaire

Aide-mémoire, exercices et applications Robert C. Dalang et Amel Chaabouni Analyse avancée pour ingénieurs Bernard Dacorogna, Chiara Tanteri Initiation aux probabilités Sheldon M. Ross

Cours d’Analyse Srishti D. Chatterji 1 Analyse vectorielle 2 Analyse complexe 3 Equations différentielles

D ANS LA COLLECTION «M ÉTHODES MATHÉMATIQUES POUR L ’ INGÉNIEUR » Introduction à la statistique

Stephan Morgenthaler Aide-mémoire d’analyse Heinrich Matzinger

Les Presses polytechniques et universitaires romandes sont une fondation scientifique dont le but est principalement la diffusion des travaux de l’Ecole polytechnique fédérale de Lausanne ainsi que d’autres universités et écoles d’ingénieurs francophones.

Le catalogue de leurs publications peut être obtenu par courrier aux Presses polytechniques et universitaires romandes,

EPFL – Centre Midi, CH-1015 Lausanne, par E-Mail à ppur@epfl.ch, par téléphone au (0)21 693 41 40, ou par fax au (0)21 693 40 27.

www.ppur.org

ISBN 2-88074-363-X

© 1998, 2000, 2004, Presses polytechniques et universitaires romandes, CH – 1015 Lausanne

Imprimé en Italie Tous droits réservés.

Reproduction, même partielle, sous quelque forme

ou sur quelque support que ce soit, interdite sans l’accord écrit de l’éditeur.

"!#%$&'(*)+-,/.*)#0$&'213$ 4657#%$%8:9;'4<,/465-#%$&+->=4++?4657#%$&4$@$ :$A!3#B6

@#%9;')++?4#C,/+3!3#*@)#D$ +-)-E(%?46$F!3#0$$E13$G@.*,/#%$9H+,/#HI3#0$!KJ(%57LE4+)$L3IM!3(%E4<N

O

(*#0$P @4<#%<#%$%Q%R@(%$E+3E4+)M)2,/(*@465-#S!3#D9H#%$$ 13$ E.%,T#0$!FJ(05-:UE4+):$PV&WJL46!3#X!KJ)

+-@!4<)LE#%F)(09H#%$@$E4Y@#C!3#%$F9*+)):L46$E$@L)9*#%$ZLE+[\+)!4<#0$Z#*)M,U@'(*,U@457#0$*Q^]S'9*)#

!#A9H#0$C(%57LE4+)$XL3I/!3(*@4

O

(*#0$_LEE4#*<#0$`a#*3DbHE@#&@(*a#*EE+-E4(*#c`G12:#-Q-d&4)$E4e8-<#0$

(057U@4<+-)$c!3#fG17a#g#*<43@457#M!3(09H@4

O

#*)7S#%$D'(%)+,/.%)#%$R!3#g!34<ha:$ 4+)i$G;U@4<+-))L4@#8

#%$A(%57LE4+)$"!3#MG12:#jL;L=a+457#B!(%9H@4

O

#*)7&#%$A'(*)+,/.*)#0$A!3#T!34<ha$E4+)>(

O

+3N

@4

O

#S#*k#%$k(05-:UE4+):$!3#SG12:#D'212:#%E=a+457#D!3(%9*E4

O

#%)7#%$'(*)+,/.*)#0$_!3#X@@)$E:+-

V O

4Y@#%$@$ #Ml:)4<#-Q:m_+-A9;'-5-#MG12:#T!KJ(057U@4<+-)F84#*I34$ E#j!#j)+-,j=@#*$E#%$&,T(*E'+3!3#0$

)2,/(%E4657#%$nVo!346$ a+-$E4Y@4<+-)F89*-$E$E(*#0$Mj9*LE(*?-+@4<#0$*QFdA4<):$ 4D+)pE#q!3#r,T(*E'+3!3#0$

!#n!34<ha(%E#%)9H#0$Rl)4<#0$*8Z!KJ(*(*,/#*)7@$&l)46$*8Z!3#

O

+,/#%$&l)46$*8Z!3#B,/(H@'+3!3#%$f$E:#09sE;L#%$%8

#*@9QCJ3E4<46$@UE4+)i!3#M9H#0$D,T(*E'+3!3#0$c)2,/(*@457#0$S)(%9H#0$E$E4<E#f/9H+-))4$@$E)9H#f!KJ+3@4<6$

!#&=$E#cE#%$S57#&WJ4)7E#*@a+6UE4+)/a+12)t,/46L#8L6M!3(*@4

O

LE4+)#*XeJ4)-@(*?-@LE4+)T)7,T(*N

@465-#f!KJ)#f[\+-)9s@4<+-)F82nE(0$ +-<E4+)u!3#f?;L):!$S$E13$G@.*,/#%$S4<)(0L4E#0$X#*R)+)r<4)(%4<@#%$%8

WJ4<)7E(%?;U@4<+-)v!KJ(%57LE4+)$R!4YhZ(*@#*)7E4#*#%$c+;!34)L4@#%$%Q

wxR#*I34$ E#i!3#i)+,n=@#*3Ip+

O

;L?-#%$T9H+-)$E-9H@(%$nVWJ(*E!#>,/LE'(%,UE4657#v#HqV6

@(%$E+3@4<+-))2,/(*@457#/!3#0$g(%57U@4<+-)$fL3Iy!(*@4

O

(*#%$" @4<#%<#%$%QP]S#T4

O

@#T)FJr-$"6

E(*E#*)7@4<+-)!KJbHE@#B)#HI3:+7$ (n#HI3'$G@4Y[C$Ef<#j$EUzG#*%QZm`L"9*+)7E@#8:4<_a+"=3M!3#

[\+-@)4<DLq#%9HE#%X#%$S)+L@4<+-)$X!#&=$E#ALl)u!3#":+-

O

+-4<X=:+-@!#*X!3#AE#%$S@+=.*,/#%$%Q

P#M9H+)7@#*)2v!#g9*#g<4

O

E#g#%$ R)#

O

#%@$E4+)u(*E+ha(%#g!i9*+;$D:+-<139*+4(f57#g#M@#HN

,/4#*gL3@#*f#*):$ #%4<?-)#j!3#%46$"<$E4#*;$A))(*#0$"3I>(*.

O

#%$&4)?-(*)4#*;$A!@#*,/4#*

9*139H#T!3#/WJ{S9H+#Tmk+-<17@#%9;')4657#/|(0!3(*;L#n!#TL$@L))#QZF#/<4

O

E#n#%$ g!4

O

46$ (T#*)~}*

9;':L4<E@#%$%879;':57#"9;':L4<E@#A(*@L)7D<4<N€,Tb%,/#f!34

O

4$E(f#*)r$E#%9HE4+)$%Q2F#%$S@#*,/4.*@#%$D$ #09sN

@4<+-)$B!3#q9;':57#/9;'L4<E@#9H+-)-@4<#%))#%)-B?-(*)(%@<#%,/#*)7g!#%$ML@?,T#%)7@$g$ 4,/#%$%8F<#0$

!#*@)4<.%E#0$`$ #09s@4<+-)$`!3(

O

#*+a#*)7X!3#0$`)+E4+)$_$C9*+,/#HI3#%$%QLCJ O

L)7 Nx!3#%E)4.*@#c$ #09sN

@4<+-)!#/9;':57#/9;'L4<E@#/9H+)7@4<#%)-M!3#0$g#HI3#*;9H469H#0$"9*+@E4?(0$*8Z!3#$ +- @#T57#T#T#%9HE#%

4$@$E#czG?#%A$ +-)i)4

O

#%i!#B9H+,/@(*'#*)$E4<+-)FQ3{C)3l:)F86!3#*@)4.*@#M$ #09sE4+)>!3#n9;'57#

9;':L4<E@#A9H+-)7E4#*)7D!3#0$C)+E#0$C=4=4<+-?;L'4657#%$C#HD9*+,/,/#*)7@4<@#%$S!3#%$ E4)(0$XLIT4)?(*N

)4<#%@$A5-4K3E4<46$E#*)7&/$E4<,n6UE4+)v)2,/(*@457#M!)$c#*;$DE;

O

L3Ir!#f@#%9;'#*;9;'#Q

P#%$R}0‚fE#%,/4<#%@$`9;'4<E@#%$_#HI3a+-$E#*)7C#%$`+-3E4$`!#&=$E#R!3#RWJ)L13$ #c)2,/(*@457#-Q

]S#0$A+-3E46$A$E+)7"9H+-)7E#*)2$A!L)$Aq<LE"!3#0$&+?469H4#*6$A!#j9*9*$@9H4#*)7E4<l:57#j!4$ N

a+)4<=#%$g!)$f<#/9H+-,/,T#%@9*#QaF#%$fv!3#*@)4#*;$"9;'4<E@#%$f$ +-)-g9H+-)$E-9H@(%$"VqWJ)L13$ #

)2,/(%E4657#n!3#0$&(05-:UE4+):$ALIi!3(%E4

O

(%#%$R:LEE4#*<#0$&#Hg$ +-)7"!3#B#%9HE@#j$"!34Yƒq9*4<#

57#/#%$"@(%9H(0!3#*)7;$*QK„+3@#H[\+46$%8aWJ+

O

;L?#nu(*E(/9H+):…Hy!#T$E+EE#T5-#n#%$f@#*,/4.*@#%$

O

$E#%9HE4+)$D!3#%$Sn!#*@)4<#%@$D9;'4Y@E#0$C@#%$ E#%)-cL=a+;!L=<#0$XS!3#0$D(*.

O

#0$C4)?(%)4#*;$S!#

@#*,/4#*&9*129*<#g)4

O

#*;$ 4<@4<@#Q

F#0$D,/LE'(%,UE4657#%$D9H+-)7E#*)2#0$X!)$D9H#f<4

O

E#f$ +-)-D#*)q?-@)!3#&:LEE4#A#*):$ #%4<?-)(*#0$

B#%$j!#*3IpL3@#*;$B3I(*E!4)-;$j!3pE#%,T4#*T9H139H#q!3#0$j$E#%9s@4<+-)$j!#q?(%)4<#r9H4<N

O

4e87?(%)4<#"E@e87,/(%9*)465-#87,/LE(%E46L3IK8-,T469H@+L@#%9;')457#-8-'213$E457#-8-4)3[\+@,U@457#-8

$E13$G@.*,/#%$T!3#>9H+-,T,n)469*LE4+)$T!3#veJ{X9H+-<#im_+17E#09;')465-#r|k(%!3(%@<#i!3#iP$E))#-Q

M9H+,/@(*'#*)$E4<+-)!3#A9H#0$`,U@'(*,U@457#0$C@#%574#*E`)#A=a+))#&9H+-))L46$@$E)9H#c@(%LN

6L=#g!3>9*L69HZ!4YhZ(*@#*)7E4#*K#*R4<)7E(%?;LZ#H&!3#gWJ<?-.*=@#A4)(%4<@#82,U@4<.%E#0$S?-(*)(%@<#*N

,/#*)7g#*)$E#*4?)(*#%$A<+-@$"!3#TuE#%,/4<.%E#j))(%#n!KJ(H@!3#%$g$@9H4#*)7E4<l:57#0$A)4

O

#*;$ 4<@4<@#%$%Q

fL)$T) $E+9*4c!#i9*+)9*4$E4+)F8`57#*657#%$T@(%$E<@U;$/$ +-)-/(*)+)9*(%$/$E)$/!3(*,/+-)$G@@LN

@4<+-)FQK]S#%:#%)!)-08:#%$fLE#*@$"$ #T$ +-)-"#HhZ+;9H(%$"!3#j,4<)7E#%)4")#n9H#% ;L4)#B@4<?-#*

,U@'(*,LE465-#g!L)$c6T[\+@,j6U@4<+-)r!3#B9*#%$c@(%$E<@U;$*Q

F#0$&LE#*@$c@4<#%))#%)-AV/@#*,/#%@9*4<#%o+)$E4<#%&#MmC@+L[\#%$@$E#*ZQnQZ]S'L E#%zG4W8"4YN

@#%9HE#*P!3#X6&9*+<#09sE4+)g,U@'(*,LE465-#%$P!#%$FmXE#0$E$E#%$Pmk+-<17@#%9;')4657#%$&)4

O

#%@$E4Y;L4E#0$

R+-,L)!3#0$*8aa+M$E#%$f#*)9*+;L?-#*,/#*)7;$&#HB$E+)y4<)7E(%Eb*gVr6u=<469*LE4+)!3#T9*#HM+N

O

;L?#-QF#BE;

O

4<!3#j!-9sG12<+-?;L'4<#M(*E(j#*ha#09s@(j

O

#09g?;L)!>$E+4)iL o!L,/#

:Q>+7$ #* E4WQ"FJ#*#/E@+

O

#T469H4Si@#%9*+)):L46$E$@L)9*#j#*j<#0$M@#*,/#*;9H4#*,/#*)7;$M!3#%$B!#*3I

3E#%@$%Q

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

}-Q<} mk+7$ 4<E4+)i!3i@+=.*,/#QgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ }

}-Q/ 0D-$ #g!3#BL?;L)?#uQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ /

}-Q1 wx)-@#*@:+-LE4+)v!3#BP?;L)?-# QgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ 1

}-Q wx)-@#*@:+-LE4+)v!KJ)#g[\+)9HE4+)>9H+-)7E4)7#f:LR)ia+17)t,/#iQfQMQgQMQgQ 

}-Q2 wx)-@#*@:+-LE4+)v!KJ3A#*@,/4Y@# QMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ 4

}-Q5 wx)-@#*@:+-LE4+)rLR4)7E#*

O

L<#0$ QMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ 6

}-Q74 {`I3#*;9H469H#%$ QMQMQfQMQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ })/

}-Q8 9&+L@#%$c=4=4<+-?;L'457#0$S#*&E#%,L;5-#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ })2

:<;>=*#@?A,-')#&B%C%ED=*#&FED AG

/Q<} A(%E4

O

(%#%$D)2,/(*@457#0$R!KJ+-@!E#/}g#H&#%E@#*R!3#f@E+-)9*LE@# QfQMQgQMQgQ }H4

/Q/ A(%E4

O

(%#%$D)2,/(*@457#0$R!KJ+-@!E#/}g#H&#%E@#*R!KJL@@+)!346$ QMQMQfQMQgQMQgQ }.6

/Q1 A(%E4

O

(%#%$D)2,/(*@457#0$R!KJ+-@!E#/}g#H&#%E@#*;$>QMQgQMQgQMQgQMQMQfQMQgQMQgQI/J/

/Q A(%E4

O

(%#%$D)2,/(*@457#0$R!KJ+-@!E#M$E:(%E4#* QMQMQgQMQgQMQgQMQMQfQMQgQMQgQI/K1

/Q2 A(%E4

O

(%#%$D)2,/(*@457#0$c#H&4)-@#*@:+-LE4+) QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQI/U

/Q5 {`I2E;La+6UE4+)i!3# R469;'L;!$E+) QgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQI/J2

/Q74 {`I3#*;9H469H#%$ QMQMQfQMQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQI/L4

/Q8 9&+L@#%$c=4=4<+-?;L'457#0$S#*&E#%,L;5-#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQM1}

NPOQ%('.=RS,-')#&B%T%EDUV=L#FEDUBWUXUSYDUZ[FEDU,S\,-')D\ NBN

1Q<}<]f(*)(%@<4<E(0$QMQfQMQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQM1J1

1Q/ mk+-4!$R!KJ)#g[\+@,j#M!3#B57!@LE@# QMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQM1*4

1Q1 |+@,j<#M!3i@#%9s;L)?-<#iQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ-‚

1Q |+@,j<#M!3# 24,/$ +-) QgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ}

1Q2 |+@,j<#0$D!# ]gL$@$GNxF#%?#%)!3@#&QMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ*/

1Q5 {`I3#*;9H469H#%$ QMQMQfQMQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQL5

1Q74^9&+L@#%$c=4=4<+-?;L'457#0$S#*&E#%,L;5-#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQI2L‚

_^`a=.B&DU'.#%bUc.de.'. #&%=*,S#@JWfg#&#%U,-')#&B%Vih>,AD.LW

j

de.'.ZV,AlkL%U#&'.#%%U=LW

j

deQ') )D\=L')*HV#&%=* me

:Q<} mk+7$ 4<E4+)i!3i@+=.*,/#QgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQI23}

:Q/ {C<4,/4<):UE4+)>!3# ]g$@$D$E&)r#*I2#%,/<#MQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQI2J/

:Q1 d&?+@4<E',/#M!KJ(*4<,/4)U@4<+-)uQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQI2K1

:Q 9&+,n=@#g!FJ+-:(%@LE4+)$Da+RWJ(%<4,/4<):UE4+)>!3#n]g$E$ QgQMQMQfQMQgQMQgQI2L4

:Q2 {C<4,/4<):UE4+)>!3# ]g$@$D

O

#%9g9;')?#%,T#%)7c!3#M4

O

+LMQMQgQMQMQfQMQgQMQgQI2K8

4<4

O

Q5 213$ E.%,T#0$R,LF9*+)!34<E4+))(%$AQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQ 5-‚

Q4 213$ E.%,T#0$&$E@!(HE#%E,/4)(%$%Q >(H@'+3!3#B!3#0$R,/+4)!3@#%$c9*E@(%$ QMQgQMQgQMQ 5

Q8 {`I3#%@9*49*#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQ 5L5

Q6 9A+LE#0$R=4=4<+-?;L'4657#%$D#*c@#*,@57#0$ QMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQ 54

m ;>=*kL+l)#@')#&B%

LU

WU;>=*kL+l)#@')#&B%VUB&Ed

23Q} "(%9*+,/:+7$ 4<E4+)

LU

QgQMQfQMQMQgQMQgQMQgQMQgQMQMQfQMQgQMQgQMQMQgQMQgQMQ 5L6 23Q7/ &E4<4<E(j!#f6!3(09H+,/a+-$E4Y@4<+-)

LU

QgQMQgQMQgQMQMQfQMQgQMQgQMQMQgQMQgQMQM4J/

23Q1 "(%9*+,/:+7$ 4<E4+)

LU

^O #09f9;':L)?-#*,/#*)7R!3#M4

O

+L QMQgQMQgQMQMQgQMQgQMQM4U

23Q oLE@49*#%$R$E12,T(*E@457#0$R!3(Hl:)4<#0$c:+7$ 4<E4

O

#0$*Q

"(%9*+,/:+7$ 4<E4+)i!3#n]S'+-<#0$! 21QfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQM4J2

23Q72 oLE@49*#%$R!3#M=:L)!3# QgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQM4K8

23Q5 {`I3#%@9*49*#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQ 8-‚

23Q4P9A+LE#0$R=4=4<+-?;L'4657#%$D#*c@#*,@57#0$ QMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQ 8L1

<` =)D\')#&B%b[.deQ')*V* &#%U=,A#&*

+ ,-a V=L'"U\U #&'.=*,-'.#&? #(m

5Q} ]f(%)(*;L4Y@(%$%Q >(H@'+3!3#%$A!3# --9H+=4K#Hg]gL:$E$ N 3#*46!3#* QMQgQMQMQgQMQgQMQ 8*2

5Q7/ A)v#*I3#*,/# QMQMQgQMQgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQ 8L6

5Q1 >(*E'+3!3#0$&!3#M@#*6UIU@4<+-)F87,T(*E'+3!3# %$ QMQMQgQMQfQMQgQMQMQgQMQgQMQ 6-‚

5Q >(*E'+3!3#0$&!3i?;!4<#%)-c#HA!v?-@-!34#*)7c9*+)UzG?( QfQMQgQMQMQgQMQgQMQ 6*/

5Q72 {`I3#%@9*49*#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQ 6L8

5Q5 9A+LE#0$R=4=4<+-?;L'4657#%$D#*c@#*,@57#0$ QMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}0‚L/

G'& =*'e %ED=*#&FED +lBDU &[kJ,S&kJD

* ?A,S&D\ +H+H "!$D%U[V,-'Q#&kL[QdeV=L'.#&FED )((m

42Q} ]f(%)(*;L4Y@(%$QMQMQgQMQgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}0‚L2

42Q7/ >(*E'+3!3#B!#MT46$@$E)9H# QMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}0‚*4

42Q1 >(*E'+3!3#B!#MT46$@$E)9H#g4)

O

#*;$ #MQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}0‚J6

42Q >(*E'+3!3#B!# --9H+=4QgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}-}}

42Q72 {`I3#%@9*49*#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}-}*

42Q5 9A+LE#0$R=4=4<+-?;L'4657#%$D#*c@#*,@57#0$ QMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}-}H4

#<f FEDU,-')#&B%U *' Qde.'. l!=FEDU,-')#&B%U %% #&%=*,S#@ B)

8Q} {X57LE4+)$&)+)v<4)(0L4E#0$+*?(*)(*;L4Y@(%$QMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}-}.6

8Q7/ >(*E'+3!3#0$&!3#Ma+4)7DlI2#,*3?-(*)(%@<4<E(0$&QgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}H/3}

8Q1 >(*E'+3!3#B!#n9&#.-c@+)v#H&,/(H@'+3!3#B!3#M6/9H+;!3# QgQMQfQMQgQMQMQgQMQgQMQu}H/U

8Q 213$ E.%,T#0$R)+-)r4<)(%L4@#%$ QgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}H/L4

8Q72 {`I3#%@9*49*#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu})1‚

8Q5 9A+LE#0$R=4=4<+-?;L'4657#%$D#*c@#*,@57#0$ QMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu})1L

<f FEDU,-')#&B%U #0/"=L%('.#&& -N(G

6Q} {X57LE4+)$A!34YhZ(*@#*)7@4<#%<#%$R!3i@#*,/4#*R+-@!3@#1*3?(*)(*;L4Y@(%$ QMQgQMQgQMQu})1*4

6Q7/ mC@+=<.%,T#0$c)2,/(*@457#%,T#%)7c,F:+7$ (0$>QMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}%-‚

6Q1 39;'(*,$R!KJ{C#* QMQgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}%}

6Q >(*E'+3!3#0$&!3# R)?#*N32g3E@T!KJ+;!3E#n/QMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}%*2

6Q72 >(*E'+3!3#B!# R)?-#HN42g3 ;T9*-$E$E465-# QgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}%L5

6Q5 213$ E.%,T#0$&!34<ha(%E#%)7E4#*6$R!3iE#%,/4<#%R+;!3E#vQgQMQMQgQMQfQMQgQMQMQgQMQgQMQu}%4

6Q74 {X57U@4<+-)$R!34<ha(%E#%)-@4<#%<#%$R!FJ+-@!3@#M$ :(%E4#* QgQMQgQMQgQMQMQfQMQgQMQgQu}%L8

6Q8 {`I3#*;9H469H#%$ QMQMQfQMQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu}H23}

6Q6 9&+L@#%$c=4=4<+-?;L'457#0$S#*&E#%,L;5-#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu}H2U

)(c;>#0/"=L%kJ %#& *' =&=%(') %U#

+"DU * +S & ,SD &##@')* DU%#&#&%.#%%* -mBm

}0‚Q} &)i@+=.*,/#ML3Ir<4,/4Y@#%$R)4!34,/#*)$E4+))#% QgQMQgQMQgQMQMQfQMQgQMQgQu}H2J2

}0‚Q7/ >(H@'+3!3#B!3#%$R!4YhZ(*@#*)9*#%$Dl)4<#0$ QgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu}H2K5

}0‚Q1 >(H@'+3!3#B!3#n]gL#* 24<) QgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu}H2L4

}0‚Q >(H@'+3!3#B!KJ(*(*,/#*)7@$Dl:)4$A!3#B!3#*?-E(T} QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu})5}

}0‚Q72 >(H@'+3!3#B!KJ(*(*,/#*)7@$Dl:)4$A!3#B!3#*?-E( / QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu})5L2

}0‚Q5 d&@+^I34<,LE4+)u:L&!34<ha(%E#%)9H#0$Dl)4#%$

!KJ)vE+-=<.%,/#g3Iu4<,/4<E#0$R)+)v4<)(0L4E# QMQMQgQMQgQMQgQMQMQfQMQgQMQgQu})5*4

}0‚Q4 {`I3#*;9H469H#%$ QMQMQfQMQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu})5J6

}0‚Q8 9&+L@#%$c=4=4<+-?;L'457#0$S#*&E#%,L;5-#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu}41

B>%c=*'ea"!$=&=%(')

%U#Z+"DU !,-++Ue#&V,-'.#%

[+S & &#@+'.#FEDU -GBm

}-}Q} mCE+-=.*,/#%$c#%<4<E465-#%$R#*D[\+-E,n6UE4+)

O

L@4LE4+))#*<# QMQMQfQMQgQMQgQu}4K2

}-}Q7/ {C<(%,/#*)7@$cl:)4$RE@4)?6L4@#%$D!3#B!3#*?-E(T} QMQMQgQMQgQMQgQMQMQfQMQgQMQgQu}46

}-}Q1 &)i#HI3#*,/#gLEE469H<4#*iQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu})8}

}-}Q {X$G@4<,LE4+)$R!KJ#*@@#*;$D#H&,/(H@'+3!3#%$&!#B!3#*?-E(g$ :(%E4#*AQMQfQMQgQMQgQu})8L2

}-}Q72 {`I3#*;9H469H#%$ QMQMQfQMQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu})8J5

}-}Q5 9&+L@#%$c=4=4<+-?;L'457#0$S#*&E#%,L;5-#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQu})6J1

A: +U+e#&,A'.#&B% U +H &n+ ,-,-lB&#FEDULW

U[, k ,A&DU (m

}H/3Q} {X57U@4<+-)v!3#M6/9;'L#*M} #HA!4YhZ(*@#*)9*#%$Dl)4<#0$oQgQMQgQMQMQfQMQgQMQgQu})6L2

}H/3Q7/ {X57U@4<+-)v!3#M6/9;'L#*M} #H&(%<(%,/#*)7@$cl:)4$ QMQgQMQgQMQMQfQMQgQMQgQu})6J8

}H/3Q1~mCE+-=.*,/#%$c@=:+-<4657#%$ / #H&#*;$RLE+^I34,/LE4+)$&QMQMQfQMQgQMQgQ /‚L/

}H/3Q &)i#HI3#*,/#gLEE469H<4#*iQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ /‚L

}H/3Q72 {`I3#*;9H469H#%$ QMQMQfQMQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ /‚L2

}H/3Q5 9&+L@#%$c=4=4<+-?;L'457#0$S#*&E#%,L;5-#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ /‚J8

-N +U+e#&,A'.#&B% U +S &(dE+l*)"#&FED*LW

f FED,-'.#%T 'Q,A%Q+"SH' *' =FED,-'.#%T %U :%(

})1Q} {X57U@4<+-)v!3#g@@)$E:+- M} #*A!34<ha(%E#%)9H#0$Dl)4#%$ QgQMQgQMQMQfQMQgQMQgQ /‚J6

})1Q7/ {X57U@4<+-)v!3#0$R+):!3#%$g} #HA!34<ha(%E#%)9H#0$Dl)4#%$gQgQMQgQMQgQMQMQfQMQgQMQgQ /}.1

})1Q1~{X57U@4<+-)$R!3#%$&+)!3#0$ / #*&(*(*,/#*)7@$Dl:)4$TQMQgQMQgQMQgQMQMQfQMQgQMQgQ /}.8

})1Q {X57U@4<+-)v!3#g@@)$E:+- M} )+)i<4)(0L4E# QMQMQgQMQgQMQgQMQMQfQMQgQMQgQ /L/L‚

})1Q72 {`I3#*;9H469H#%$ QMQMQfQMQgQMQgQMQMQgQMQgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ /L/J/

})1Q5 9&+L@#%$c=4=4<+-?;L'457#0$S#*&E#%,L;5-#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ /L/K5

_ +U+e#&,A'.#&B% U +S &ZUckL%(?BkK'.#% #0/"DU)#&B% ::

}%Q} &)i@+=.*,/#M!3#M9H+-)

O

#%9HE4+)3Nx!34<ha$E4+)

$G;UE4+))L4E#f#HA!4YhZ(*@#*)9*#%$Dl)4<#0$ QMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ /L/K6

}%Q7/ &)i@+=.*,/#M!3#M9H+-)

O

#%9HE4+)3Nx!34<ha$E4+)

$G;UE4+))L4E#f#H&(%<(%,/#*)7@$Dl)4$ QgQMQfQMQgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQ /J1L

}*:Q1~mC@+=<.%,T#0$c=46!34<,/#%)$ 4+))#*6$&!3#M9H+-)

O

#%9HE4+)3Nx!34<ha:$ 4+) QgQMQMQgQMQgQMQ /J1*4

}*:Q {`I3#%@9*49*#%$ QgQMQMQgQMQgQMQgQMQMQfQMQgQMQgQMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQ /J1J6

}*:Q2C9A+LE#0$R=4=4<+-?;L'4657#%$D#*c@#*,@57#0$ QMQgQMQMQgQMQfQMQgQMQMQgQMQgQMQ /L

(R R c

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

3a+-$E+)$S5-#AWJ+-)

O

#%4<#A9;'#*;9;'#%C)qa+12)t,/#

p

!3#"!3#%?@(

n ≥ 0

^574W8-a+

!#%$

O

<#%@$

t 0

8

t 1

8

t 2

8

. . .

⁸

t n

!346$ E4)9s@#%$X!+))(%#%$%8@#*))#&#%$

O

<#%@$

p 0

8

p 1

8

p 2

8

. . .

⁸

p n

@#%$Ea#%9s@4

O

#*,/#*)7089LJ#%$ NxVUNx!34<@#

p(t j ) = p j

a+

0 ≤ j ≤ n.

^{0 }-Q<}21}

A)#M,L)4.*@#ML:L@#*,/,/#*)7&$ 4,/<#M!3#M@(%$E+:!3E#g9H#ME+-=<.%,/#"#0$GA!KJ(%9*E4@#

p(t) = a 0 + a 1 t + a 2 t ² + · · · + a n t ⁿ

^{0 }Q7/31}

+54

a 0

8

a 1

8

a 2

8

. . .

⁸

a n

$ +-)7P!3#%$9H+2#Hƒq9H4#*)7;$F574-!3#

O

E+-)-Fb*E@#C!3(H@#*@,/4<)(0$

0

9*4<@#*,/#%)-08

$E4F#%$R9*+7#*ƒq9H4#*)7@$

a j

8

0 ≤ j ≤ n

$E+)7R9H+-))2$RL+;$D<#ga+17)t,/#

p

#0$G&9*+))261HQF#%$

0

n + 1

1D@#*6U@4<+-)$

0

}-Q<}21S$%J(%9H@4

O

#*)7RL+;$ *

a 0 + a 1 t j + a 2 t ² _j + a 3 t ³ _j + · · · + a n t ⁿ _j = p j , 0 ≤ j ≤ n.

^{0 }-Q171}

mX4$@57#g<#0$

O

<#%;$

t j

#*

p j

8

0 ≤ j ≤ n

82$E+)7R9H+-))2#%$%8-#%$D@#*6U@4<+-)$

0

}Q151C[\+@,T#%)7

) $ 13$ E.%,T#q!#

(n + 1)

(%57LE4+)$BV

(n + 1)

4)9H+-))2#%$

a 0

8

a 1

8

a 2

8

. . .

⁸

a n

Q A)#

,)4<.%E#M!34<hZ(*@#*)7E#B!KJ(%9*E4@#

0

}Q151D#%$ R/$E4

O

)7E#Q

3+4<

T

⁶

(n + 1) × (n + 1)

,LE@49*#g!(Hl)4#M *

T =



 

 

 



1 t 0 t ² ₀ t ³ ₀ . . . t ⁿ ₀ 1 t 1 t ² ₁ t ³ ₁ . . . t ⁿ ₁ 1 t 2 t ² ₂ t ³ ₂ . . . t ⁿ ₂ 1 t 3 t ² ₃ t ³ ₃ . . . t ⁿ ₃

· · · · . . . · 1 t n t ² _n t ³ _n . . . t ⁿ _n



 

 

 

 .

;>=

%#@')#&B% W98;:=<?>-@=ACBD:FEG>IH<KJ

T

J>MLONQPRSPFLTBMAVUJW@3JYX6PFE @3JBMRZ:=E @5JIP=>>:[UAV\]J^PF<3_

`

:=ACE6LV>

t 0

^a

t 1

^a

t 2

^a

. . .

^a

t n

^b

24

~a

^#H

~ p

^{$ +-)-R#%$}

(n + 1)

^NO #%9s@#*;$c9H+-<+-))#%$c$E4

O

)7@$ *

~a =



 

 

  a 0

a 1

a 2

QQQ

a n



 

 

 

, p ~ =



 

 

  p 0

p 1

p 2

QQQ

p n



 

 

  ,

)+-$ca+

O

+)$c(09H@4<@#

0

}Q151D$E+$D[\+-E,/#g,UE@469H4#*<#*

T~a = ~ p.

^{0 }Q71}

dA4<)$E4W8-#A@+=<.%,T#"9H+):$ 46$G;L)7SVB9;'#*;9;'#*C#Aa+12)t,/#

p

^{$ELE46$G[4$@L)7 0} }Q}[1`a#*c$E#

@(%!34<@#MV/E(0$ +-!3@#f#B$E12$ E.%,/#M<4)(%4<@#

0

}-Q?1s89J#0$GEN€VLN€!4<@#fV!(HE#%E,/4)#*

~a

^46$E57#

T

^#H

~ p

$ +-)7R9*+))2$%Q

R(0$ +-!3@#")i$E12$ E.%,/#"4<)(%L4@#f!3#

(n + 1)

(%57LE4+)$cV

(n + 1)

4)9H+-))2#%$D)FJ#%$

-$D)#f-9;'#A@E4

O

4<#

0

9;':LFQ28A2n#H 571sQ3Pn,/(HE'+2!#g57#g)+-$

O

#%)+)$c!3#g!3(09H@4<@#

a+g@E+-

O

#%"#a+17)t,/#

p

^)PJ#0$GM-$g)#/=:+-))#,/(H@'+3!3#Q fL):$gi$E4Y@#)+:$

@(%$E#*)7@+)$D)#M,/(*E'+3!3#M$&-$G@9H4#*:$ #f:+-A9*+)$ E@4E#f<#ga+12)t,/#

p

^Q

. #.&.

wx_#%$ A[-9H4<#/!3#j@(%$E+:!3E#j<#nE+-=.*,/#

0

}Q}[1R+;$@5-#ME+-3E#0$"#%$

O

L#*;$

p j

$E+)7

(%?-L#%$AV *(%E+u$@L[_)#8Z574#0$G"lI3(*#jVo}Q 3+4<

k

)#*)7E4#*"!+))(j#*)7E@#

0

^#*

n

^#H

$Ea+-$E+)$M57#/WJ+-)pL4<

p k = 1

^#*

p j = 0

^a+

j 6 = k

^{Q 2+-4Y}

ϕ k

6v[\+)9HE4+) !3#

t

!3(*l)4#ML

ϕ k (t) = (t − t 0 )(t − t 1 ) · · · (t − t k−1 )(t − t k+1 ) · · · (t − t n )

(t k − t 0 )(t k − t 1 ) · · · (t k − t k−1 )(t k − t k+1 ) · · · (t k − t n ) .

^{0 }-Q251}

]S6L4E#%,/#*)7%80<#D)2,/(*;U@#*_!3#

ϕ k

#%$ k)n@+3!34<_!#

n

^@#*@,/#%$

(t − t j )

^{O #09}

j 6 = k

^#H

#0$GX!3+-)9c)/a+17)t,/#R!3#&!3#*?-E(

n

^#*)

t

Q-P#R!3(*)+-,/4<)LE#%X!3#

ϕ k

#%$ C)#&9*+)$ @)-@#

#*&4<K#0$GA<+-@$S[9*4<#M!3#

O

(%E4<l#%c57#

(i) ϕ k

#0$GR)ia+17)t,/#M!3#B!3#*?-E(

n, (ii) ϕ k (t j ) = 0

^$E4

j 6 = k, 0 ≤ j ≤ n, (iii) ϕ k (t k ) = 1.

d 9;':57#g:+-4<)7

t k

8:)+:$&

O

+-)$c!3+)9M-$E$E+39H4(g)>:+-<12)t-,/#

ϕ k

!#B!3#*?-E(

n

^{O L6L)7}

)i#%)

t k

#H!%(*@+T3Ir3E@#%$ca+4)7@$

t j

8

j 6 = k

^Q

F#0$&a+12)t,/#%$

ϕ 0

8

ϕ 1

8

ϕ 2

8

. . .

⁸

ϕ n

$ +-)7A<4)(0L4E#%,T#%)7&4<)!(*a#*)!)7@$%Q{C)o#*ha#*

$E4

α 0

8

α 1

8

α 2

8

. . .

⁸

α n

$E+)7

(n + 1)

)+,n=@#%$B@(*#%$BE#%$T57#

P n

j=0 α j ϕ j (t) = 0

⁸

∀ t ∈ R

^8:L+;$Sa+

t = t k

)+$&+=3@#*)+-)$ *

0 = X n j=0

α j ϕ j (t k )

| {z }

0

$E4

j6=k 1

$E4

j=k

= α k ,

#*&L&9*+)$E(%57#%)7D@+$c#%$

α k

8

k = 0, 1, . . . , n

$ +-)7c46!3#%)-@457#%,/#*)7R)7$%Q 9A+L@+)$n,L4)-@#*))7

P n

WJ#%$E9*#

O

#09s@+@4<#%X[\+-E,/(u:LnE+:$j#%$Ta+17)t,/#%$T!3#

!#*?@(v4<)3[\(%E4#*T+- (%?-LRV

n

Q_wxR#%$ /=4<#%) 9H+-))2 57#

P n

#0$G)~#%$E-9H#

O

#%9HE+@4#*

!#g!4<,/#*):$ 4+)

(n + 1)

#H&57#M$@j=-$ #M9%L)+-)457#f#%$ &!3+))(*#fL

1, t, t ² , t ³ , . . . , t ⁿ

^Q

P#[L4<i57#

ϕ 0 , ϕ 1 , ϕ 2 , . . . , ϕ n

$ +-4<#%)-i!3#%$va+12)t,/#%$i!3# !3#%?@(

n

4)(%4<@#*,/#*)7 4)!3(%:#%)!)-;$D,T+-)7E@#"57#g9H#%$D!3#%E)4#*;$`[\+-E,/#%)-cL$@$E4:)#"=-$ #f!3#

P n

Q7dA4<):$ 4Z)+-$

-!3+E#*@+):$S6!3(*l)4<E4+)>$ 4

O

L)7@#1*

;>=

%#@')#&B% W: 8;:=<?>Z@=ACBD:FEG> H<KJ

ϕ 0 , ϕ 1 , ϕ 2 , . . . , ϕ n

JM>ML-NQPP=>JY@3JP FBDPFE7J@5J

P n

P2>>: UAV\J PF<3_

`

:=ACE6LV>

t 0 , t 1 , t 2 , . . . , t n

^b

f e + &YW mXE#%)+)$

n = 2

⁸

t 0 = − 1

⁸

t 1 = 0

⁸

t 2 = 1

QPT=-$ #M!3#BL?-@)?#f!3#

P 2

$@$E+29*4<(%#RL3I/:+-4<)7;$

− 1

⁸

0

^#*

1

#%$ C[\+-E,/(*#ALC#%$X:+-<12)t-,/#%$

ϕ 0 , ϕ 1 , ϕ 2

!3(*l)46$

:L

ϕ 0 (t) ≡ (t − t 1 )(t − t 2 ) (t 0 − t 1 )(t 0 − t 2 ) = 1

2 t(t − 1) = 1 2 t ² − 1

2 t;

^{0 }Q551}

ϕ 1 (t) ≡ (t − t 0 )(t − t 2 )

(t 1 − t 0 )(t 1 − t 2 ) = − (t + 1)(t − 1) = 1 − t ² ;

^{0 }Q4F1}

ϕ 2 (t) ≡ (t − t 0 )(t − t 1 )

(t 2 − t 0 )(t 2 − t 1 ) = 1

2 (t + 1)t = 1 2 t ² + 1

2 t.

^{0 }Q851}

P#%$c?-@'#%$c!3#

ϕ 0

8

ϕ 1

8

ϕ 2

$EReJ4)-@#*

O

L<#

[ − 1, +1]

)4657#*,/#%)-A$E+)7cE#%@(%$E#*)7E(0$

!L)$R6nl?E#/}Q}Q

-0.2 0 0.2 0.4 0.6 0.8 1

-1 0 1

ϕ 0

ϕ 1

ϕ 2

X #R\W"W

0c$E#f!3#BL?-@)?#f!3#

P 2

$@$ +39*4<(%#g3Iua+4)7@$

− 1

⁸

0

^#H

1

^Q

;O. &$ * )O Y. )& .

&#

O

#%)+):$CL@+=<.%,T#

0

}Q}[1`9*+)$E46$G;L)7XVB9;'#%@9;'#%`)qa+12)t,/#

p

!#A!3#*?-E(

n

574Z@#*))#f!#%$

O

L#*;$S!+))(%#%$

p 0

8

p 1

8

p 2

8

. . .

⁸

p n

#%)r!3#0$D:+-4<)7@$c!346$ E4)9s;$D!+))(0$

t 0

8

t 1

8

t 2

8

. . .

⁸

t n

Q

2+-4Y

ϕ 0 , ϕ 1 , ϕ 2 , . . . , ϕ n

6T=$E#f!#BPL?-@)?#"!3#

P n

-$E$E+39H4(*#gL3Ir:+-4<)7@$

t 0

8

t 1

8

t 2

8

. . .

⁸

t n

QdA<+-@$c#Ma+17)t,/#

p

9;'#*;9;'(g#%$ R!(Hl)4PL *

p(t) = p 0 ϕ 0 (t) + p 1 ϕ 1 (t) + · · · + p n ϕ n (t) = X n j=0

p j ϕ j (t).

^{0 }-Q671}

{C)y#HhZ#H08K46$E57#

p

#0$Gg)#/9H+-,j=4)L46$E+)>4<)(0L4E#T!3#

(n + 1)

:+-<12)t-,/#%$

ϕ 0

8

ϕ 1

8

ϕ 2

8

. . .

⁸

ϕ n

!3#&!3#%?@(

n

^8<+-@$

p

#%$ C4YN€,/b*,/#R)/a+17)t,/#R!3#&!3#*?-E(

n

^89J#0$GEN€VLN€!4<@#

p ∈ P n

QjJLE@#qLE%8$ 4S)+$j3E44$E+)$B<#0$B@+@4(HE(0$B!3#0$Ba+17)t,/#%$

ϕ j

8k)+:$

O

+)$Da+

k = 0, 1, 2, . . . , n

^*

p(t k ) =

X n j=0

p j ϕ j (t k )

| {z }

0

$E4

j6=k 1

$E4

j=k

= p k

⁰

}Q}%‚71

574F#0$GR=4#*)i6TE#%LE4+)

0

}Q}[1HQ

wxX#0$GB4,/:+- ;L)7j!3#q@#*,@57#%g57#)+$B

O

+)$M9*+)$ E@4<M#HI349*4Y@#*,/#*)7j)#

$E+3E4+)o!o@+=.*,/#

0

}Q}[1c#*f9H#09H4k:+-A)PJ4,/:+- @#n57#*#%$

O

<#%@$

p 0 , p 1 , . . . , p n

!3+-))(%#%$%Q]S#09H4Z,/+)7E@#"57#g#f$E13$G@.*,/#

0

}Q71SjE+LzG+;$D)#g$ +-<3@4<+-)

~a

^a+c)FJ4<,TN

a+EE#u57#%

~

p

#HT4<):$ 4S>,LE@49*#r!3#C)!3#%E,/+):!3#

T

#%$ n@(*?-4<.%E#-QP$ +-<3@4<+-)

0

}Q651D!3i@+=.*,/#

0

}-Q<}21X#0$GA!+)9f)4657#Q

f e +iBW:

„E+-

O

#%X)r:+-<12)t-,/#f!3#g!3#%?@( /n5-4a#%)

t 0 = − 1

^{O L3}

p 0 = 8

^82#%)

t 1 = 0

^{O 3}

p 1 = 3

^#HR#*)

t 2 = 1

^{O 3}

p 2 = 6

^Q

nJE.0$g9H#574`@(%9H.0!3#8Z)+-$g

O

+-)$

p(t) = 8ϕ 0 (t) + 3ϕ 1 (t) + 6ϕ 2 (t)

⁺³⁴

ϕ 0

8

ϕ 1

#H

ϕ 2

$E+)7&!3+))(%$c

0

}-Q571€N

0

}Q851HQA9&+-$c+=3@#*)+-)$R!3+-)91*

p(t) = 8 1

2 t ² − 1 2 t

+ 3(1 − t ² ) + 6 1

2 t ² + 1 2 t

= 4t ² − t + 3.

. &O$ * ^O "##.= O S"Y.

$ )& "#%$ *Z ;,/.

2+-4YD)#R[\+-)9s@4<+-)

f : R → R

9H+-)-@4<)2#&!+))(%#&#Hc$ +-4Y

t 0

8

t 1

8

t 2

8

. . .

⁸

t n

8

(n + 1)

a+4)7@$R!346$G@4<):9s@$A!3+))(%$%Q

9&+-$T9;'#*;9;'+):$B,L4)7E#*):L)7nVo4)7E#%Ea+#*

f

^{Ln) a+17)t,/#}

p

^!#r!3#%?@(

n

3Iq:+-4<)7;$

t j

8

0 ≤ j ≤ n

^829J#0$GEN€VLN€!4<@#A)+-$R9;'#%@9;'+-)$S)ra+12)t,/#

p

!3#M!3#*?-E(

n

@#*P57#

p(t j ) = f (t j ), 0 ≤ j ≤ n.

^0}Q}}[1

24

f (t)

#%$ M!3+))(*#8K<+-@$"#*)y:+7$E)-

p j = f (t j )

⁸

0 ≤ j ≤ n

^{#*f#%)y$E4O )-M9H#/574}

#0$Gf[L4<g!L)$f6r$E#%9HE4+) }-Q18:)+-$f+=3@#*)+-)$

p(t) = P n

j=0 p j ϕ j (t)

^+34#%$

ϕ j

8

0 ≤ j ≤ n

8a[\+@,T#%)7"6q=-$ #T!3#TL?-@)?#B!3#

P n

-$E$E+39H4(*#j3I>:+-4<)7;$

t 0 , t 1 , t 2 , . . . , t n

Q

T$E+3@4<+-)v!3i@+=<.%,T#

0

}Q}}[1S#0$G&!3+-)9g!3(Hl)4<#M *

p(t) = X n j=0

f (t j )ϕ j (t) ∀ t ∈ R.

^0}Q})/51

;>=

%#@')#&B% WN E@=ACBDPZH<KJINQJ

`

:FN=E FRSJ

p

^{@5\ EKA ` P=B bSJ>MLNACEKLJB ` :=NQPFEKL @5J}

f

^@5J;@5JFBD\

n

^{P=<3_ `} :FACEKLT>

t 0 , t 1 , t 2 , . . . , t n

^b

f e + &YWN 3+4<

f

6j[\+-)9s@4<+-)v!(Hl)4#gL

f (t) = e ^t

^{Q„E+- O #%SWJ}4)7E#%Ea+6L)7&!3#

f

!3#M!3#%?@(

2

LIu:+-4<)7;$

− 1

⁸

0

^#H

1

^Q

34K)+$D@#*@#*)+)$Sj[\+@,j<#

0

}-Q<}H/31s87)+-$c

O

+):$

p(t) = e ⁻¹ ϕ 0 (t) + e ⁰ ϕ 1 (t) + eϕ 2 (t)

⁺³⁴

ϕ 0 , ϕ 1 , ϕ 2

$ +-)7c!3+-))(0$S:L

0

}-Q571s8

0

}-Q7431C#*

0

}Q851HQd&4)$ 4F!+)9")+:$S+-=3E#*N

)+)$

p(t) = 1 e

1 2 t ² − 1

2 t

+ (1 − t ² ) + e 1

2 t ² + 1 2 t

= 1

2e − 1 + e 2

t ² +

e 2 − 1

2e

t + 1.

/l?-E#r}Q7//,/+)7E@#B#j?-@'#j!3#n[\+)9HE4+)

f

#*f$E+)o4)7E#*@a+6L)7"!3#T!3#*?-E(

2

^3Iua+4)7@$

− 1

⁸

0

^#H

1

^Q

0 0.5 1 1.5 2 2.5 3

-1 0 1

f p

X #R\WlW: wx)7E#%Ea+6U@4<+-)u!3#A6M[\+-)9s@4<+-)

f

!3(*l)4#":L

f (t) = e ^t

S)r:+-<12)t-,/#

p

!#B!3#*?-E(

2

L3Ir:+-4<)7@$

− 1

⁸

0

^#*

1

^Q

3+4<v,L4)7E#%)L)7i)#y[\+)9HE4+)

f : [a, b] → R

9H+-)-@4<)2#y#Ho!3+-))(*# $Er)

4)7E#%

O

L#A[\#%E,/(

[a, b]

^{Q 2+4<}

n

)v#*)7E4#*ca+-$E4Y@4Y[k#HR9H+-)$ 46!3(%E+-)$X#g9*$D+54u#%$D:+-4<)7;$

t j

8

j = 0, 1, 2, . . . , n

8C$E+)7/(%5746!34$ E@4=(%$!L)$

[a, b]

^8C9LJ#%$ NxVUNx!34<@#

t j = a + jh

⁸

j = 0, 1, 2, . . . , n

^{8a O #%9}

h = (b − a)/n

^{Q 2+4<}

p

^WJ4)7E#*@a+6L)7f!3#

f

!3#/!3#*?-E(

n

^L3I

a+4)7@$

t 0 , t 1 , . . . , t n

57#i)+:$n)+LE#%E+-)$

p n

:+-/=4#*) ,/+)7E@#*57FJ4<&!3(%:#%)! !3#

WJ#*)7E4#*

n

9;'+46$E4FLi!3(* 0Q jJLE.0$

0

}Q})/51s8

p n

#0$G&!3(*l)4F *

p n (t) = X n j=0

f (t j )ϕ j (t),

⁰}-Q<})151

+54

ϕ 0 , ϕ 1 , . . . , ϕ n

#%$ C6g=$E#R!3#&P?;L)?-#S!3#

P n

$@$E+29*4<(%#RV

t 0 , t 1 , . . . , t n

Q $")/:#%3

,/+-)-@E#%R<#g@(%$E<@U&$E4

O

)7 *

=SYBW&6<

`3`

:2>:FEG> H<KJ

f

^>:=ACL

(n + 1)

^{:FA> U:FEKLACE} RZJE6L)@5\BMA=P NQJ >M< B N ACE6LJB=P=NCNQJ

[a, b]

b N:=B>^>MA

p n

J>ML @3\ E6A

`

PFB

b

E :F<?>)P2:FEG>

t∈[a,b] max | f (t) − p n (t) | ≤ 1 2(n + 1)

b − a n

(n+1)

t∈[a,b] max | f ⁽ⁿ⁺¹⁾ (t) |

^0}Q}*?1

:

f ⁽ⁿ⁺¹⁾ (t) = d ⁿ⁺¹ f (t)/dt ⁿ⁺¹

^b

CJ4)(*?7L4Y@(

0

}Q}*?1#%$ C)#R#%$ E4,U@4<+-)/!FJ#%E@#*`#%)7E@#D6f[\+):9sE4+)

f

#HX$E+)/4<)7@#*EN a+6L)7

p n

!#A!3#*?-E(

n

^3ITa+4)7@$

t 0 , t 1 , t 2 , . . . , t n

(%574E(%LEE46$`!L):$

[a, b]

^{Q7d @4<+-E4}

)+-$f:+-@E4+)$fa#*)$E#*B57#9*#H @#/#*@E#%M9H+)

O

#*@?#

O

#*;$%(*@+r+;$@5-#

n

^{E#*):! O #*;$}

WJ4)3l)4F46$E57#M)+$R

O

+)$

n→∞ lim 1 2(n + 1)

b − a n

(n+1)

= 0.

{C) @(%<4<E(-8C9H#HEE#>Uƒ@,UE4+)~#%$ q$ +-

O

#%)7n[$E$E#>9*

max t∈[a,b] | f ⁽ⁿ⁺¹⁾ (t) |

^a#*3

9*E+@E#_@E.0$F;L46!3#*,/#%)-k

O

#%9

n

Q^]S#X'(*)+-,/.*)#C#0$Gk4<$G@E(D!L):$FWJ#*I2#%,/<#S$ 4

O

L)7%Q

f e +iBW$_ ` D%URB

2+-4Y

f (t) = 1/(1 + 25t ² )

^57#AWJ+)r9*+)$E46!3.*@#"$EDeJ4<)7@#*EN

O

L#

[ − 1, +1]

Q2Pn[\+-)9s@4<+-)

f (t)

#0$GR4)3l)4,/#*)7A!3(%E4

O

L=#M$ cWJ4)7E#%

O

L#

[ − 1, +1]

#*

| f ⁽ⁿ⁾ (1) |

^{!3# O} 4<#%)-c@E.0$S;L4!3#%,/#*)7D?-@)!q+;$@5-#

n

^{@#*)! O #%@$SWJ}4)3l)4WQBl?-@#

}-Q1/,/+)7@E#B$E+)i4)-@#*@:+-)7

p n

!3#j!#*?@(

n

L3Ii:+-4<)7;$

t j = − 1 + 2j/n

⁸

j = 0

⁸

1

⁸

. . .

⁸

n

^8:+-

n = 5

^#H

n = 10

^Q

-0.5 0 0.5 1 1.5 2

-1 -0.5 0 0.5 1

p f 5

p 10

X #R\WBWN

wx)7E#%Ea+6L)7B!3#

f (t) = 1/(1 + 25t ² )

!3#u!3#%?@( 2v#*u}%‚i

O

#%9/!3#%$B:+-4<)7@$

(05-4<@(* @4$%Q

9&+-$v+=$E#*

O

+-)$q57#-8c

O

+46$ 4)L?-#!3#0$u#*I7@E(%,/4Y@(%$i!3#eJ4<)7@#*

O

<#

[ − 1, +1]

⁸

WJ4)7E#%Ea+6L)7rE(0$ #%)-@#!#y?-@)!3#%$r+-$@9H4LE4+)$

0

4)$G;L=44Y@(%$i)2,/(*@457#0$]1HQ 9&+:$

9*+)9*<+-)$P!+)9S5-PJ4-)PJ#0$Gk$4<):!3457(D!KJ4<)7E#%Ea+#*k)#X[\+)9HE4+)B:LP)ja+17)t,/#

!3#B!#*?@(

n

^{(%<# O} (g#*)i!3#%$&:+-4<)7@$

t 0 , t 1 , . . . , t n

(05-4!346$ E@4<=(%$%Q

m`L&9*+)7E@#8$E4K)+:$R9;'+46$ 46$@$ +-)$D<#0$c:+-4<)7@$A!34Y;$&!3#M„k9;'#*=2139;'#Hh

t j = a + (b − a) 2

1 + cos (2j + 1)π 2(n + 1)

, j = 0, 1, 2, . . . , n,

a+D9*+)$ E@4E#RWJ4)7E#%Ea+6L)7

p n

!3#

f

^{82L+;$`WJ#%E@#*}

max t∈[a,b] | f (t) − p n (t) |

^E#%)!

O

#*;$ *(%E+T<+-@$@57#

n

^{@#*)! O #%@$DWJ}4)3l)4e8a9H+,/,/#g<#M,/+-)-@E#f6Tl?@#T}QQ

-0.5 0 0.5 1 1.5 2

-1 -0.5 0 0.5 1

f p 10

X #R\W BW$_ wx)7E#%Ea+6L)7!3#

f (t) = 1/(1 + 25t ² )

!3#>!3#*?-E(}%‚ O

#09u!3#0$a+4)-;$!3#

„k9;'#%=7139;'#*hkQ

;O. &$ * )O . & ,/O.

P#%$AE+-=.*,/#%$"!KJ4)7E#%Ea+6U@4<+-)o5-#B)+:$

O

#%)+):$A!3#j9*+)$E46!3(*@#*A$E+)7"!#%$

! "# $%&"(' ")'#+* 4<6$g[\+-)-M4)7E#*

O

#%)4M<#0$

O

<#%;$M!3#:+N

12)t,/#0$n#*q!3#u[\+-)9s@4<+-)$T#*) 9H#% ;L4)$ja+4)7@$%8`,L46$j)#v@4<#%))#*)7-$T9*+,/3@#v!3#0$

!(*@4

O

(*#%$%QUwx#*I34$ E#&!KJLE@#%$_@+=.*,/#0$C!KJ4<)7E#%Ea+6U@4<+-)na+C<#0$E57#%$_#%$

O

<#%;$C!3#

p(t)

#HA!#f6/!3(*@4

O

(*#

p ⁰ (t)

$E+)7R!3+))(*#0$c#*)i9H#*E@4<):$Sa+4)7@$%Q$")u:L@<#M!)$R9H#M9*-$

!FJ4)7E#*@a+6UE4+)>!KJ3&#%E,/4<E#-Qm_+&4<$ E@#*R)+LE@#g@+a+-$%83)+$&E(0$ #%)7E+):$c)>$ #%

#*I3#*,/#"574#0$GSeJ4<)7@#*@:+-LE4+)!KJ3&#*@,/4Y@#AX!3#0$X9H=457#0$

0

:+-<12)t-,T#0$X!3#"!3#*?-E(

171sQ

3+4<

t 0 < t 1

!3#*3I a+4)7@$/!3+-))(%$n#H$E+4<

p 0 , p 1 , p ⁰ ₀ , p ⁰ ₁

57LE@#u)+-,j=@#%$n@(*#*6$

!+))(0$*Q9&+:$R9;'#*;9;'+):$S)va+12)t,/#

p

!3#B!3#%?@( 1T@#*F57#

p(t 0 ) = p 0 , p(t 1 ) = p 1 ,

⁰}-Q<}H231

p ⁰ (t 0 ) = p ⁰ ₀ , p ⁰ (t 1 ) = p ⁰ ₁ ,

⁰}-Q<})551 +54

p ⁰ (t)

#%$ &/!3(%E4

O

(%#f!#

p

i:+-4<)7

t

^Q

P#%$_9H+-)!34<E4+)$

0

}Q})231P4,Ta+-$E#*)7_

O

L#*`!3#

p

^#*)

t 0

#H

t 1

* <#0$`9H+-)!34<E4+)$

0

}-Q<})551

4,/:+7$ #%)7j

O

<#%n!3#6o!3(%E4

O

(%#

p ⁰

^!#

p

^#*)

t 0

#*

t 1

Q&)pa+12)t,/#u!3#r!3#*?-E( 1

$%J(%9H@4<A$ +-$DT[\+-E,/#

p(t) = a 0 + a 1 t + a 2 t ² + a 3 t ³ ;

)+$>

O

+)$v!3+):9lI2(p!)$

0

}-Q<}H231H8

0

}Q}.571r57LE@#9*+)!4Y@4<+-)$i:+->!3(H@#*@,/4<)#%

57LE@#r9*+7#*ƒq9H4#*)7@$

a 0

8

a 1

8

a 2

#*

a 3

Q9A+$j:+-E@4+)$j@,T#%)#*j<#rE+-=.*,/#rV>6

@(%$E+3@4<+-)/!FJ)r$ 13$ E.*,/#&4)(%4<@#&!3#

4

(057U@4<+-)$CV

4

4)9H+-))2#%$

a 0

8

a 1

8

a 2

#*

a 3

Q 24

)+$",/+)7E@+)$"57#n9*#n$E12$ E.%,/#j4<)(0L4E#T/E+-UzG+@$")#T$ +-<3@4<+-)oa+f)FJ4<,/a+EE#

57#%<#%$

O

<#%;$

p 0 , p 1 , p ⁰ ₀ , p ⁰ ₁

8L+;$c9H#* E#B$E+3@4<+-)r#0$GR)4657#Q m_+-f9*+)$ E@4E#

p

8a)+$f9H+-,/,T#%)…H+-)$&:L"9*+)$ E@4E#j)#T=$E#

ϕ 0 , ϕ 1 , ψ 0 , ψ 1

!#%$Ra+12)t,/#%$R!#g!#*?@( 1/$@$ +39H4(*#gLIu:+-4<)7;$

t 0

#H

t 1

Q

4C1 $")v9*+)$ E@4<

ϕ 0

E#*F57#

ϕ 0

$E+4<R)i:+-<12)t-,T#M!3#B!#*?@( 1T#H

ϕ 0 (t 0 ) = 1, ϕ ⁰ ₀ (t 0 ) = ϕ 0 (t 1 ) = ϕ ⁰ ₀ (t 1 ) = 0.

$")

O

(%E4<l#M57#geJ+)> *

ϕ 0 (t) = − (t − t 1 ) ² (2t + t 1 − 3t 0 )

(t 0 − t 1 ) ³ .

^0}Q}H431

441"#M,/b*,/#8+)i9H+):$G@E4<

ϕ 1

E#*5-#

ϕ 1

$E+4<R)i:+-<12)t-,/#g!#B!3#*?-E(1/#H

ϕ 1 (t 1 ) = 1, ϕ ⁰ ₁ (t 1 ) = ϕ 1 (t 0 ) = ϕ ⁰ ₁ (t 0 ) = 0.

$")v+=3@4<#%)7 *

ϕ 1 (t) = − (t − t 0 ) ² (2t + t 0 − 3t 1 )

(t 1 − t 0 ) ³ .

^0}Q}.871

44<4C1 $")i9H+):$G@E4<

ψ 0

E#*5-#

ψ 0

$E+4<R)i:+-<12)t-,/#g!#B!3#*?-E(1T#H

ψ ⁰ 0 (t 0 ) = 1, ψ 0 (t 0 ) = ψ 0 (t 1 ) = ψ ⁰ 0 (t 1 ) = 0.

$")

O

(%E4<l#M57#geJ+)> *

ψ 0 (t) = (t − t 1 ) ² (t − t 0 )

(t 0 − t 1 ) ² .

^0}Q}.671

4O

1A#M,/b*,/#-83+)>9H+-)$G@E4Y

ψ 1

E#%P57#

ψ 1

$E+4<R)i:+-<12)t-,/#B!3#M!3#*?-E( 1T#H

ψ ⁰ ₁ (t 1 ) = 1, ψ 1 (t 1 ) = ψ 1 (t 0 ) = ψ ⁰ ₁ (t 0 ) = 0.

$")v+=3@4<#%)7 *

ψ 1 (t) = (t − t 0 ) ² (t − t 1 )

(t 1 − t 0 ) ² .

^0}Q7/L‚71

9A+$`a+

O

+):$

O

(%E4<l#*C57#&9*#%$C57LE@#R:+-<12)t-,/#%$

ϕ 0 , ϕ 1 , ψ 0 , ψ 1

$E+)7C4<)(0L4E#%,/#*)7

4)!3(%:#%)!L)7;$*Q

;>= %#@')#&B%BW$_ OJM>-H<KPFLTB]J

`

:=N=E =RZJM>

ϕ 0 , ϕ 1 , ψ 0 , ψ 1

:=BMRZJEKL< E J P2>JW@3J

P 3

H<KJ

E :=<?>P

`3`

JNCNQJB]:=EG> ]P=>J;@ JBMR;ACLJ;@5J;L `

J;U< AVH<KJ P2>>: UAV\J

t 0

JL

t 1

^b

fL)$n>l?@#o}-Q28P)+-$n

O

+)$BE#%E(0$ #%)7E(

ϕ 0

8

ϕ 1

8

ψ 0

#H

ψ 1

$EjWJ4<)7E#%

O

<#

[t 0 , t 1 ]

)457#%,T#%)7%Q

d_1-)7M9*+)$ E@4<Mv=-$ #!KJ3&#%E,/4<E#

ϕ 0 , ϕ 1 , ψ 0 , ψ 1

!3#%$ga+12)t,/#%$M!#/!#*?@( 1

-$E$E+39H4(*#ML3Iia+4)-;$

t 0 , t 1

8:)+:$

O

(%E4<l+):$c[9*4<#*,/#*)7"57#B$E4

p

#%$ A<#j:+-<12)t-,T#B!#

!3#%?@(

3

^!(Hl)4PL

p(t) = p 0 ϕ 0 (t) + p 1 ϕ 1 (t) + p ⁰ ₀ ψ 0 (t) + p ⁰ ₁ ψ 1 (t),

^0}Q7/3}21

<+-@$

p

^$ELE46$G[4YR#%$R@#*6U@4<+-)$

0

}-Q<}H231

0

}-Q<})551sQ

24`)+-$f@#*)+-)$A,4<)7E#%)L)7g)#n[\+)9HE4+)

f

)#T[\+46$g9H+-)-@4<),/#%)-M!3(%E4

O

L=#

$ERWJ4)7E#*

O

L<#

[t 0 , t 1 ]

#*A$ 4K)+-$R9H+-)$ E@4$E+):$S#Ma+17)t,/#

p

^!3(Hl)4FL

p(t) = f (t 0 )ϕ 0 (t) + f (t 1 )ϕ 1 (t) + f ⁰ (t 0 )ψ 0 (t) + f ⁰ (t 1 )ψ 1 (t),

<+-@$g)+-$B!34@+)$M57#

p

^#0$GBWJ4<)7E#%Ea+6L)7j!KJ3&#%E,/4<E#q!3#

f

B!3#%$n9H=4657#%$j$

[t 0 , t 1 ]

QS9&+-$R

O

+-)$D)U@E#%<#*,/#%)-

p(t 0 ) = f (t 0 ), p(t 1 ) = f (t 1 ),

p ⁰ (t 0 ) = f ⁰ (t 0 ), p ⁰ (t 1 ) = f ⁰ (t 1 ).

-0.2 0 0.2 0.4 0.6 0.8 1

t 0 t 1

ϕ 0 ϕ 1

ψ 0

ψ 1

X #&RUW Wm

0c-$ #M!KJ3&#%E,/4<E#B!3#fG12:#B9*=4657#Q

;O. &$ * )O $& ;O. &**.

XJ4)7E#%Ea+6U@4<+-)B!KJ)#`[\+):9sE4+)jLk!3#%$P:+-<12)t-,T#0$P!3#D!3#*?-E(`(%<#

O

(C#%)j!3#0$Fa+4)-;$

(0574!4$ E@4<=(%$Ka#*#*)?#*):!3E#%F!3#%$F4<)$ @=4<4<E(0$K)2,/(*@457#0$F9H+-,T,/#`)+-$KWJ O +)$

O

!L)$`6M$E#%9s@4<+-)r}-Q:Q "#R<:$*8LWJ4<)7E#%Ea+6U@4<+-)!KJ)#c[\+)9HE4+)/C!3#%$`a+12)t,/#0$C!3#

!#*?@(&(%<#

O

(&)FJ#%$ X-$KzG$ E4<l(%#A+;$E57#R6g[\+):9sE4+)qVB4<)7E#%Ea+#*S)FJ#%$ X$XE(%?4.*@#Q

]"J#%$ &6n;L46$E+)ua+&-57#*<#gWJ4)7E#%Ea+6U@4<+-)uR4<)7E#%

O

<#%$c#0$G&$E+

O

#*)7R3E4<46$E(*#Q

3+4<

f

)#T[\+)9HE4+) 9H+-)-@4<)2#/!3+-))(*#/$Eg)y4)7E#%

O

L#

[a, b]

#HB$E+4<

0

N + 1

¹

a+4)7@$

x 0 = a < x 1 < x 2 < x 3 < . . . < x N = b

^{!L)$jeJ4<)7@#* O <#}

[a, b]

^QPm_+

9;':57#v4<)7E#%

O

<#

[x i , x i+1 ]

8_4<c#%$ :+7$E$E4=<#v!3#>9;'+46$E4<

0

n − 1

1j:+-4<)7@$T4)7E(*@4#*;$

(0574<@(*:LEE46$c)+L@(%$

x i,1 < x i,2 < x i,3 < . . . < x i,n−1 .

{X) a+-$@L)7

t 0 = x i

8

t j = x i,j

O

#%9

1 ≤ j ≤ n − 1

⁸

t n = x i+1

8_)+-$T:+-

O

+-)$

4)7E#%Ea+#*

f

^LI/a+4)7@$

t j

8

0 ≤ j ≤ n

82LD)ua+12)t,/#g!3#g!3#*?-E(

n

9H+-,T,/#")+$

WJ O

+):$X[4YA!L)$c6$ #09s@4<+-)y}-Q:Q fL):$R/$E4<E#M)+-$R!3(Hl:)4$@$E+)$

h = max

0≤i≤N−1 | x i+1 − x i | ,

#*)+$T9H+-)$ E@4$E+):$B)#r[\+):9sE4+)

f h : x ∈ [a, b] −→ f h (x) ∈ R

@#*<#i57#

f h

@#%$ E@#*4)7E#/Vr9*#Hg4<)7@#*

O

<#

[x i , x i+1 ]

$E+4<DzG$G@#*,/#*)7M9H#/a+17)t,/#T!FJ4)7E#*@a+6UE4+)

!#B!3#*?-E(

n

^Q

;>=

%#@')#&B% Wm E @FACBDP H<KJ

f h

J>LWN ACE6LJB

`

:FNQP=E6L^@5JY@5J FBD\

n

^{` PFBZACEKLJB=P=NCNJM>Y@5J}

NP :FE ULTAV:FE

f

^b

9A+$R!(*,/+)7E@+):$c<#g@(%$EY;U&$E4

O

)- *

U=H BW: :FACL

n

^{< E JE6LTAVJB `} :=>MACLTA

@3:=E6E \

a

>:=ACL

f : [a, b] → R

^{< E J :=E UL AV:=E}

@5:=E6E \JYH<KJE :=<?> >M<

`3`

:2>:=EK>

(n + 1)

^:=A>YU:FEKLACE RZJEKLW@3\BMA2P NQJY>M< BSN ACE6LJB2PFNCNQJ

[a, b]

JLW>:FACL

f h

>:=E ACEKL JB

`

:=NP=EKL@5J @3J 3BD\

n

^{` P=BYACEKL} JB=P=NCNQJ>

b N:=B>ACN JD_5A>MLJ< E J

U:FEG>MLPFEKL J

C

^{ACE @3\ ` JE @5P=EKLJ @=< UG:=A_ @5JM>}

x i

^a

1 ≤ i ≤ N − 1

^{SL JNCNQJ H<KJ}

x∈[a,b] max | f (x) − f h (x) | ≤ Ch ⁿ⁺¹ .

^0}-Q/L/31

;>=%.'Q,-'.#%

{C)>E4<46$E)7&<#g@'(*+-E.%,T#}-Q<}M$E&WJ4)7E#%

O

L#

[x i , x i+1 ]

#*)i<4#*i#*A69H#B!#MeJ4<)7@#*EN

O

L#

[a, b]

83)+:$R+=3@#*)+-)$ *

t∈[x max i ,x i+1 ] | f (t) − f h (t) | ≤ 1 2(n + 1)

x i+1 − x i

n

(n+1)

t∈[x max i ,x i+1 ] | f ⁽ⁿ⁺¹⁾ (t) | .

dA4<)$E4F)+-$R

O

+-)$

t∈[x max i ,x i+1 ] | f (t) − f h (t) | ≤ C

0≤j≤N−1 max | x j+1 − x j | n+1

0

}Q7/K171

+54r6/9H+):$G;L)7E#

C

#%$ A!3+))(*#gL

C = 1

2(n + 1)n ⁽ⁿ⁺¹⁾ max

t∈[a,b] | f ⁽ⁿ⁺¹⁾ (t) |

#*k+54

i = 0, 1, 2, . . . , N − 1

Q%dA4<)$E4

0

}-Q/L/31K#0$Gk)#c9H+-)$ (057#*)9*#C4,T,/(0!34LE#D!3#

0

}Q7/K171sQ

&)#f4)7E#%E@(H;U@4<+-)q!3q@'(*+-E.%,T#T}Q7/M#0$Gc6j$E4

O

)-@#Q 24Z+)r$ #g!3+))#")u#*)7@4<#%

a+-$E4Y@4Y[

n

#Hc$ 4:+)/@#*)!

N

^a+4)7@$

x 1 , x 2 , . . . , x N

O

#%9

N

!#&$C#%)<:$C?;L):!T!#

[-…H+)BV"9H#D5-#

h

$E+4<k!#X$#*)B$:#*E4<%8UL+;$

max x∈[a,b] | f (x) − f h (x) |

^{9H+) O #*@?#}

O

#*;$ *(*@+/<+-@$@57#

h

^{@#*)! O #*;$} *(*@+Q3m`&#HI3#*,/#8a$ 4P+)i:+7$ #

h = (b − a)/N

^#HA$E4

x i = a + ih

^{O #09}

i = 0, 1, 2, . . . , N

^8+)iL;

x∈[a,b] max | f (x) − f h (x) | ≤ Ch ⁿ⁺¹ .

{C)i;U@457#-83+)v@#*)!3;

N

^?-@)!r#H

n

^{a#H@4Y 0}

n = 1

^{+[/n+ 1n+-i71sQ}

f e +iBWm ]S+-)$ 46!3(%E+-)$R#n9*-$&+54

f (x) = x ^1.7 + 0.1e ^3x sin(13x)

⁸

a = 0

⁸

b = 0.8

⁸

N = 4

⁰a+&#%$c=a#%$E+4)$&!#Mnl?-E#21s8

x 0 = 0

⁸

x 1 = 0.2

⁸

x 2 = 0.4

⁸

x 3 = 0.6

⁸

x 4 = 0.8

^Q24

n = 1

^8K4C)FJ1yi-$M!3#a+4)-B4)7E(%E4#*B3I4)7E#*

O

L<#0$

[x i , x i+1 ]

^#*

WJ4)7E#%Ea+6U@4<+-) $ET9;'57#q4)7E#%

O

L#q$ #u[L4<jLT!3#%$j:+-<12)t-,/#%$j!#u!3#*?-E(}Q

l?-@#T}-Q5n,/+)7E@#g<#M?-@'#g!3#MWJ4)7E#*@a+6L)7RLc4)-@#*

O

L<#0$*Q

24

n = 2

8)+:$j9;'+-4$E4$@$E+)$ga+n:+-4<)7j4)7E(%E4#*nV

[x i , x i+1 ]

#ua+4)7j,/4<4<#%

x i,1 = ^{x i +x} ₂ ⁱ⁺¹

^#HcWJ4)7E#*@a+6UE4+)r$ c9;'-5-#&4<)7@#*

O

<#"$E#"[\#*;BS!3#0$Sa+12)t,/#0$

!3#B!#*?@( /3QPnl?-@#T}-Q74j,/+)7@E#g#f?-@'#M!3#gWJ4)7E#*@a+6L)7RLR4)7E#*

O

L<#0$*Q

CJ4)-@#*@:+-LE4+)>!3#TPL?-@)?#f&4<)7@#*

O

<#%$A,/#HA#%)>(

O

4!#*)9*#j!3#%$"$E3@$A!3#j6

!3(%E4

O

(%#A@#*,/4.*@#"#%)v9;':57#Aa+4)7

x i

⁰

l?:QK}Q5j#*f}-Q7431sQ &)#f,)4<.%E#g!3#M9H+-)$ E@4<@#

) 4)7E#*@a+6L)7MLM4)7E#*

O

L<#0$g$M4$@$ #/#0$Gj!KJ3@4<4$E#*jeJ4<)7@#*@:+-LE4+) !KJ3A#*@,/4Y@#

O

#%9&!3#0$c9H=4657#%$D$ER9;'57#"4<)7E#%

O

<#

[x i , x i+1 ]

⁰$E#%9H%QZ}Q7231HQ3]S6L4E#%,/#*)7D$E4

f

^#%$

)#g[\+)9HE4+)

C ¹

^{$E&WJ4)7E#* O L<#}

[a, b]

#*A$ 4W8$EA9;'-57#g4<)7E#%

O

<#

[x i , x i+1 ]

^8:)+:$

4)7E#*@a++)$

f

^LM) a+12)t,/#!3#q!3#%?@( 1>9*+,/,/#T)+$gWJ O

+)$g!3( z Vv[L4<j!L):$

6y$E#%9HE4+) }Q72$E/eJ4)-@#*

O

L<#

[t 0 , t 1 ]

8C<+-@$n#*) 9;'57#r:+-4<)7

x i

6[\+)9HE4+) 574

4)7E#*@a+#

f

^{E#%)!i O <#%}

f (x i )

#H"$@!3(%E4

O

(%#ME#%,/4<.%E#M@#*):!i

O

L#*

f ⁰ (x i )

^Q

XJ4)7E#*@a+6L)7&L4)$ 4F9*+)$ E@4<R#%$ c)#g[\+)9HE4+)

C ¹

^$E

[a, b]

^{0l?:QF}Q851HQ}

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 0.2 0.4 0.6 0.8

a = x 0 x 1 x 2 x 3 b = x 4

x ^1.7 + 0.1e ^3x sin(13x) interpolant

X #&RUWlBW

wx)7@#*@:+-LE4+)u:LR4)-@#*

O

L<#0$R!3#

f

&!3#%$ca+12)t,/#0$R!3#M!3#*?-E(/}Q

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 a = x 0 x 0,1 x 1 x 1,1 x 2 x 2,1 x 3 x 3,1 b = x 4

x ^1.7 + 0.1e ^3x sin(13x) interpolant

X #&RUWlBWG

wx)7@#*@:+-LE4+)u:LR4)-@#*

O

L<#0$R!3#

f

&!3#%$ca+12)t,/#0$R!3#M!3#*?-E(n/3Q

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 0.2 0.4 0.6 0.8

a = x 0 x 1 x 2 x 3 b = x 4

x ^1.7 + 0.1e ^3x sin(13x) interpolant

X #R\WZW#

wx)7@#*@:+-LE4+) n4)7E#%

O

L#%$T!3#

f

:L/!3#%$n:+-<12)t-,/#%$T!KJ3A#*@,/4Y@#v9*3N

=457#0$*Q

. & .

f e*HkL#&kL W

$") $Ea+-$E#o57#>eJ+) 9*+))4$@$E#v#%$

O

<#%;$

f (t)

^!KJ)#i[\+-)9s@4<+-) 9*+)7E4)2#&L3I

O

L#*;$C!3#

t

#*)7@4<.%E#0$C$ #%#*,/#*)7%879LJ#%$ NxVUNx!34E#c+)$Ea+-$E#A9H+-))2#%$_#%$

O

L#*@$

f (k), k = 0, ± 1, ± 2, . . .

24

t ∈ R

⁸

t

)+-)v#%)7E4#*08+)>!3(*l)4<A)#B@+^I24,U@4<+-)

p(t)

^!3#

f (t)

#%)i4<)7E#%Ea+6L)7 6o[\+-)9s@4<+-)

f

:L/)~:+-<12)t-,T#i!3#i!3#*?-E(>1y3I 57U@E#va+4)-;$T#*)7E4#*;$T<#0$n<$

@+39;'#%$c!3#

t

^Q

{X9*E4E#g)><?-+@4Y@',/#g574W83a+

t

!3+))(83[\+-@)4Y

p(t)

^Q

j

D\')#&B%

2+-4Y

t ∈ R

⁸

t

)+-)j#%)-@4<#%_#HC$ +-4Y

k = E[t]

" @4<#S#*)7E4.*@#D!#

t

QUmC46$@5-#

t ∈ ]k, k+1[

4P$%J?4<c!3#B!(Hl)4&#g:+-<12)t-,T#

p

!3+-)7c#M?;L'#":$@$ #gD#%$&:+-4<)7@$

k − 1, f (k − 1)

, k, f (k)

, k + 1, f(k + 1)

, k + 2, f (k + 2) .

,-,AD BW& d&?+-E4<E',/#f[\+E)46$@$E)-

p = p(t)

VTLEE4&!3#

t

^Q

R#@'"U ZB%('.,S#@

ψ 0 (x) := − 1

6 x(x − 1)(x − 2) ψ 1 (x) := 1

2 (x + 1)(x − 1)(x − 2) ψ 2 (x) := − 1

2 (x + 1)x(x − 2) ψ 3 (x) := 1

6 (x + 1)x(x − 1)

F4E#

t k := E[t]

24

t = k p := f k ;

24<)+)

p := f k−1 ∗ ψ 0 (t − k) +f k ∗ ψ 1 (t − k) +f k+1 ∗ ψ 2 (t − k) +f k+2 ∗ ψ 3 (t − k)

A(*l)4<E4+)o!3#0$D[\+)9HE4+)$

ψ 0

8

ψ 1

8

ψ 2

8

ψ 3

]DL69HF!3#

k

34

t

#0$GR#%)-@4<#%

24)+)v+)>9%L69H#

p(t)

]S+-)3[\+@,/(*,/#*)7&LIuE(0$ Y;U@$R!#f6$E#%9s@4<+-)y}-Q:8

p

#%$ c#Ma+17)t,/#M!3(Hl)4FL

p(t) =f (k − 1)ϕ 0 (t) + f (k)ϕ 1 (t)

+ f (k + 1)ϕ 2 (t) + f (k + 2)ϕ 3 (t),

⁰

}-Q/L71

+54

ϕ 0

8

ϕ 1

8

ϕ 2

8

ϕ 3

#0$GB6i=-$ #!3#uL?-@)?#T!3#%$Ma+12)t,/#0$B!3#q!3#%?@( 1>-$E$E+39H4(*#

3Ia+4)7@$

k − 1

⁸

k

⁸

k + 1

⁸

k + 2

QK{C)3E4<46$@L)7MeJ(*?7L4Y@(

0

}Q7231s8K)9%L69HC$E4<,/#

!+))#,*

ϕ 0 (t) = − 1

6 (t − k)(t − k − 1)(t − k − 2), ϕ 1 (t) = 1

2 (t − k + 1)(t − k − 1)(t − k − 2), ϕ 2 (t) = − 1

2 (t − k + 1)(t − k)(t − k − 2), ϕ 3 (t) = 1

6 (t − k + 1)(t − k)(t − k − 1).

{Cha#09sE+)$A<#j9;':L)?-#*,/#*)7A!3#

O

E46L=#

x = t − k

^QmX4$@57#

t ∈ ]k, k + 1[

^8<+-@$

x ∈ ]0, 1[

#*&)+$R!(Hl)46$@$ +-)$c<#0$D[\+)9HE4+)$

ψ 0

8

ψ 1

8

ψ 2

8

ψ 3

L

ψ 0 (x) = ϕ 0 (x + k), ψ 1 (x) = ϕ 1 (x + k), ψ 2 (x) = ϕ 2 (x + k), ψ 3 (x) = ϕ 3 (x + k),

$E+4<&#*):9H+@#

ψ 0 (x) = − 1

6 x(x − 1)(x − 2), ψ 1 (x) = 1

2 (x + 1)(x − 1)(x − 2), ψ 2 (x) = − 1

2 (x + 1)x(x − 2), ψ 3 (x) = 1

6 (x + 1)x(x − 1).

XJ(%?-<4<E(

0

}-Q/L71D$%J(09H@4YR,L4)7E#*):L)7 *

p(t) =f (k − 1)ψ 0 (t − k) + f (k)ψ 1 (t − k) + f (k + 1)ψ 2 (t − k) + f (k + 2)ψ 3 (t − k).

XJ<?-+@4Y@',/#>9H+@@#%$E:+-)!L)7/#%$ q@(%$E#*)7E(>!)$/<#i;L=#% }Q}QCF#0$@,T.*E@#%$

!FJ#%)-@E(%#q$ +-)7

t ∈ R

^{#*B<#0$ O L#*;$}

f k = f (t k )

QkF#@,T.*E@#!3#q$ +- @4<#

p

^#0$GB6

O

<#%

p(t)

^Q

f e*kJ#&kLBW: 2+4<

t 0 < t 1

!3#%3Iy)+-,j=@#%$BE(%#*6$B!346$G@4<)9H@$M#*n$E+4<

ε

^@#*S57#

0 <

ε < t 1 − t 0

Q

}Qc{`I349*4Y@#*&)va+12)t,/#

p ε

!3#M!3#%?@(1n@#*P57#

p ε (t 0 ) = p ε (t 0 + ε) = 1,

p ε (t 1 ) = p ε (t 1 + ε) = 0.

/Q 24

ϕ(t) = lim

ε→0 p ε (t)

8_,T+-)7E@#*/5-#

ϕ(t 0 ) = 1

⁸

ϕ(t 1 ) = ϕ ⁰ (t 0 ) = ϕ ⁰ (t 1 ) = 0

8X#*v4<)$E4

ϕ

#%$ u)#o[\+):9sE4+) !#o=$E#!3#0$qa+12)t,/#%$r!3#!3#%?@(i1pa+

eJ4<)7@#*@:+-LE4+)i!KJ3A#*@,/4Y@#

0

$E#%9s0QF}Q7231HQ

j

D\')#&B%

}-Q 2+4<

ϕ 0

8

ϕ 1

8

ϕ 2

8

ϕ 3

T=-$ #B!#BPL?-@)?#g!3#

P 3

$@$ +39H4(*#gLIra+4)-;$

t 0

8

t 0 + ε

⁸

t 1

8

t 1 + ε

Q3{C)i3@4<46$E)-&#gE(0$ Y;U

0

}-Q671D)+$R

O

+)$ *

p ε (t) = ϕ 0 (t) + ϕ 1 (t).

nJ3E@#gLE%83WJ(*?-<4<E(

0

}Q7231c!3+-))#*

ϕ 0 (t) = (t − t 0 − ε)(t − t 1 )(t − t 1 − ε) ( − ε)(t 0 − t 1 )(t 0 − t 1 − ε) , ϕ 1 (t) = (t − t 0 )(t − t 1 )(t − t 1 − ε)

ε(t 0 − t 1 + ε)(t 0 − t 1 ) .

{C) E(0!346$E)7r#%$v!3#*3I [\;9HE4+)$v9H4<N€!#%$@$ $vV ) !3(%)+,/4)U@#*v9H+,/,n)F8c)+:$

+-=3E#%)+)$ *

p ε (t) = (t − t 1 )(t − t 1 − ε) ε(t 0 − t 1 )(t 0 − t 1 − ε)(t 0 − t 1 + ε)

×

(t 0 − t 1 − ε)(t − t 0 ) − (t 0 − t 1 + ε)(t − t 0 − ε)

= (t − t 1 )(t − t 1 − ε)(3t 0 − t 1 − 2t + ε) (t 0 − t 1 )(t 0 − t 1 − ε)(t 0 − t 1 + ε) .

/Q3m_&!3(Hl:)4Y@4<+-)>!3#

ϕ

^{8)+:$R O +):$ *}

ϕ(t) = lim

ε→0 p ε (t) = (t − t 1 ) ² (3t 0 − t 1 − 2t) (t 0 − t 1 ) ³ ,

#*A!3+)983#*)>!3(%E4

O

L)7%83)+-$c+=E#*)+)$ *

ϕ ⁰ (t) = 2(t − t 1 )(3t 0 − t 1 − 2t) − 2(t − t 1 ) ² (t 0 − t 1 ) ³

= 6 (t − t 1 )(t 0 − t) (t 0 − t 1 ) ³ .

9A+$n

O

+):$M!3+)9q=4<#%)

ϕ(t 0 ) = 1

⁸

ϕ(t 1 ) = ϕ ⁰ (t 0 ) = ϕ ⁰ (t 1 ) = 0

8k9H#u574D@+

O #

57#

ϕ

#%$ D)#"!3#0$D57LE@#R[\+):9sE4+):$S!3#f=-$ #f!3#%$S:+-<12)t-,/#%$D!KJ3A#*@,/4Y@#f!3#f!3#*?-E( 1 -$E$E+39H4(*#gL3Iua+4)7@$

t 0

#*

t 1

⁰

$E#%9s0QF}Q7231HQ

f e*HkL#&kL WN 3+4<

f

)#T[\+-)9s@4<+-)9*+)7E4)2#!3+))(*#$ MeJ4<)7@#*

O

<#

[ − 1, +1]

^#H

$E+4<

p

<#Ma+12)t,/#M!3#j!#*?@(n/T5-4F4<)7@#*@:+-<#

f

#*)i<#0$Ra+4)-;$

− 1, 0, +1

Q7{CI2E4,/#*

R +1

−1 p(t)dt

^#%) [\+):9sE4+) !3#

f ( − 1), f(0)

^8k#H

f (+1)

^Q "(*@4Yl#%T57#ro[\+@,j#r4<):$ 4 +-=3E#%)7#g9*+W)9H46!3#M

O

#09A6n[\+@,j#M!3# 24,/$ +-)v!#M/$E#%9s@4<+-)>1Q:Q

j

B&DU'.#&B%

{X)>3E4<46$@L)7R<#M@(%$E<@U

0

}Q651s83#Ma+17)t,/#

p

#0$GA!(Hl)4PL

p(t) = f ( − 1)ϕ 0 (t) + f (0)ϕ 1 (t) + f (+1)ϕ 2 (t),

+54

ϕ 0

8

ϕ 1

8

ϕ 2

#%$ S6j=-$ #"!3#fL?-@)?#&!3#

P 2

$@$ +39H4(*#"L3Ia+4)-;$

− 1, 0, +1

^#Hc#%$

#*I3<469H4<E(%#B!L):$ceJ#HI3#*,/#}Q}QA9A+$R

O

+)$c!3+-)9

Z +1

−1

p(t)dt = f ( − 1) Z +1

−1

ϕ 0 (t)dt + f (0) Z +1

−1

ϕ 1 (t)dt + f (+1) Z +1

−1

ϕ 2 (t)dt.

A)>9*L69HP$ 4,/#g!+))#

Z +1

−1

ϕ 0 (t)dt = 1 3 ,

Z +1

−1

ϕ 1 (t)dt = 4 3 ,

Z +1

−1

ϕ 2 (t)dt = 1 3 ,

#*&L&9*+)$E(%57#%)7

Z +1

−1

p(t)dt = 1 3

f ( − 1) + 4f (0) + f (+1) .

wx$ #%,j=#M!3+)9g):UEE#%F!KJ@+39;'#*

R +1

−1 f (t)dt

R/57L)7@4Y@(

J (f )

^!(Hl)4#M

J (f ) = 1

3

f ( − 1) + 4f (0) + f (+1) .

fL):$u#9;'4Y@E#c1 )+$rLa#*#*@+)$6 57)-@4Y@(

J (f )

^{% "# "}

QFm`Lj9*+)$ E@9HE4+)F8K6v[\+@,j#q!3#q57-!3@LE@#

J (f )

4<)7@.*?@#/#HI9HE#%,T#%)7B<#0$

a+12)t,/#0$R!3#B!3#*?-E(M!3#%3Irv$E#*)$&+34

Z +1

−1

q(t)dt = J(q)

a+"@+3ga+17)t,/#

q

!3#T!#*?@(n!#*3IKQP]S#HEE#n[\+@,j#T!#T57-!3@LE@#j$%Ja#*<#%@

[\+-E,n#g!# 24<,/$E+)

0

$ #09s%QA1Q?1sQ

^O. (M(*&)$#S"Y. . &.-, &S"#.

9A+$r

O

+-)$TE(0$ #%)7E(o!3#%3I~#*I3#*,/#%$u!FJ4)7E#*@a+6UE4+) a+12)t,/46L#8XVp$@

O

+-4<

WJ4<)7E#%Ea+6U@4<+-)r!3#gP?;L)?-#&#HcWJ4)7E#*@a+6UE4+)r!KJ3A#*@,/4Y@#Q3wxZ#HI346$G@#")LE@#*#*,/#*)7

!FJ3E@#%$DG12a#%$&!KJ4)-@#*@:+-LE4+) *34<)7@#*@:+-LE4+)rE@4<?-+)+-,T(*E@457#-83$ 4)#%$%83#H;9LQ

d @4Y@E#/!KJ#HI3#*,/#8F9*+)$E4!(*@+)$AeJ4<)7@#*@:+-LE4+)y!FJ)#n[\+-)9s@4<+-)

f : [a, b] → R

:Lg)#T[\+):9sE4+) 57#/)+:$fa#*<#%E+-)$g$ 4)#9H=4657#T#HB57#/)+$f)+E#%E+-)$

S

^Q

34

t 0 = a < t 1 < . . . < t n = b

^8$E+)7

(n + 1)

a+4)7@$&!+))(0$&!)$RWJ4)7E#*

O

L<#

[a, b]

<+-@$g4X#0$Gn:+7$E$E4<=<#q!3#/@E+-

O

#%M)#[\+-)9s@4<+-)

S : [a, b] → R

9*+)7E4)2#8K@#*<#r5-#

S(t 0 ) = f (t 0 )

⁸

. . .

⁸

S(t n ) = f (t n )

8P!3#r!3(*@4

O

(*#0$"E#%,T4.*@#/#Hj$E#%9*+)!3#q9*+)7E4)2#%$%8

a+12)t,/46L#o!3#o!3#%?@( 1p$Eq9;'-57#i4<)7E#%

O

<#

[t j−1 , t j ]

⁸

1 ≤ j ≤ n

^{0O +4}

#*I3#*,/#/L27/K8Pa+R<:$R!3#B!3(H;L4$1sQ

CJ#HI3#%,T<#!#>p$E#%9HE4+) }Q ,T#* ;L)7u#*) ( O

4!3#%)9H#>6p!34

O

#%E?-#*)9*#v!3#eJ4<)7@#*EN

a+6U@4<+-)o#%$ "WJ#*I3#*,/#T!3# R)?-#Q &)E;L4<E#%,T#%)7",U@'(*,U@457#T9H+-,T<#*g!3#T9H#*

#*I2#%,/<#M$E#gE@+

O

#f!L)$ }.5€Q

CJ4)-@#*@:+-LE4+)vLA4<)7E#%

O

<#%$c#0$G&)>!3#0$R4)?@(%!34#*)7;$R!3#B6n,/(*E'+3!3#j!#%$R(%<(*N

,/#*)7;$Kl)46$

0

9;'FQ}0‚RVM}*?1sQ*{X)BLEE469H<4#*F#C@'(*+-E.%,/#R}Q7/c#%$ P#C?#%)E#S!3#C@(%$EY;U;$

@'(*+-E4657#%$M574S:#%E,/#HT!3#,/+)7E@#*M6o9H+-)

O

#*@?#%)9H#/!3#6i,/(HE'+2!#u!3#%$B(*(*,/#*)7@$

l)4$%Q

9&+-$TE#

O

4<#%)!3@+)$B$E/9H#% ;L4)$n-$ a#%9H@$T!3#uWJ4<)7E#%Ea+6U@4<+-) a+17)t,/4<#v!L):$

#%$R9;'4<E@#%$ /n#H 1Q

"#y)+-,j=@#*I~+?-49*4<#%$r$E9*4<#%)-@4Yla5-#%$q?-@)! =<469

0

Lv#HI3#*,/# L#

T M

⁸

oLE'#%,UE469*

T M

^{8 UE6L=}

T M

1=:+-@!3#%)7u<#0$q@+=<.%,T#0$q!KJ4<)7@#*@:+-LE4+)FQD|4)<#*N ,/#*)76 !3#0$n+?-49*4<#%$/!3#o]S+):9H#*E4+) d"$E$E46$G@(*#vL $";!34<):UE#%;$

0

]Dd $^1

3@4<46$ #%)-A!#%$c,/(H@'+3!3#%$R46$@$ #0$R!3#gT@'(*+-E4#M!3#geJ4)-@#*@:+-LE4+)/J5€Q