Introduction à l'analyse numérique by EPFL Press

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Cet ouvrage présente une introduction aux notions mathématiques nécessaires à l’utilisation des méthodes numériques employées dans les sciences de l’ingénieur. ! La plupart des phénomènes physiques, chimiques ou biologiques, issus de la technologie moderne, sont régis par des systèmes complexes d’équations aux dérivées partielles. La résolution numérique de ces systèmes d’équations au moyen d’un ordinateur nécessite des connaissances approfondies en mathématiques. Ce livre a donc pour but de fournir au lecteur les notions mathématiques de base qui lui permettront d’aborder ce sujet. ! L’ouvrage s’adresse tout particulièrement aux étudiants du 1er cycle universitaire en sciences de l’ingénieur, en physique et en mathématiques, ainsi qu’à tous ceux qui désirent s’initier à la simulation numérique et au calcul scientifique. ! Cette troisième édition constitue le compagnon indispensable du cours en ligne (MOOC) du même nom, que le lecteur pourra suivre au travers des liens renvoyant à chacune des vidéos.

De nationalité française, Marco Picasso est né en 1963 en Italie. Il obtient un diplôme d’ingénieur ECAM de Lyon en 1986, puis le DESS d’ingénierie mathématiques et calcul scientifique de l’Université de Besançon en 1987. En 1988, il entreprend un travail de recherche dans le groupe du Professeur Jacques Rappaz, en collaboration avec le département des matériaux de l’Ecole Polytechnique Fédérale de Lausanne. En 1992, il soutient sa thèse de doctorat concernant la simulation numérique des traitements de surface par laser. Depuis 1993, il est responsable du calcul scientifique au sein de la chaire d’analyse et simulation numérique du département de mathématiques de l’EPFL. Actuellement, il est chargé de cours pour l’enseignement de l’analyse numérique aux ingénieurs.

Presses polytechniques et universitaires romandes

Introduction à l’analyse numérique Jacques Rappaz Marco Picasso

Jacques Rappaz Marco Picasso

De nationalité suisse, Jacques Rappaz est né en 1947 à Lausanne. Il obtient un diplôme d’ingénieur physicien à l’Ecole Polytechnique Fédérale de Lausanne en 1971 et soutient sa thèse de doctorat consacrée à l’approximation spectrale d’opérateurs provenant de la physique des plasmas en 1976. Après sa thèse, il poursuit ses recherches en analyse numérique à l’Ecole Polytechniques de Paris où il séjourne trois ans. De retour à l’EPFL, il occupe un poste d’adjoint scientifique au département de mathématiques et oriente une partie de ses recherches vers des applications industrielles. En 1985, il est nommé professeur d’analyse numérique à l’Université de Neuchâtel. Depuis 1987, il est professeur à l’EPFL où il enseigne l’analyse et l’analyse numérique. Sa recherche est orientée sur les aspects théoriques et pratiques de la résolution numérique des équations aux dérivées partielles. Il dirige plusieurs projets en collaboration avec les milieux industriels et il est auteur ou co-auteur de nombreuses publications dans ce domaine.

Introduction à l’analyse numérique

Presses polytechniques et universitaires romandes

Dans la collection

Recueil d’exercices et aide-mémoire vol. 2

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Analyse avancée pour ingénieurs Troisième édition

Bernard Dacorogna Chiara Tanteri

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Introduction à l’analyse numérique

Introduction à l’analyse numérique Jacques Rappaz Marco Picasso

Presses polytechniques et universitaires romandes

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En 2012, nous avons réalisé un cours en ligne (MOOC) basé sur les chapitres 1, 2, 3, 4/5/6, 8, 9 et 10 du livre. Ce cours est disponible sur coursera.org, il contient des vidéos (6 à 10 vidéos de 10 minutes pour chaque chapitre), des quiz, et des exercices. Les vidéos sont disponibles également sur : KWWS JR HSIO FK DQDO\VH QXPHULTXH YLGHRV Tout au long de ce livre, des symboles sont reproduits dans la marge; ils renvoient aux vidéos du MOOC. Par exemple !"! renvoie à la vidéo 1 du chapitre 1.

Finalement, nous avons aussi développé quelques applets java pour illustrer certains des chapitres du livre. Des symboles sont reproduits dans la marge; ils renvoient aux applets correspondantes. Par exemple # renvoie au fichier : analyse-numerique.org/node3.html qui contient l’applet en question.

LU LU LU LU

n≥0 p0 p1 p2 . . .

t0 t1 t2 . . . tn pn p(tj ) = pj

0 ≤ j ≤ n.

p(t) = a0 + a1 t + a2 t2 + · · · + an tn a0 a1 a2 . . . an aj 0 ≤ j ≤ n n+1 a0 + a1 tj + a2 t2j + a3 t3j + · · · + an tnj = pj ,

0 ≤ j ≤ n. a 0 a1 a2 . . . an

(n + 1) × (n + 1)  1 1  1 T = 1  · 1

t0 t1 t2 t3 · tn T

t0 t1 t2 . . . tn

tj pj 0 ≤ j ≤ n (n + 1) (n + 1)

t20 t21 t22 t23 · t2n

t30 t31 t32 t33 · t3n

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

 tn0 tn1   tn2  . tn3   · tnn

p !

(n + 1) 

 a0  a1      !a =  a2  ,     an



 p0  p1      p! =  p2  ,     pn

T!a = p!. p !a T

p! (n + 1)

(n + 1)

p p

pj k pk = 1

ϕk (t) =

pj = 0

0 j $= k

ϕk

(t − t0 )(t − t1 ) · · · (t − tk−1 )(t − tk+1 ) · · · (t − tn ) . (tk − t0 )(tk − t1 ) · · · (tk − tk−1 )(tk − tk+1 ) · · · (tk − tn ) ϕk n

n t

(i) ϕk (ii) ϕk (tj ) = 0 (iii) ϕk (tk ) = 1.

ϕ0 ϕ1 α0 α1 α2 . . . αn ∀t ∈ R t = tk

ϕk tj j $= k ϕ2 . . . ϕn (n + 1)

n ( j=0

j $= k

n, j $= k, 0 ≤ j ≤ n,

tk tk

(t−tj ) ϕk

αj ϕj (tk ) = αk , ) *+ , 0 1

j"=k j=k

j=0

αj ϕj (t) = 0

αk k = 0, 1, . . . , n Pn n

1, t, t2 , t3 , . . . , tn n Pn

(n + 1) ϕ0 , ϕ1 , ϕ2 , . . . , ϕn

ϕ0 , ϕ1 , ϕ2 , . . . , ϕn t0 , t1 , t2 , . . . , tn

n = 2 t0 = −1 t1 = 0 t2 = 1 −1 0 1

ϕ0 , ϕ1 , ϕ2

(t − t1 )(t − t2 ) 1 1 1 = t(t − 1) = t2 − t; (t0 − t1 )(t0 − t2 ) 2 2 2 (t − t0 )(t − t2 ) ϕ1 (t) ≡ = −(t + 1)(t − 1) = 1 − t2 ; (t1 − t0 )(t1 − t2 ) (t − t0 )(t − t1 ) 1 1 1 ϕ2 (t) ≡ = (t + 1)t = t2 + t. (t2 − t0 )(t2 − t1 ) 2 2 2 ϕ0 (t) ≡

ϕ0 ϕ1 ϕ2

[−1, +1]

1 0.8

ϕ1

0.6 ϕ0

0.4

ϕ2

0.2 0 -0.2 -1

0 P2

1 −1 0

p p0 p1 p2 . . . pn t0 t1 t2 . . . tn

ϕ0 , ϕ1 , ϕ2 , . . . , ϕn t2 . . . tn

p(t) = p0 ϕ0 (t) + p1 ϕ1 (t) + · · · + pn ϕn (t) = p ϕ2 . . . ϕn p ∈ Pn

t0 t1

Pn p n (

pj ϕj (t).

j=0

(n + 1) n

ϕ0 ϕ1 n ϕj

k = 0, 1, 2, . . . , n p(tk ) =

n ( j=0

pj ϕj (tk ) = pk ) *+ , 0 1

j"=k j=k

p 0 , p 1 , . . . , pn !a p!

t1 = 0 ϕ1

p1 = 3

t2 = 1

t0 = −1 p0 = 8 p2 = 6 p(t) = 8ϕ0 (t) + 3ϕ1 (t) + 6ϕ2 (t) ϕ0

ϕ2 p(t) = 8

. . 1 2 1 1 2 1 t − t + 3(1 − t2 ) + 6 t + t 2 2 2 2

= 4t2 − t + 3.

f :R→R

t0 t1 t2 . . . tn (n + 1) f

p p

tj 0 ≤ j ≤ n p(tj ) = f (tj ), f (t)

0 ≤ j ≤ n.

n n

pj = f (tj ) 0 ≤'j ≤ n n p(t) = j=0 pj ϕj (t) ϕj 0 ≤ Pn t0 , t1 , t2 , . . . , tn

j≤n p(t) =

n ( j=0

f (tj )ϕj (t)

∀t ∈ R.

p f

t0 , t1 , t2 , . . . , tn f (t) = et

f f

eϕ2 (t)

−1 0

1 p(t) = e−1 ϕ0 (t) + e0 ϕ1 (t) +

ϕ0 , ϕ1 , ϕ2

p(t) = =

1 e -

. . 1 2 1 1 2 1 t − t + (1 − t2 ) + e t + t 2 2 2 2 . . 1 e 2 e 1 −1+ t + − t + 1. 2e 2 2 2e f

−1 0

1 3 f p

2.5 2 1.5 1 0.5 0 -1

1 f (t) = et

f 2

[a, b] tj j = 0, 1, 2, . . . , n j = 0, 1, 2, . . . , n t0 , t1 , . . . , tn n

−1 0 n

1 f : [a, b] → R [a, b]

h = (b − a)/n

tj = a + jh n

pn pn

pn (t) =

n (

f (tj )ϕj (t),

j=0

ϕ0 , ϕ1 , . . . , ϕn

t0 , t1 , . . . , tn

f [a, b]

(n + 1)

max |f (t) − pn (t)| ≤

t∈[a,b]

1 2(n + 1)

b−a n

.(n+1)

max |f (n+1) (t)|

t∈[a,b]

f (n+1) (t) = dn+1 f (t)/dtn+1 pn

f [a, b] n

t0 , t1 , t2 , . . . , tn

1 n→∞ 2(n + 1) lim

b−a n

.(n+1)

maxt∈[a,b] |f (n+1) (t)|

[−1, +1] |f (n) (1)| ... n

n=5

= 0.

f (t) = 1/(1 + 25t2 ) f (t)

[−1, +1] n

pn n = 10

tj = −1 + 2j/n j = 0 1

2 f p5 p10

1.5 1 0.5 0 -0.5 -1

-0.5

0.5

f (t) = 1/(1 + 25t2 )

[−1, +1]

tj = a +

t0 , t1 , . . . , tn (b − a) 2

. (2j + 1)π 1 + cos , 2(n + 1)

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

maxt∈[a,b] |f (t) − pn (t)|

n 2 f p10

1.5 1 0.5 0 -0.5 -1

-0.5

0.5

f (t) = 1/(1 + 25t2 )

p& (t)

p(t)

p0 , p1 , p&0 , p&1

t0 < t 1 p p(t0 ) = p0 , p& (t0 ) = p&0 , p& (t)

p(t1 ) = p1 , p& (t1 ) = p&1 ,

t p&

p t0

t0 t1

p(t) = a0 + a1 t + a2 t2 + a3 t3 ; a0 a1 a2

a3 4

a 0 a1 a2

p0 , p1 , p&0 , p&1 p

ϕ0 , ϕ1 , ψ0 , ψ1 t0

ϕ0

ϕ0 ϕ&0 (t0 ) = ϕ0 (t1 ) = ϕ&0 (t1 ) = 0.

ϕ0 (t0 ) = 1,

ϕ0 (t) = − ϕ1

ϕ1 (t) = − ψ0

ψ0& (t0 )

ϕ1

ϕ&1 (t1 )

ϕ1 (t1 ) = 1,

ψ0

(t − t1 )2 (2t + t1 − 3t0 ) . (t0 − t1 )3

= 1,

(t − t0 )2 (2t + t0 − 3t1 ) . (t1 − t0 )3 ψ0 (t0 ) = ψ0 (t1 ) = ψ0& (t1 ) = 0.

ψ0 (t) = ψ1 ψ1& (t1 ) = 1,

= ϕ1 (t0 ) = ϕ&1 (t0 ) = 0.

(t − t1 )2 (t − t0 ) . (t0 − t1 )2 ψ1

ψ1 (t1 ) = ψ1 (t0 ) = ψ1& (t0 ) = 0.

ψ1 (t) =

(t − t0 )2 (t − t1 ) . (t1 − t0 )2 ϕ0 , ϕ1 , ψ0 , ψ1 ϕ0 , ϕ1 , ψ0 , ψ1

P3 t0

ϕ0 ϕ1 ψ0

ψ1

[t0 , t1 ] ϕ0 , ϕ1 , ψ0 , ψ1 t0 , t1

3 p(t) = p0 ϕ0 (t) + p1 ϕ1 (t) + p&0 ψ0 (t) + p&1 ψ1 (t), p f [t0 , t1 ]

p(t) = f (t0 )ϕ0 (t) + f (t1 )ϕ1 (t) + f & (t0 )ψ0 (t) + f & (t1 )ψ1 (t), p

[t0 , t1 ] p(t0 ) = f (t0 ), p& (t0 ) = f & (t0 ),

p(t1 ) = f (t1 ), p& (t1 ) = f & (t1 ).

Ï&#x2022;0

Ï&#x2022;1

0.8 0.6 0.4 Ï&#x2C6;0

0.2 0 -0.2

Ï&#x2C6;1

f x0 = a < x1 < x2 < x3 < . . . < xN = b [xi , xi+1 ]

[a, b]

N +1 [a, b]

nâ&#x2C6;&#x2019;1

xi,1 < xi,2 < xi,3 < . . . < xi,nâ&#x2C6;&#x2019;1 . t0 = xi tj = xi,j 1 â&#x2030;¤ j â&#x2030;¤ n â&#x2C6;&#x2019; 1 tn = xi+1 f tj 0 â&#x2030;¤ j â&#x2030;¤ n n h=

max

0â&#x2030;¤iâ&#x2030;¤N â&#x2C6;&#x2019;1

[xi , xi+1 ]

|xi+1 â&#x2C6;&#x2019; xi |,

fh : x â&#x2C6;&#x2C6; [a, b] â&#x2C6;&#x2019;â&#x2020;&#x2019; fh (x) â&#x2C6;&#x2C6; R

n fh

n (n + 1) [a, b]

f : [a, b] â&#x2020;&#x2019; R n xi 1 â&#x2030;¤ i â&#x2030;¤ N â&#x2C6;&#x2019; 1

max |f (x) â&#x2C6;&#x2019; fh (x)| â&#x2030;¤ Chn+1 .

xâ&#x2C6;&#x2C6;[a,b]

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

max

t∈[xi ,xi+1 ]

|f (t) − fh (t)| ≤ C

xi+1 − xi n

max

.(n+1)

0≤j≤N −1

max

t∈[xi ,xi+1 ]

|f (n+1) (t)|.

.n+1 |xj+1 − xj |

C C=

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

i = 0, 1, 2, . . . , N −1

x1 , x2 , . . . , xN

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

xi = a + ih

N maxx∈[a,b] |f (x)−fh (x)| h = (b − a)/N

max |f (x) − fh (x)| ≤ Chn+1 .

x∈[a,b]

0.8 N = 4 x4 = 0.8

n=1

f (x) = x1.7 + 0.1e3x sin(13x) a = 0 b = x0 = 0 x1 = 0.2 x2 = 0.4 x3 = 0.6 [xi , xi+1 ]

n=1

n = 2 xi,1 =

[xi , xi+1 ]

xi +xi+1 2

xi C1

[xi , xi+1 ]

[a, b]

[xi , xi+1 ]

f f

[t0 , t1 ] f (xi )

xi C

[a, b]

f & (xi )

1.2 1 0.8

x1.7 + 0.1e3x sin(13x) interpolant

0.6 0.4 0.2 0 -0.2 -0.4 0 a = x0

0.2 x1

0.4 x2

0.6 x3

0.8 b = x4

0.5 0.6 x2,1 x3

0.7 0.8 x3,1 b = x4

0.6 x3

0.8 b = x4

f 1.2 1 0.8

x1.7 + 0.1e3x sin(13x) interpolant

0.6 0.4 0.2 0 -0.2 -0.4 0 a = x0

0.1 0.2 x0,1 x1

0.3 0.4 x1,1 x2 f

1.2 1 0.8

x1.7 + 0.1e3x sin(13x) interpolant

0.6 0.4 0.2 0 -0.2 -0.4 0 a = x0

0.2 x1

0.4 x2 f

f (t) t f (k), k = 0, ±1, ±2, . . . t∈R t

p(t)

f (t)

f t t

t∈R t

p(t)

k = E[t] p

/ 0 k − 1, f (k − 1) ,

/ 0 k, f (k) ,

/ 0 k + 1, f (k + 1) ,

t ∈]k, k+1[

/ 0 k + 2, f (k + 2) .

p = p(t)

1 ψ0 (x) := − x(x − 1)(x − 2) 6 1 ψ1 (x) := (x + 1)(x − 1)(x − 2) 2 1 ψ2 (x) := − (x + 1)x(x − 2) 2 1 ψ3 (x) := (x + 1)x(x − 1) 6 t

ψ0 ψ1 ψ2 ψ3

k := E[t] t=k

p := fk ; p(t) p

:= fk−1 ∗ ψ0 (t − k) +fk ∗ ψ1 (t − k) +fk+1 ∗ ψ2 (t − k) +fk+2 ∗ ψ3 (t − k)

p p(t) =f (k − 1)ϕ0 (t) + f (k)ϕ1 (t) + f (k + 1)ϕ2 (t) + f (k + 2)ϕ3 (t), ϕ0 ϕ1 ϕ2 ϕ3 k−1 k k+1 k+2 1 ϕ0 (t) = − (t − k)(t − k − 1)(t − k − 2), 6 1 ϕ1 (t) = (t − k + 1)(t − k − 1)(t − k − 2), 2 1 ϕ2 (t) = − (t − k + 1)(t − k)(t − k − 2), 2 1 ϕ3 (t) = (t − k + 1)(t − k)(t − k − 1). 6 x = t−k ψ0 ψ1 ψ2 ψ3

x ∈]0, 1[ ψ0 (x) = ϕ0 (x + k), ψ2 (x) = ϕ2 (x + k),

t ∈]k, k + 1[

ψ1 (x) = ϕ1 (x + k), ψ3 (x) = ϕ3 (x + k),

1 ψ0 (x) = − x(x − 1)(x − 2), 6 1 ψ1 (x) = (x + 1)(x − 1)(x − 2), 2 1 ψ2 (x) = − (x + 1)x(x − 2), 2 1 ψ3 (x) = (x + 1)x(x − 1). 6

p(t) =f (k − 1)ψ0 (t − k) + f (k)ψ1 (t − k)

+ f (k + 1)ψ2 (t − k) + f (k + 2)ψ3 (t − k).

p(t)

t∈R

fk = f (tk )

t0 < t 1 ε < t 1 − t0

ε pε pε (t0 ) = pε (t0 + ε) = 1, pε (t1 ) = pε (t1 + ε) = 0.

ϕ(t0 ) = 1 ϕ(t1 ) = ϕ& (t0 ) = ϕ& (t1 ) =

ϕ(t) = lim pε (t) ε→0

ϕ0 ϕ1 ϕ2 ϕ3 t1 t1 + ε

t0 t0 + ε

P3 pε (t) = ϕ0 (t) + ϕ1 (t).

(t − t0 − ε)(t − t1 )(t − t1 − ε) , (−ε)(t0 − t1 )(t0 − t1 − ε) (t − t0 )(t − t1 )(t − t1 − ε) ϕ1 (t) = . ε(t0 − t1 + ε)(t0 − t1 ) ϕ0 (t) =

pε (t) =

(t − t1 )(t − t1 − ε) ε(t0 − t1 )(t0 − t1 − ε)(t0 − t1 + ε) . × (t0 − t1 − ε)(t − t0 ) − (t0 − t1 + ε)(t − t0 − ε) (t − t1 )(t − t1 − ε)(3t0 − t1 − 2t + ε) . (t0 − t1 )(t0 − t1 − ε)(t0 − t1 + ε) ϕ ϕ(t) = lim pε (t) = ε→0

(t − t1 )2 (3t0 − t1 − 2t) , (t0 − t1 )3

2(t − t1 )(3t0 − t1 − 2t) − 2(t − t1 )2 (t0 − t1 )3 (t − t1 )(t0 − t) =6 . (t0 − t1 )3

ϕ& (t) =

ϕ(t0 ) = 1 ϕ(t1 ) = ϕ& (t0 ) = ϕ& (t1 ) = 0 ϕ t0

f p 1 +1 p(t)dt −1

f (−1), f (0)

f f (+1)

[−1, +1] −1, 0, +1

p p(t) = f (−1)ϕ0 (t) + f (0)ϕ1 (t) + f (+1)ϕ2 (t), ϕ0 ϕ1 ϕ2 2

p(t)dt = f (−1)

−1

ϕ0 (t)dt + f (0)

−1

+1 −1

−1, 0, +1

1 ϕ0 (t)dt = , 3

ϕ1 (t)dt + f (+1)

−1

p(t)dt =

−1

ϕ2 (t)dt.

−1

4 ϕ1 (t)dt = , 3

ϕ2 (t)dt =

−1

1 , 3

4 13 f (−1) + 4f (0) + f (+1) . 3 1 +1 −1

J(f ) =

f (t)dt

J(f )

4 13 f (−1) + 4f (0) + f (+1) . 3 J(f ) J(f )

q(t)dt = J(q) −1

t0 = a < t 1 < . . . < t n = b S(t0 ) = f (t0 )

. . . S(tn ) = f (tn )

f : [a, b] → R S [a, b]

(n + 1) S : [a, b] → R

[tj−1 , tj ] 1 ≤ j ≤ n

TM TM

f f&

x0 ∈ R f (x0 + h) − f (x0 ) h f (x0 ) − f (x0 − h) = lim h→0 h f (x0 + h/2) − f (x0 − h/2) = lim . h→0 h

f & (x0 ) = lim

h→0

h ∆h f (x0 ) h

∇h f (x0 ) h

δh f (x0 ) h

∆h f (x0 ) = f (x0 + h) − f (x0 ), def

∇h f (x0 ) = f (x0 ) − f (x0 − h), def

δh f (x0 ) = f (x0 + h/2) − f (x0 − h/2). def

h>0 ∆h f

∆h f : R → R ∆h f (x) = f (x + h) − f (x) ∇h δh h > 0

∆h ∇h

δh

∆h ∇h

δh

∆h α

f, g : R → R

∆h (αf + βg)(x) = αf (x + h) + βg(x + h) − αf (x) − βg(x) = α∆h f (x) + β∆h g(x)

∀x ∈ R.

∇h

δh !

f x0

1 f (x0 + h) = f (x0 ) + f & (x0 )h + f && (ξ)h2 , 2 ξ

[x0 , x0 + h] 5 5 5 & 5 5f (x0 ) − ∆h f (x0 ) 5 = 1 |f && (ξ)|h. 5 5 2 h h0 > 0

f :R→R

x0 ∈ R C

5 5 5 & 5 5f (x0 ) − ∆h f (x0 ) 5 ≤ Ch, 5 5 h C=

ξ ∈ [x0 , x0 + h0 ]

∀h ≤ h0 .

1 max |f && (x)|. 2 x∈[x0 ,x0 +h0 ]

1 && 2 |f (ξ)|

h ≤ h0

≤C !

∆h f (x0 ) !

∇h f (x0 )

f & (x0 ) f

δh f (x0 ) h

- .2 - .3 h f &&& (ξ) h + , 2 3! 2 - .2 - .3 h f && (x0 ) h f &&& (η) h & f (x0 − h/2) = f (x0 ) − f (x0 ) + − , 2 2! 2 3! 2 f (x0 + h/2) = f (x0 ) + f & (x0 )

h f && (x0 ) + 2 2!

Î¾

[x0 , x0 + h/2]

Î·

[x0 â&#x2C6;&#x2019; h/2, x0 ]

5 5 5 &&& 5 2 &&& 5 & 5 5 5 5f (x0 ) â&#x2C6;&#x2019; Î´h f (x0 ) 5 = 5 f (Î¾) + f (Î·) 5 h 5 5 5 5 8 h 6 &&& &&& |f (Î¾)| + |f (Î·)| h2 â&#x2030;¤ . 2 24 h0 C=

h0 > 0

â&#x2C6;&#x2020;h f (x0 ) h

1 max |f &&& (x)|, 24 xâ&#x2C6;&#x2C6;[x0 â&#x2C6;&#x2019;h0 /2,x0 +h0 /2]

f :Râ&#x2020;&#x2019;R

x0 â&#x2C6;&#x2C6; R C

5 5 5 & 5 5f (x0 ) â&#x2C6;&#x2019; Î´h f (x0 ) 5 â&#x2030;¤ Ch2 , 5 5 h

Î´h f (x0 ) h

â&#x2C6;&#x20AC;h â&#x2030;¤ h0 . f

f & (x0 ) h

h2 â&#x2C6;&#x2020;h f (x0 )/h

â&#x2C6;&#x2021;h f (x0 )/h

5 5 5 & 5 5f (x0 ) â&#x2C6;&#x2019; â&#x2C6;&#x2020;h f (x0 ) 5 5 5 h f & (x0 )

f & (x0 )

5 5 5 & 5 5f (x0 ) â&#x2C6;&#x2019; â&#x2C6;&#x2021;h f (x0 ) 5 5 5 h h 1 h Î´h f (x0 )/h

h 5 5 5 & 5 5f (x0 ) â&#x2C6;&#x2019; Î´h f (x0 ) 5 5 5 h

c c = 1/3 c

c˜

c˜ = 0.333333

c˜

N N

0.333333 = 0.333333 100 34.2456 = 0.342456 102 0.000345033 = 0.345033 10−3 3.42550 1018 = 0.342550 1019

c |c − c˜|

c˜ N c

η = 10

−N

|c − c˜| ≤ 5|c|η c = 1/3 c˜ = 0.333333 η = 10−6

c |c − c˜| = 13 10−6 ≤ 5|c|η

c f & (x0 )

∆h f (x0 )/h ∆h f (x0 )/h

f (x) = x2 x0 = 7 h = 0.06

0.01 h = 0.06

h = 0.01

∆h f (x0 ) (7.06)2 − (7.00)2 49.8 − 49.0 = , , 13.3, h 0.0600 0.0600 ∆h f (x0 ) (7.01)2 − (7.00)2 49.1 − 49.0 h = 0.01 : = , , 10.0. h 0.0100 0.0100 h = 0.06 :

f & (x0 ) = 14 ∆h f (x0 )/h

h = 0.01

h = 0.06

∆h f (x0 ) (7.06000)2 − (7.00000)2 49.8436 − 49.0000 = = = 14.0600, h 0.0600000 0.0600000 ∆h f (x0 ) (7.01000)2 − (7.00000)2 49.1401 − 49.0000 h = 0.01 : = = = 14.0100. h 0.0100000 0.0100000 h = 0.06 :

η |c − c˜|

c˜ ∼

|c − c˜| ≤ 5|c|η

|c|η

∆h f (x0 ) f (x0 + h) − f (x0 ) = def h h h = 10−7

h 10

−8

... f (x0 + h) ∼ η |f (x0 + h)|;

f (x0 )

∼ η |f (x0 )|;

∆h f (x0 ) 4 ∼ η |f (x0 + h)| + |f (x0 )| , 2η |f (x0 )|; 3

∆h f (x0 )/h

∼ 2η

|f (x0 )| . h ∆h f (x0 )/h

2η |f (x0 )|/h er

∆h f (x0 )/h

5 5 5 2ηf (x0 )/h 5 2η|f (x0 )| 2η|f (x0 )| 5 5= er ∼ 5 , & ; ∆h f (x0 )/h 5 |f (x0 + h) − f (x0 )| |f (x0 )|h h

f (x0 ) = 49 f & (x0 ) = 14 er , h = 0.01 10−3 er = 10−3 −5 η , 10 /7 , 10−6

∇h f (x0 )/h

δh f (x0 )/h

7η . h ∆h f (x0 )/h N = 6

f : R→R

Eth = 12 |f && (x0 )|h

x0 ∈ R h f & (x0 ) f & (x0 ) , ∆h f (x0 )/h

Eah = 2η|f (x0 )|/h f & (x0 ) η

η ∆h f (x0 )/h

E h = Eah + Eth Eh Eh =

1 && |f (x0 )| |f (x0 )|h + 2η . 2 h h

g g(x) = ax +

b x

a = 12 |f && (x0 )| b = 2η|f (x0 )| x¯

∀x ∈ R, h

g(x)

x>0

b g & (x) = a − 2 x 6 g & (¯ x) = 0 x ¯ = b/a g && (¯ x) > 0 g(x) x>0 Eh 7 η|f (x0 )| h=2 . |f && (x0 )|

p p=2

x ¯

x ¯ h

|f (x0 )| |f && (x0 )|

f & (x0 )

∆h f (x0 )/h N

p · 10−N/2 &

f & (x0 )

f (x0 ) , ∇h f (x0 )/h

δh f (x0 )/h Eh ,

1 &&& |f (x0 )| |f (x0 )|h2 + 2η 24 h

f & (x0 ) f & (x0 ) ,

h=2

3η|f (x0 )| |f &&& (x0 )|

.1/3

1 m−1 ∆m f ), h f = ∆h (∆h m−1 ∇m f ), h f = ∇h (∇h

δhm f = δh (δhm−1 f ).

3 4 δh2 f (x) =δh δh f (x) = δh (f (x + h/2) − f (x − h/2)) =δh f (x + h/2) − δh f (x − h/2) =f (x + h/2 + h/2) − f (x + h/2 − h/2)

− [f (x − h/2 + h/2) − f (x − h/2 − h/2)] =f (x + h) − 2f (x) + f (x − h). m=1

m ∆m h ∇h

δhm f

C m+1

C m+2

x0 ∈ R ∆m h f (x0 ) , hm

∇m h f (x0 ) , hm

f (m) (x0 )

m h h

δhm f (x0 ) hm f

m x0 ∈ R

h0 > 0

C 5 5 m 5 (m) 5 5f (x0 ) − ∆h f (x0 ) 5 ≤ Ch 5 5 m h 5 5 m 5 (m) 5 5f (x0 ) − ∇h f (x0 ) 5 ≤ Ch 5 hm 5

f : R → R

∀h ≤ h0 , ∀h ≤ h0 .

(m + 1)

m x0 ∈ R

h0 > 0

f : R → R

C 5 5 m 5 (m) 5 5f (x0 ) − δh f (x0 ) 5 ≤ Ch2 5 5 m h f && (x0 )

f IV (x0 )

m=2

(m + 2)

∀h ≤ h0 .

m=4

f (x0 + h) − 2f (x0 ) + f (x0 − h) , h2 f (x0 + 2h) − 4f (x0 + h) + 6f (x0 ) − 4f (x0 − h) + f (x0 − 2h) f IV (x0 ) , . h4 f && (x0 ) ,

h h m

h 0, 1, 2, 3, . . .

f :R→R xj = x0 + jh

x0 ∈ R j =

m ∆h f (x0 ) ∆2 f (x0 ) (x − x0 ) + h 2 (x − x0 )(x − x1 ) h 2!h ∆3h f (x0 ) + (x − x0 )(x − x1 )(x − x2 ) + · · · 3!h3 ∆m f (x0 ) + h m (x − x0 )(x − x1 ) · · · (x − xm−1 ). m!h pm m pm (x0 ), pm (x1 ), . . .

pm (x) = f (x0 ) +

pm (x0 ) = f (x0 ), ∆h f (x0 ) f (x1 ) − f (x0 ) (x1 − x0 ) = f (x0 ) + h = f (x1 ), h h ∆h f (x0 ) ∆2 f (x0 ) pm (x2 ) = f (x0 ) + (x2 − x0 ) + h 2 (x2 − x0 )(x2 − x1 ) h 2h 2 = f (x0 ) + ∆h f (x0 ) · 2 + ∆h f (x0 ) = f (x0 ) + 2(f (x1 ) − f (x0 )) + (f (x2 ) − 2f (x1 ) + f (x0 )) = f (x2 ). pm (x1 ) = f (x0 ) +

pm (xj ) = f (xj ) j = 0, 1, 2, . . . , m pm x0 , x1 , x2 , . . . , xm

m f

(m + 1) m

dm ∆m h f (x0 ) p (x) = . m dxm hm

pm j = 0, 1, 2, · · · , m

xj = x0 + jh (i)

∆h f (x0 ) ∆2 f (x0 ) (x − x0 ) + h m (x − x0 )(x − x1 ) h 2!h ∆m f (x ) 0 + · · · + h m (x − x0 )(x − x1 ) · · · (x − xm−1 ); m!h dm ∆m h f (x0 ) p (x ) = . m 0 m dx hm

(m + 1)

pm max

x∈[x0 ,x0 +mh]

|f (x) − pm (x)| ≤

1 hm+1 max |f (m+1) (x)|. 2(m + 1) x∈[x0 ,x0 +mh] pm x = x0

pm (x) = f (x0 ) +

(ii)

q2 x0 − h x0

∇m h

δhm

x0 + h

d2 δh2 f (x0 ) f (x0 + h) − 2f (x0 ) + f (x0 − h) q (x ) = = . 2 0 2 2 dx h h2 f q2 (x0 , f (x0 ))

f h

f q2

x0 − h x0 x0 + h f

x0 − h

x0 x0 + h f - .2 - .3 h f &&& (x0 ) h + 2 3! 2 - .4 - .5 IV V f (x0 ) h f (ξ) h + + , 4! 2 5! 2 - .2 - .3 h h f && (x0 ) h f &&& (x0 ) h & f (x0 − ) = f (x0 ) − f (x0 ) + − 2 2 2! 2 3! 2 - .4 - .5 IV V f (x0 ) h f (η) h + − , 4! 2 5! 2 h h f && (x0 ) f (x0 + ) = f (x0 ) + f & (x0 ) + 2 2 2!

[x0 , x0 + h2 ]

[x0 − h2 , x0 ]

f (x0 + h/2) − f (x0 − h/2) f &&& (x0 ) 2 f V (ξ) + f V (η) 4 = f & (x0 ) + h + h h 24 5!25 δh f (x0 ) δh f (x0 ) f &&& (x0 ) 2 = f & (x0 ) + h + O(h4 ), h 24 O(h4 ) h

h/2 δh/2 f (x0 ) f &&& (x0 ) h2 = f & (x0 ) + + O(h4 ). h/2 24 4

δh f (x0 ) 8δh/2 f (x0 ) − = −3f & (x0 ) + O(h4 ) h h

f & (x0 ) =

8δh/2 f (x0 ) − δh f (x0 ) + O(h4 ). 3h δh

8δh/2 f (x0 ) − δh f (x0 )

= 8f (x0 + h/4) − 8f (x0 − h/4) − f (x0 + h/2) + f (x0 − h/2),

8f (x0 + h/4) − 8f (x0 − h/4) + f (x0 − h/2) − f (x0 + h/2) 3h f & (x0 )

6 8 ... h δh/4 f (x0 ) δh/8 f (x0 ) . . .

h4 f & (x0 )

m = 4 C

f 6 ∀h ≤ h0

5 5 4 5 (4) 5 5f (x0 ) − δh f (x0 ) 5 ≤ Ch2 . 5 5 4 h

3 / / 0 04 / 0 δh4 f (x0 ) = δh δh3 f (x0 ) = δh δh δh2 f (x0 ) = δh2 δh2 f (x0 ) . δh2 δh4 f (x0 ) = δh2 f (x0 + h) − 2δh2 f (x0 ) + δh2 f (x0 − h) = f (x0 + 2h) − 4f (x0 + h) + 6f (x0 ) − 4f (x0 − h) + f (x0 − 2h).

f x0 (2h)1 (2h)2 (2h)3 + f && (x0 ) + f &&& (x0 ) 1! 2! 3! 4 5 (2h) (2h) (2h)6 + f (4) (x0 ) + f (5) (x0 ) + f (6) (η1 ) , 4! 5! 6! 1 2 3 (2h) (2h) (2h) f (x0 − 2h) = f (x0 ) − f & (x0 ) + f && (x0 ) − f &&& (x0 ) 1! 2! 3! (2h)4 (2h)5 (2h)6 (4) (5) (6) + f (x0 ) − f (x0 ) + f (η2 ) , 4! 5! 6! h1 h2 h3 f (x0 + h) = f (x0 ) + f & (x0 ) + f && (x0 ) + f &&& (x0 ) 1! 2! 3! h4 h5 h6 (4) (5) (6) + f (x0 ) + f (x0 ) + f (η3 ) , 4! 5! 6! 1 2 3 h h h f (x0 − h) = f (x0 ) − f & (x0 ) + f && (x0 ) − f &&& (x0 ) 1! 2! 3! 4 5 h h h6 + f (4) (x0 ) − f (5) (x0 ) + f (6) (η4 ) , 4! 5! 6! f (x0 + 2h) = f (x0 ) + f & (x0 )

η1 ∈ ]x0 , x0 + 2h[ η2 ∈ ]x0 − 2h, x0 [ η3 ∈ ]x0 , x0 + h[

η4 ∈ ]x0 − h, x0 [

δh4 f (x0 ) = f (4) (x0 )h4 - 3 4 4 3 4. 64 (6) + f (η1 ) + f (6) (η2 ) − f (6) (η3 ) + f (6) (η4 ) h6 . 6! 6! h0 > 0

C= h ≤ h0

5 4 5 5 δh f (x0 ) 5 (4) 2 5 5 − f (x ) 0 5 ≤ Ch . 5 h4

x0 ∈ R

f : R → R h>0

g(x) = f (x0 ) + ξ0 ∈ [x0 , x1 ]

5 5 17 5 (6) 5 max 5f (x)5 . 90 x∈[x0 −2h0 ,x0 +2h0 ]

x1 = x0 + h x2 = x0 + 2h

∆h f (x0 ) ∆2 f (x0 ) (x − x0 ) + h 2 (x − x0 )(x − x1 ). h 2h

g(xj ) = f (xj ) ξ1 ∈ [x1 , x2 ]

j = 0, 1, 2

f & (ξ0 ) = g & (ξ0 ) ,

f & (ξ1 ) = g & (ξ1 ).

r η ∈ [ξ0 , ξ1 ]

r(x) = f (x) − g(x) r&& (η) = 0 2 x 2 x r&& (x) = r&&& (t)dt = f &&& (t)dt. η

|f (x) − g(x)| ≤ 2h3 max |f &&& (t)| t∈[x0 ,x2 ]

x ∈ [x0 , x2 ].

g(x0 ) = f (x0 )

∆h g(x1 ) = f (x0 ) + ∆h f (x0 ) = f (x0 + h) = f (x1 ),

g(x2 ) = f (x0 ) + 2∆h f (x0 ) + ∆2h f (x0 ) = 2f (x0 + h) − f (x0 ) + ∆2h f (x0 ). ∆2h 3 4 ∆2h f (x0 ) = ∆h f (x0 + h) − f (x0 ) 3 4 3 4 = ∆h f (x0 + h) − ∆h f (x0 )

= f (x0 + 2h) − 2f (x0 + h) + f (x0 ),

g(x2 ) = f (x0 + 2h) = f (x2 ) r r(x) = f (x) − g(x) r(x0 ) = r(x1 ) = r(x2 ) = 0 ∃ξ0 ∈ [x0 , x1 ] ∃ξ1 ∈ [x1 , x2 ]

r& (ξ0 ) = 0

f & (ξ0 ) = g & (ξ0 ),

r& (ξ1 ) = 0

f & (ξ1 ) = g & (ξ1 ).

r& (ξ0 ) = r& (ξ1 ) = 0

r& r&& (η) = 0.

∃η ∈ [ξ0 , ξ1 ] r&&& &&

r (x) = r (x) − r (η) =

r&&& (t)dt.

r&&& (t) = f &&& (t)−g &&& (t) =

g f &&& (t) &&

r (x) =

f &&& (t)dt.

x ∈ [x0 , x1 ] r(x0 ) = 0

x ∈ [x1 , x2 ]

f (x) − g(x) = r(x) = r(x) − r(x0 ) =

|f (x) − g(x)| ≤

|r (s)|ds ≤

r& (s)ds.

|r& (s)|ds ≤ h max |r& (s)|. x0 ≤s≤x1

r& (ξ0 ) = 0 2 r& (s) = r& (s) − r& (ξ0 ) =

s ∈ [x0 , x1 ]

r&& (t)dt.

ξ0

|r& (s)| ≤ |

ξ0

|r&& (t)|dt| ≤

|r&& (t)|dt ≤ h max |r&& (t)|. x0 ≤t≤x1

t ∈ [x0 , x1 ] r&& (t) =

f &&& (u)du,

|r&& (t)| ≤ |

|f &&& (u)|du| ≤

|f &&& (u)|du ≤ 2h max |f &&& (u)|. x0 ≤u≤x2

|f (x) − g(x)| ≤ 2h3 max |f &&& (s)|. x0 ≤s≤x2

x ∈ [x0 , x2 ] ξ ∈ [x0 , x]

f f (x) = G(x) +

f &&& (ξ) (x − x0 )3 , 6

G G(x) = f (x0 ) + f & (x0 )(x − x0 ) +

f && (x0 ) (x − x0 )2 . 2

x ∈ [x0 , x2 ] f &&& (ξ) (x − x0 )3 | 6 (x2 − x0 )3 ≤ max |f &&& (t)| x0 ≤t≤x2 6 4 = h3 max |f &&& (t)|. 3 x0 ≤t≤x2

|f (x) − G(x)| = |

g G

f x0

f [x0 , x0 + 2h] x0

n=2

x0 +2h

f g

f : x ∈ [a, b] → f (x) ∈ R [a, b] 2

f (x)dx.

[a, b] [xi , xi+1 ] i = 0, 1, 2, . . . , N − 1 xi i = 0, 1, 2, . . . , N a = x0 < x1 < x2 < x3 < · · · · · · < xN −1 < xN = b. h=

max

0≤i≤N −1

|xi+1 − xi | N xi

b−a N

xi = a + ih,

f (x)dx =

N −1 2 xi+1 ( i=0

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

f (x)dx.

xi+1

f (x)dx

[−1, +1]

t=2

x − xi −1 xi+1 − xi

x ∈ [xi , xi+1 ]

t ∈ [−1, +1]

xi+1

x = xi + (xi+1 − xi )

t+1 , 2

xi+1 − xi f (x)dx = 2

gi (t)dt,

−1

gi . t+1 gi (t) = f xi + (xi+1 − xi ) , 2 1 +1 −1

[−1, +1]

t ∈ [−1, +1].

g(t)dt g

[−1, +1]

J(g) =

def

M M M J(g)

g β

M (

ωj g(tj )

j=1

−1 ≤ t1 < t2 < · · · < tM ≤ 1 ω1 ω2 . . . ωM M 1 +1 −1 g(t)dt

[−1, +1]

J(αg + β,) = αJ(g) + βJ(,).

M =2 t1 = −1, t2 = +1, ω1 = 1, ω2 = 1 J(g) = g(−1) + g(1). J(g)

1 +1 −1

g(t)dt

J(g)

g(t)

t −1 = t1

1 = t2 [−1, +1]

1 xi+1 xi

1 +1 −1

f (x)dx

gi (t)dt

J(gi )

. M xi+1 − xi ( tj + 1 ωj f xi + (xi+1 − xi ) . 2 2 j=1 1b a

Lh (f ) =

N −1 ( i=0

f (x)dx

. M xi+1 − xi ( tj + 1 ωj f xi + (xi+1 − xi ) . 2 2 j=1

t1 = −1 t2 = 1 ω1 = ω2 = 1 Lh (f ) =

N −1 ( i=0

4 xi+1 − xi 3 f (xi ) + f (xi+1 ) . 2 Lh (f )

t x0

x3 1b a

1b a

f (x)dx M

[xi , xi+1 ]

Lh (f ) t1 , t2 , . . . , tM

f (x)dx

N =4

ω 1 , ω 2 , . . . , ωM [a, b] Lh (f )

J(g) !

J(g) =

M (

ωj g(tj )

j=1

1 +1 −1

r≥0

g(t)dt

J(p) =

p(t)dt

−1

≤r J(·)

Lh (f )

1b a

J(g) =

M (

f (x)dx

ωj g(tj )

j=1

1 +1 −1

g(t)dt [a, b]

Lh (f )

h r+1 C

f [a, b]

52 5 5 b 5 5 5 f (x)dx − Lh (f )5 ≤ Chr+1 . 5 5 a 5 Lh (f ) p

p(t) = αt + β α, β ∈ R 2

p(t)dt = J(p).

−1

1 +1 −1

r=1 [a, b] i = 0, 1, 2, . . . , N [a, b]

h = (b−a)/N xi = a+ih

f 52 5 5 b 5 5 5 f (x)dx − Lh (f )5 ≤ Ch2 , 5 5 a 5

1b a

g(t)dt

f (x)dx

N h 1b a

f (x)dx h r

Lh (f )

ωj 1 ≤ j ≤ M

J(·)

M [−1, +1] −1 ≤ t1 < t2 < t3 < · · · < tM ≤ 1

J(g) =

ω 1 , ω 2 , . . . , ωM

j=1 ωj g(tj )

r ϕ1 , ϕ2 , . . . , ϕM ϕj

t1 t2 . . . tM M −1

PM−1

ϕj (t) =

(t − t1 )(t − t2 ) · · · (t − tj−1 )(t − tj+1 ) · · · (t − tM ) , (tj − t1 )(tj − t2 ) · · · (tj − tj−1 )(tj − tj+1 ) · · · (tj − tM )

j = 1, 2, . . . , M

g : t ∈ [−1, +1] → g(t) ∈ R M −1 t1 , . . . , tM

g˜

g˜(t) =

M (

g(tj )ϕj (t).

j=1

1 +1 −1

g˜(t)dt =

−1

g(t)dt M (

−1

g(tj )

ωj =

g˜(t)dt

ϕj (t)dt,

−1

j=1

1 +1

ϕj (t)dt

−1

J(g) =

j=1

1 +1

ωj g(tj )

−1

t1 < t 2 < · · · < t M M ϕ1 , ϕ2 , · · · , ϕM

[−1, +1]

J(g) =

M (

PM−1

ωj g(tj )

j=1

M −1 ωj =

ϕj (t)dt,

j = 1, 2, . . . , M.

−1

J(·) M −1 J(p) =

M ( j=1

ωj p(tj ) =

−1

p(t)dt,

g(t)dt

p ∈ PM−1 J(ϕk ) =

p = ϕk k = 1, 2, . . . , M

M (

ωj ϕk (tj ) =

j $= k

ϕk (t)dt.

−1

j=1

ϕk (tj ) = 0

ϕk (tk ) = 1 2 +1 ωk = ϕk (t)dt. −1

M −1 M −1 t1 , t2 , . . . , tM

p PM−1 p(t) =

M (

p(tj )ϕj (t).

j=1

M (

p(t)dt =

−1

j=1

M (

p(tj )

ϕj (t)dt

−1

p(tj )ωj = J(p).

j=1

ω1 ω2 . . . ωM 'M t1 , t2 , . . . , tM j=1 ϕj (t) M t 1 t2 . . . tM M (

ωj =

−1

j=1

−1

M −1 1

  2 M (   ϕj (t) dt =

dt = 2,

−1

j=1

2 M = 2 t1 = ϕ1 ϕ2

t2 = +1 t1 t2 ϕ1 (t) =

ω1 =

t − t2 (1 − t) = t1 − t 2 2

ϕ2 (t) =

ω2 =

−1

ϕ1 (t)dt = 1

t − t1 (t + 1) = . t2 − t 1 2 +1

−1

ϕ2 (t)dt = 1,

M −1 r

M −1

(M = 1) t1 = 0. t1 = 0

P0 ϕ1 (t) = 1,

ω1 =

∀t ∈ [−1, +1].

ϕ1 (t)dt = 2

−1

J(g) = 2g(0). 1 +1 −1

g(t)dt

[−1, +1]

p ∈ P1

p(t) = αt + β 1 +1

α, β ∈ R

g(0)

−1

p(t)dt = 2β = 2p(0)

g(t)

t −1

0 = t1

+1 [−1, +1]

Lh (f ) =

N −1 ( i=0

(xi+1 − xi )f

xi + xi+1 2

52 5 5 b 5 5 5 f (x)dx − Lh (f )5 ≤ Ch2 . 5 5 a 5 [xi , xi+1 ] ξi

f (ξi )

[xi , xi+1 ]

t2 = 0 t3 = +1

ϕ1 ϕ2 ϕ3 1 2 (t − t), 2

ϕ1 (t) =

ω1 =

ϕ1 (t)dt =

−1

1 , 3

ϕ2 (t) = 1 − t2 ,

ω2 =

ϕ2 (t)dt =

−1

J(g) =

M = 3 t1 = −1

ϕ3 (t) =

4 , 3

ω3 =

1 2 (t + t). 2

ϕ3 (t)dt =

−1

1 . 3

1 4 1 g(−1) + g(0) + g(1). 3 3 3 1/3

2/3

Lh (f ) =

N −1 ( i=0

2 g(t) = t3

xi+1 − xi 6

. . xi + xi+1 f (xi ) + 4f + f (xi+1 ) . 2

J(g) = 0

1 +1 −1

g(t)dt =

1 +1 −1

52 5 5 b 5 5 5 f (x)dx − Lh (f )5 ≤ Ch4 . 5 5 a 5 h4

Lh (f ) h

3 3

t dt = 0

1b a

f (x)dx

t1 , t2 , . . . , tM 1 +1 â&#x2C6;&#x2019;1 p(t)dt 1b a

J(p) = p

Lh (f )

j=1

f (x)dx

r 1b a

f (x)dx

Lh (f )

M 1

LM (t) =

2M M !

dM 2 (t â&#x2C6;&#x2019; 1)M . dtM

tâ&#x2C6;&#x2C6;R L0 (t) = 1,

L1 (t) = t,

L2 (t) =

3t2 â&#x2C6;&#x2019; 1 , 2

Â·Â·Â·

L0 L1 L2 . . .

L0 L1 L2 . . . L0 L1 . . . LM PM 1 +1 i $= j Li (t)Lj (t)dt = 0 â&#x2C6;&#x2019;1 LM ] â&#x2C6;&#x2019; 1, +1[

Lj (t)

L 0 , L 1 , L 2 , . . . , LM PM i>j 2+1 Li (t)Lj (t)dt

Ï&#x2030;j p(tj )

1 2(i+j) i!j!

â&#x2C6;&#x2019;1

2+1

â&#x2C6;&#x2019;1

di 2 dj (t â&#x2C6;&#x2019; 1)i j (t2 â&#x2C6;&#x2019; 1)j dt i dt dt

<5 5t=1 j 5 diâ&#x2C6;&#x2019;1 2 5 i d 2 j5 5 (t â&#x2C6;&#x2019; 1) j (t â&#x2C6;&#x2019; 1) 5 5 iâ&#x2C6;&#x2019;1 (i+j) dt 2 i!j! dt 1

t=â&#x2C6;&#x2019;1

â&#x2C6;&#x2019;

2+1

â&#x2C6;&#x2019;1

= j+1 diâ&#x2C6;&#x2019;1 2 i d 2 j (t â&#x2C6;&#x2019; 1) j+1 (t â&#x2C6;&#x2019; 1) dt . dtiâ&#x2C6;&#x2019;1 dt

(t − 1)

(t2 − 1)i

i 1 t = −1

t=1

2+1 Li (t)Lj (t)dt =

−1

(−1) 2(i+j) i!j!

2+1

−1

(i − 1)

di−1 2 dj+1 (t − 1)i j+1 (t2 − 1)j dt. i−1 dt dt

j 2+1 Li (t)Lj (t)dt

−1

(−1)j 2(i+j) i!j!

2+1

−1

(−1)j (2j)! 2(i+j) i!j!

2j di−j 2 i d (t − 1) (t2 − 1)j dt 2j dti−j dt ) *+ , (2j)!

2+1

−1

di−j 2 (t − 1)i dt dti−j

5 5t=1 5 (−1) (2j)! 55 di−j−1 2 i5 (t − 1) = 0. 5 5 i−j−1 (i+j) 2 i!j! dt t=−1 j

= t1 , t2 , . . . , ts

−1

LM s≤M

+1 LM

p(t) = (t − t1 )(t − t2 )(t − t3 ) . . . (t − ts ) p ∈ Ps p p(t)LM (t) ≥ 0 ∀t ∈ [−1, +1] p(t)LM (t)

tj 1 ≤ j ≤ s p(t)LM (t) ≤ 0 ∀t ∈ [−1, +1]

2+1 p(t)LM (t)dt $= 0.

−1

α0 , α1 , α2 , . . . , αs p(t) =

s (

αj Lj (t)

j=0

2+1 2+1 2+1 s ( p(t)LM (t)dt = αj Lj (t)LM (t)dt = αs Ls (t)LM (t)dt. j=0

−1

1 +1

−1

p(t)LM (t)dt LM t1 , t2 , . . . , tM !

−1

s=M

J(g) =

M (

ωj g(tj )

j=1

M t1 < t 2 < · · · < t M M

ω 1 , ω 2 , . . . , ωM 2 ωj =

ϕj (t)dt,

j = 1, 2, . . . , M,

−1

ϕ1 , ϕ2 , . . . , ϕM

PM−1

≥1

r = 2M − 1 J(g) =

j=1

ωj g(tj ) 2M − 1

p t∈R

p˜(t) =

M (

p(tj )ϕj (t),

j=1

ϕ1 , ϕ2 , . . . , ϕM t1 , t2 , . . . , tM M −1 M

PM−1 p˜

q(t) = p(t) − p˜(t) q t1 , t2 , . . . , tM v

t1 , t2 , . . . , tM q

q(tj ) = 0 M

∀t ∈ R.

2M − 1 j = 1, 2, . . . , M q

v(t) = (t − t1 )(t − t2 )(t − t3 ) · · · (t − tM ) w

M −1

q(t) = v(t)w(t) v

∀t ∈ R.

M M

v(t) = αLM (t)

∀t ∈ R,

LM α

∀t ∈ R.

M −1 w(t) =

M−1 (

β0 , β1 , β2 , . . . , βM−1 ∈ R

βk Lk (t).

k=0

q(t)dt =

−1

v(t)w(t)dt = α

−1

M−1 (

βk

p˜(t)dt,

k=0

q 2

p(t)dt =

−1

LM (t)Lk (t)dt = 0. −1

+1 −1

p˜ 2

p(t)dt =

−1

M (

p(tj )

ϕj (t)dt =

−1

j=1

M (

ωj p(tj ) = J(p).

j=1

! ωj j = 1, 2, ..., M ϕ2j

+1 −1

ϕ2j (t)dt = J(ϕ2j ) =

M (

2M − 2 ωk ϕ2j (tk ) = ωj .

k=1

M p

2M 2

−1

p(t)dt $= J(p).

p(t) = 1 +1 −1

p(t)dt > 0

j=1 (t

− tj )2

J(p) = 0

M f

[a, b]

Lh (f ) 52 5 5 b 5 5 5 f (x)dx − Lh (f )5 ≤ Ch2M , 5 5 a 5

xi i = 0, 1, . . . , N

C [a, b]

L1 (t) = t

t1 = 0 h2

L2 (t) = 12 (3t2 − 1)

1 t1 = − √ 3

1 t2 = √ . 3

ϕ1 ϕ2

t1 t2 √ 3t + 1 ϕ2 (t) = 2

P2 √ 1 − 3t ϕ1 (t) = 2

ω1 =

ϕ1 (t)dt = 1

ω2 =

−1

ϕ2 (t)dt = 1.

−1

3 3 √ 4 √ 4 J(g) = g −1/ 3 + g 1/ 3 , Lh (f ) =

N −1 ( i=0

< ? @ √ xi+1 − xi 3−1 f xi + √ (xi+1 − xi ) 2 2 3 ? @= √ 3+1 +f xi + √ (xi+1 − xi ) . 2 3

f C 52 5 5 b 5 5 5 f (x)dx − Lh (f )5 ≤ Ch4 . 5 5 a 5

α t2 = −α t3 = α t4 = +1

0<α<1 ω 1 , ω2 , ω3 , ω4

J(g) =

4 (

ωj g(tj ),

j=1

[−1, +1]

t1 = −1

ω1 , ω2 , ω3 , ω4 p α α

α 1 +1

J(p) =

−1

r>3 ω1 , ω2 , ω3 , ω4

ωi =

J(p) =

p(t)dt

1 +1 −1

p(t)dt

ϕi (t)dt,

−1

ϕi i = 1, 2, 3, 4 t1 t2 t3 t4

t+α −1 + α t+1 ϕ2 (t) = −α + 1 ϕ1 (t) =

ω1 =

t−α t−1 · , −1 − α −1 − 1 t−α t−1 · · , −α − α −α − 1 ·

ϕ1 (t)dt =

−1 1

ω2 =

ϕ2 (t)dt =

−1

ω4 = ω1

1 1 − 3α2 · , 3 1 − α2 1 2 · . 3 1 − α2

ω3 = ω2

J(p)

p p r−1

p(t) = atr + q(t)

r a∈R J(p) = a

4 (

ωj trj + J(q)

j=1

p(t)dt = a

−1

J(p) = J(q) =

1 +1 −1

q(t)dt

+1 r

t dt +

−1

1 +1 −1

q(t)dt.

−1

p(t)dt

p q

J(tr ) =

r−1 +1

−1

tr dt.

3 4

J(t ) =

4 (

ωj t4j

j=1

1 +1

J(t4 )

ω1 = ω4 =

1 6

−1

t4 dt =

−1

2 . 5

√ α = 1/ 5

t4 dt

ω2 = ω3 =

J(g) =

2 1 − 3α2 + 2α4 = , 3 1 − α2

5 6

. - 3 3 √ 4. √ 4 1 5 g(−1) + g(1) + g −1/ 5 + g 1/ 5 . 6 6

t5 5

J(t ) = 0 =

t5 dt.

−1

t6 . . √ 46 3 √ 46 1 5 3 26 6 6 J(t ) = (−1) + (1) + −1/ 5 + 1/ 5 = 6 6 75 2 1 2 $= t6 dt = . 7 −1 6

r=5

t3 = α

0 < α ≤ 1 ω 1 , ω2 , ω3

t1 = −α t2 = 0

J(g) =

3 (

ωj g(tj ),

j=1

[−1, 1] ω1 ω2 ω3 p

α J(p) =

J(p) = 11

−1

p(t)dt

−1

p(t)dt p

ωi =

ϕi (t)dt,

−1

ϕi i = 1, 2, 3 t1 t2 t3

t t−α · , α 2α t+α t−α ϕ2 (t) = · . α −α ϕ1 (t) =

ω1 =

ϕ1 (t)dt =

1 , 3α2

ϕ2 (t)dt =

−2 + 2. 3α2

−1 1

ω2 =

−1

ω3 = ω1

J(p)

p J(t3 ) =

1 +1 −1

3 t3 dt

J(t3 ) = ω1 (−α)3 + ω2 · 0 + ω3 α3 = 0, 2

1 A 4 Bt=+1 t t=−1 = 0. 4

t3 dt =

−1

1 +1 −1

J(t4 ) =

4 t4 dt J(t4 ) = ω1 (−α)4 + ω2 · 0 + ω3 α4 = 2ω1 α4 , 1 +1 −1

t4 dt = 2/5

ω1 α4 = 1/5

J(t5 ) =

α=

t5 dt = 0

−1

α =

6 3/5

L3 L3 (t) =

. 5 3 t t2 − . 2 5

6 3/5

t6= 0 α = 3/5

t=±

6 3/5

3 r = 2·3−1 = 5

−1

+1 −1 +1 LM M = 2 & 2 M = 3 √ √ L3 (t) = (15t − 3)/2 t = −1/ 5 t = 1/ 5

M = 1 L&3 2M − 1

TM TM

A!x = !b. A !b

N ×N

N N

N 

a11  a21  A=  aN 1 N

a12 a22

··· ···

aN 2

···

xj  a1N a2N   ,  aN N

aij 1 ≤ i, j ≤ N bj 1 ≤ j ≤ N 1≤j≤N 

 b1  b2   !b =   ,   bN

 x1  x2    !x =   .   xN

x1 , x2 , . . . , xN  a11 x1 + a12 x2 + · · · + a1N xN     a21 x1 + a22 x2 + · · · + a2N xN    

aN 1 x1 + aN 2 x2 + · · · + aN N xN

= b1 , = b2 , = bN .



A aij = 0 i≤N

i, j

1≤j<

1≤i<j≤N A

A A aii 1, 2, . . . , N

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

A xN , xN −1 , . . . , x1 xN = bN i = N − 1, N − 2, . . . , 3, 2, 1 xi = bi −

N (

aij xj .

j=i+1

A A!x = !b

N =3



A!x = !b

4 8 A = 3 8 2 9

 12 13 , 18

  4 !b =  5  . 11

  4x1 + 8x2 + 12x3 = 4, 3x1 + 8x2 + 13x3 = 5,  2x1 + 9x2 + 18x3 = 11. a11 = 4 x1 + 2x2 + 3x3 = 1.

3 2   x1 

+ 2x2 2x2 5x2

+ 3x3 + 4x3 + 12x3

= 1, = 2, = 9.

x2 + 2x3 = 1. 5   x1

2x2 x2

+ +



3x3 2x3 2x3

= = =

1, 1, 4,

3x3 2x3 x3

= = =

1, 1, 2.

2   x1

+ 2x2 x2



+ +

x3 , x2 , x1

x3 = 2,

x2 = −3,

x1 = 1.

A(i) i

!b(i)

A(i)

 1  0  0  0   =        

(i)

a12 1 0 0

(i)

a13 (i) a23 1 0

a14 (i) a24 (i) a34 1

··· ··· ··· ···

(i)

···

(i)

···

aN i

 (i) a1N (i)  a2N  (i)  a3N   (i) a4N    .    (i)  aiN     (i) aN N

A(i)

!b(i+1)

A(i)

i (i+1)

aij

(i)

= aij /aii ,

(i)

aii

j = i + 1, i + 2, . . . , N.

(i+1)

(i)

= bi /aii . A(i) k = i + 1, i + 2, . . . , N k = i + 1, i +

k (i) aki

(i+1)

··· ··· ··· ···

aii

!b(i)

akj

(i)

1 0

(i+1)

2, . . . , N

a1i (i) a2i (i) a3i (i) a4i

(i)

(i+1)

= akj − aki ∗ aij (i+1)

(i)

j = i + 1, i + 2, . . . , N.

(i)

(i+1)

= bk − aki ∗ bi

i (i) akj i + 1 ≤ k, j ≤ N aij := aij /aii ,

j = i + 1, i + 2, . . . , N, aij /aii

aij

i=1 ···

N −1

aij 1 ≤ i, j ≤ N

bj 1 ≤ j ≤ N

aij 1 ≤ i < j ≤ N

bj 1 ≤ j ≤ N

                      

i=1

N −1

p := 1/aii  j =i+1  aij := p ∗ aij

xi i N

b := p ∗ bi i k =i+1 N     j =i+1 N     akj := akj − aki ∗ aij  bk := bk − aki ∗ bi

i k aki

k aki

p := 1/aN N

bN := p ∗ bN

i = 1, 2, 3, . . . N −1  1 0  0    0

a12 1 0

a13 a23 1

··· ··· ···

···

 a1N a2N   a3N     1 !x



 x1  x2     x3        xN



 b1  b2     b3        bN

!b  

i = N − 1 1( N ' bi := bi − aij ∗ bj

− 1)

j=i+1

aii = 0 aii

k×k

aij 1 ≤ i, j ≤ k 1 ≤ k ≤ N Ak

k = 1, 2, . . . , N

A A(i)

i (i) Ak

k 1≤k≤N

(i)

Ai = aii . A

Ai                     

(1)

A1 = A1 , (1) (2) A2 = a11 A2 , (1) (2) (3) A3 = a11 a22 A3 , (1) (2)

(i−1)

Ai = a11 a22 . . . ai−1,i−1

(1) (2) (N ) a11 , a22 , . . . , aN N

Ai $= 0

(i)

Ai ,

i = 1, 2, . . . , N !

N −1H (

I (N − i) + 1 + (N − i)(N − i + 1) + 1

i=1

N −1H (

I2 (N − i + 1) + 1

i=1

= =

N 2 + (N − 1)2 + (N − 2)2 + · · · + 22 + 12 N (

j2.

j=1

N (

j2 =

j=1

N3 N2 N + + , 3 2 6 Nm = N 3 /3 + O(N 2 )

N N

N →∞

j2 =

N (

j=1

1 x2 dx + j − . 3 j−1

N 2 ( j=1

= =

x2 dx +

j−1

N ( j=1

j−

N 3

(N + 1)N N − 2 3 0 3 2 N N N + + . 3 2 6 x2 dx +

N 3 /3

O(N 2 )

i=1

  0x1 5x1  6x1

+ x2 + 2x2 + 8x2

+ + +

3x3 3x3 x3

= = =

1, 4, 1,

  6x1 5x1  0x1

+ 8x2 + 2x2 + x2

+ + +

x3 3x3 3x3

= = =

1, 4, 1.

N −1

p := 1/aii A

aii

i = 1, 2, . . . , N N (

aij xj = bi ,

j=1

ri > 0

max ri | aij |= 1.

1≤j≤N

k := i

m := abs(aii )

m =| aii |



j =i+1 N    s := abs(aji )    m<s  k := j m := s k=i

i m k m=0 k=i xi



j=i    t := aij    aij := akj  akj := t

aij

akj

t := bi

bi := bk bk := t

i N2 N3

4.218613x1 3.141592x1

+ +

6.327917x2 4.712390x2

= =

10.546530 7.853982. A

x1 = x2 = 1. →  

↓

4.218611x1  3.141594x1 ↑

+ 6.327917x2 + 4.712390x2

x1 = −5,

= 10.546530 = 7.853980. ↑

x2 = +5.

Ox1 , x2

A yj 1 ≤ j ≤ N

N ×N

N 

1 !y 1= 

N ( j=1

1/2

yj2 

A ||| A |||= max y "=0 %

1 A!y 1 . 1 !y 1

N 1 A!x 1≤||| A ||| · 1 !x 1 .

A AT A

N ×N A

ω ||| A |||=

√

AT A ω 1 , ω 2 , ω 3 , . . . , ωN ϕ ! j $= 0 AT A ωj AT A! ϕj = ωj ϕ !j 1 A! ϕj 12 = ϕ ! Tj AT A! ϕj = ωj ϕ ! Tj ϕ ! j = ωj 1 ϕ ! j 12 T ωj ≥ 0 j = 1, 2, . . . , N A A ω = max ω 1≤j≤N j √ ω D N ×N ω 1 , ω 2 , . . . , ωN N ×N Q AT A = QT DQ.

1 A!y 12 !y T AT A!y y!T QT DQ!y = max = max T T 2 T y"=0 1 y % y "=0 % y "=0 ! % !1 y! y! y Q Q!y 'N 2 !z T D!z j=1 ωj zj = max T = max 'N . 2 z "=0 ! % % z "=0 z !z j=1 zj

||| A |||2 = max

N ( j=1

ωj zj2 ≤

N (

ωzj2 = ω

j=1

N (

zj2 .

j=1

N N (

ωj zj2 = ω

j=1

ω = ωk !

N (

zj2

j=1

zj = 0 ∀j $= k |||A|||2 = ω

zk = 1

A A

N ×N χ(A) =||| A ||| · ||| A−1 |||

A−1

A A A λ1 , λ2 , . . . , λN

N ×N

χ(A) =

max | λj |

1≤j≤N

min | λj |

1≤j≤N

AT A = A2

A T

A A

λ21 , λ22 , . . . , λ2N ||| A |||= max |λj |. 1≤j≤N

A−1

−1 −1 λ−1 1 , λ2 , . . . , λN

||| A−1 |||= max |λ−1 j |= 1≤j≤N

1 . min |λj |

1≤j≤N

! N

A!x = !b. ! δb

!b ! A!y = !b + δb,

! !y = !x + δx ! 1δx1/1! x1

! = δb. ! Aδx ! !b1 1δb1/1

A ! δb N ! = δb ! A!x = !b Aδx

N ×N

!b ! δx

N N

! ! 1δx1 1δb1 ≤ χ(A) 1!x1 1!b1

χ(A)

A−1

! ! ! 1δx1 1A−1 δb1 1δb1 = ≤ |||A−1 ||| . 1!x1 1!x1 1!x1 1!b1 = 1A!x1 ≤||| A ||| ·1!x1, 1 ||| A ||| ≤ . 1!x1 1!b1 ! !x !b

χ(A)

! !b1 1δb1/1

! 1δx1/1! x1

! !b1 1δb1/1 χ(A)

! 1δx1/1! x1 ! ! χ(A)·1δb1/1b1 ! ! !b1 1δx1/1! x1 = χ(A) · 1δb1/1 −p η = 10 ! 1δx1/1! x1

χ(A) · η = 10log10 χ(A) · 10−p = 10log10 χ(A)−p . p [p − log10 χ(A)] [·] χ(A)

107

M ×N N !x M

A!x = !b N A!x , !b 1 A!x − !b 1

!x N N

1 A!x − !b 1≤1 A!y − !b 1 . A!x = !b

A N !x

M ×N

(M ≥ N )

AT A!x = AT !b B = AT A

B N

N ×N !z T B!z

!z T B!z = !z T AT A!z =1 A!z 12 . !z T B!z = 0

A!z = 0

A AT A

N !z = 0 N

!x AT A!x = AT !b. !x

!x 1 A!x − !b 1<1 A!y − !b 1 . !z = !x − !y

!z $= 0

y! $= !x

1 A!y − !b 12 =1 (A!x − !b) − A!z 12 3 4T 3 4 = (A!x − !b) − A!z (A!x − !b) − A!z

=1 A!x − !b 12 −!z T AT (A!x − !b) − (A!x − !b)T A!z + !z T AT A!z =1 A!x − !b 12 −2!z T AT (A!x − !b) + !z T AT A!z =1 A!x − !b 12 −2!z T (AT A!x − AT !b)+ 1 A!z 12 .

!x 1 A!y − !b 12 =1 A!x − !b 12 + 1 A!z 12 . A

A!z $= 0 p!

1A!y −!b1 p!

!z $= 0 1A! p − !b1 ≤ p ! = !x 1A!x −!b1 < 1A! p −!b1

!y !x p! ! A

AT A A!x = !b N

AT A!x = AT !b.

M ×N

N N

!r (!y ) = A!y − !b. !r

A!x = !b

∀!y ∈ RN , AT A!x = AT !b

1 !r (!x) 1≤1 !r (!y ) 1,

p 1 , p 2 , . . . , pM M (

M (

pi ri2 (!y )

i=1

p 1 , p 2 , . . . , pM

i=1

A!x = !b M ×M D=

M ( i=1

ri2 (!y ),

√ √ √ ( p1 , p2 , . . . , pM ),

pi ri2 (!y ) = !rT (!y )D2!r (!y ) = 1 D(A!y − !b) 12 . N

1 DA!x − D!b 1 A DA !b

AT D2 A!x = AT D2!b

D!b

M > N !y

A!x = !b

p 1 , p 2 , . . . , pM

A N ×N λ1 , λ2 , λ3 , . . . λN | λ1 |≥| λ2 |≥| λ3 | ≥ · · · ≥| λN | . ϕ ! 1, ϕ !2, . . . , ϕ !N A! ϕj = λj ϕ !j , ε ! = ε! δb ϕN

! δx

1 ≤ j ≤ N. ! δb

A!x = !b

!b = ϕ !1 ! ! Aδx = δb

! ! 1δx1 1δb1 = χ(A) 1!x1 1!b1 χ(A)

A !x

! δx

A!x = ϕ !1

!x =

1 ϕ !1 λ1

! = ε ϕ δx !N , λN

! ! 1δx1 |λ1 | 1ε! ϕN 1 |λ1 | 1δb1 = = . 1!x1 |λN | 1! ϕ1 1 |λN | 1!b1 A

χ(A) =

|λ1 | . |λN |

y(t) = αe−βt ,

! = ε! Aδx ϕN

β t = t1 , t2 , . . . , tN

y(t) = y1 , y2 , . . . , yN ln y(t) = ln α− βt α ˜ β

α ˜ = ln α ln yi = α ˜ − βti ,

1 ≤ i ≤ N. α ˜

β A!x = !b  1 1  A=  1

 t1 t2     tN



 N AT A =  N '

i=1

α ˜

−β

!x =

α ˜ −β



 ln y1  ln y2   !b =   .   ln yN

AT A!x = AT !b  N  ' ln y i   i=1  AT !b =  N '  ti ln yi

 ti  i=1  N  ' t2i N '

i=1

AT A

N ×N

A 

h = π/(N + 1)

2 −1 −1 2   A=  

−1



   .  −1 2 −1 −1 2

sin(α − β) + sin(α + β) = 2 sin α cos β,

A λk = 2 − 2 cos kh,

1 ≤ k ≤ N,

ϕ ! k = (sin kh, sin 2kh, . . . , sin N kh)T ,

1 ≤ k ≤ N.

A χ(A) χ(A) = N

1 − cos N h 1 + cos h = . 1 − cos h 1 − cos h 1 − h2/2

cos h

O(N )

χ(A) , 4/h2 = A

A!x = !b !r (!x) = A!x − !b 1 · 1∞

1·1

1!y1∞ = max1≤i≤M |yi | N !x 1A!x − !b1∞ ≤ 1A!y − !b1∞ , N

1!r (!x)1

!r(!x)

LU A

N ×N A!x = !b

N ×N A = LU

U U

A(i)

i A(i) S (i)

i 

S (i)

(i)

1 0 0 1  0 0       =        

(i)

= 1/aii

(i)

0 0 1



0 (i) si (i) si+1 (i) si+2

1 0

(i)

sN (i)

(i)

sk = −aki /aii

         i        

k = i + 1, i + 2, . . . , N (i) aii S (i) A(i)

A(i+1) = S (i) · A(i) . A(N +1)

U = A(N +1) = S (N ) A(N ) = S (N ) S (N −1) A(N −1) = . . . = S (N ) S (N −1) . . . S (1) A(1) . A(1) = A L(j) = [S (j) ]−1

S (j) A = L(1) L(2) L(3) . . . L(N ) U.

A = LU, L(1) L(2) L(3) . . . L(N )

L LU

A A = L 1 U1 = L 2 U2 , L1 L2 L1

U1 U2 L2

1 U1

A U2

−1 L−1 2 L 1 = U2 U1 .

L−1 2 L1

U2 U1−1 1 U2 U1−1 −1 L2 L1 U2 U1−1 L2 = L1 U2 = U1 !

L−1 2 L1

A A

i, j ≤ 3

3 ,ij

 a11 a21 a31 )

a12 a22 a32 *+ A

 a13 a23  a33 ,

3 1≤j≤i≤3 

,11 ,21 ,31 )

0 ,22 ,32 *+

  a11 a21  a31

aij 1 ≤

uij 1 ≤ i < j ≤ 3

 0 0 ,33 ,

U   ,11 ,21  . ,31

  1 u12 u13 0 1 u23  . 0 0 1 ) *+ , U

U A a12 = ,11 u12

a13 = ,11 u13 ,

,11 u12 = a12 /,11 ,

u13 = a13 /,11 . L

U L

L U ,21 u12 + ,22 = a22

,31 u12 + ,32 = a32

,22 = a22 − ,21 u12

,32 = a32 − ,31 u12 . U

U ,21 u13 + ,22 u23 = a23 u23 = (a23 − ,21 u13 )/,22 .

L U

,33 ,33 = a33 − ,31 u13 − ,32 u23 . A

k−1 (2 ≤ k < N ) U i

N ×N k−1 k

L k L (k ≤ i)

U k

L L

aik = ,i1 u1k + ,i2 u2k + · · · + ,i,k−1 uk−1,k + ,ik ,ik = aik − k L

k−1 (

,ij ujk .

j=1

U U (k + 1 ≤ i)

aki = ,k1 u1i + ,k2 u2i + · · · + ,kk uki

uki

  k−1 ( 1  = aki − ,kj uji  . ,kk j=1

A 

,11  ,21   ,31    ,N 1

u12 ,22 ,32

u13 u23 ,33

··· ··· ···

,N 2

,N 3

···

N ×N

 u1N u2N   u3N  .   ,N N

A A

LU LU

LU N3

LU m A !x (() = !b (() ,

, = 1, 2, . . . , m,

LU aij 1 ≤ i, j ≤ N

aij 1 ≤ j ≤ i ≤ N

aij 1 ≤ i < j < N  

i=2

a1i := a1i /a11



k=2

             

1 N −1

1 L

k−1 '

akk := akk − akj ∗ ajk j=1  i=k+1 N   k−1  a := a − ' a ∗ a ik ij jk  ik  j=1  4 k−1 ' 1 3  aki := aki − akj ∗ aji akk j=1 N' −1 aN N := aN N − aN j ∗ ajN

A U ,kk

U ,N N

j=1

((−1)

, = 2, 3, . . . , m !b (() m=2

!x (1) , !x (2) , . . . A !x(1) = !b(1) ,

A!x(2) = 1!x(1) 12 !x(1) , !b(1)

!b(2) = 1!x(1) 12 !x(1)

N m A

m A LU!x(() = !b(() ,

, = 1, 2, . . . , m.

y!(() = U!x(() , , = 1, 2, . . . , m L!y (() = !b(() ,

U!x(() = !y (() .

LU LU

N ×N U L

det(A) = det(L) · det(U ) det(U ) = 1 det(L) = ,11 .,22 .,33 . . . ,N N L det(A) =

N L

,jj ,

j=1

det(A) A

N! N N ! = 100! , 10158

N = 100

109 10149

200

3 10141

100! LU

A A

100

A (1 ≤ k < N )

akk

j > k A

LU A

A!x = !b !b

A L LU

pk = j k

!" T

A=A

N ×N

!y A!y ≥ 0

!y A!y = 0

!y = 0

A !z

N ×N

Ak k !z T Ak !z

y! =

!z 0

Ak N

}k } (N − k)

!z T Ak !z = y! T A!y . (ii) !z T Ak !z ≥ 0

(iii) !z T Ak !z = 0 !

!z = 0

A L

A = LLT

A LU

˜ A = LU U Ak

N ×N ˜ L ˜k L Uk

˜ k Uk Ak = L

˜ L

˜ k A L ,˜kk 1 ≤ k ≤ N Ak

det Ak = ,˜11 .,˜22 .,˜33 . . . ,˜kk > 0. k = 1, 2, 3, . . . , N ,˜jj > 0,

j = 1, 2, . . . , N.

N ×N -M . M M M D= ,˜11 , ,˜22 , ,˜33 , . . . , ,˜N N def

E 3 4 ,˜11 , ,˜22 , ,˜33 , . . . , ,˜N N .

E = D2 = N ×N

ˆ L

ˆ = LE ˜ −1 L

ˆ A = LEU A ˆT . A = AT = U T E L

ˆT L UT E A ˆ T = U. L

ˆ L ˆ T = LDD ˆ ˆT . A = LE L

N ×N

L ˆ L = LD A = LLT .

A = MMT M LLT = M M T LT M −T = L−1 M M

−T

M LT

M −T

LT M −T

L−1 M L−1 M ,jj mjj = , mjj ,jj ,jj

1 ≤ j ≤ N, N ×N L−1 M = I

mjj

I M =L !

L A = LLT

L LU

˜ L

k akj = ajk

√ akk

A!x = !b, A A = LLT L!y = !b,

LT !x = y!.

(aij )1≤j≤i≤N

A (aij )1≤j≤i≤N

a11 :=             

A = LLT

√ a11

,11

i=2

ai1 := ai1 /a11

akk   

aN N

k =2 N −1 3 k−1 ' 2 41/2 := akk − akj

L ,kk

j=1

i=k+1 N 4 k−1 ' 1 3 aik := aik − aij ∗ akj akk j=1 3 41/2 N' −1 2 := aN N − aN j

L ,N N

j=1

N ×N N aij = 0

, ,

aij 1 ≤ i, j ≤ N

i, j

1 ≤ i, j ≤ N

| i − j |≥ , , ,=1 ,=2 LU LLT A

N ×N LU

, LLT

, LU

LLT

N ,2

LLT

LU A

 ∗   ∗   ∗   ∗   ∗   A = 0   0   0   0   0  0

∗

∗ ∗ " ,

∗

2 −1   −1 2   A=    



  0   0   0   0   0   ∗   ∗   ∗   ∗  ∗ !

# ,

∗

N ×N 



     .   −1 2 −1  −1 2

−1

A d!

A = LLT !e

1â&#x2030;¤jâ&#x2030;¤N

ej 1 â&#x2030;¤ j â&#x2030;¤ N â&#x2C6;&#x2019; 1 ï£® d1 ï£¯ ï£¯e ï£¯ 1 ï£¯ ï£¯ L=ï£¯ ï£¯ ï£¯ ï£¯ ï£°

L ï£¹

dN â&#x2C6;&#x2019;1 eN â&#x2C6;&#x2019;1

LLT d21 = 2; e22 d1 =

e1 d1 = â&#x2C6;&#x2019;1;

d23

ï£º ï£º ï£º ï£º ï£º ï£º. ï£º ï£º ï£º ï£»

= 2;

â&#x2C6;&#x161; 2

e21 + d22 = 2;

e3 d3 = â&#x2C6;&#x2019;1;

e2 d2 = â&#x2C6;&#x2019;1;

e2N â&#x2C6;&#x2019;1 + d2N = 2,

Â·Â·Â·;

j = 1, 2, . . . , N â&#x2C6;&#x2019; 1 ej = â&#x2C6;&#x2019;1/dj

dj+1 = LLT

d1 = ï£®

M 2 â&#x2C6;&#x2019; e2j . A

â&#x2C6;&#x161; 2

j =1 N â&#x2C6;&#x2019;1 ï£¯ ï£¯ ï£¯ ej := â&#x2C6;&#x2019;1/dj ï£° M dj+1 := 2 â&#x2C6;&#x2019; e2j . ej dj =

N Ã&#x2014;N

j+1 j

dj ej = â&#x2C6;&#x2019;

j . j+1

N Ã&#x2014;N

Aâ&#x2C6;&#x2019;1 B

(bij )1â&#x2030;¤i,jâ&#x2030;¤N

(bij )1â&#x2030;¤i,jâ&#x2030;¤N B

(bij )1â&#x2030;¤i,jâ&#x2030;¤N C

N Ã&#x2014;N AC = B

C = Aâ&#x2C6;&#x2019;1 B !j 1â&#x2030;¤jâ&#x2030;¤N B

(bij )1â&#x2030;¤i,jâ&#x2030;¤N Aâ&#x2C6;&#x2019;1 B

!j C

C j

!j = B !j 1 ≤ j ≤ N AC

AC = B A = LLT

!j = B !j 1 ≤ j ≤ N LX !j = X !j 1 ≤ j ≤ N L C (bij )1≤i,j≤N

B X

A−1 B bij 1 ≤ i, j ≤ N

bij 1 ≤ i, j ≤ N

di 1 ≤ i < N

ei 1 ≤ i < N − 1

d1 = 

√ 2

L i=1

N −1

   ei := −1/di  6 di+1 := 2 − e2i  j=1 N  b := b1j /d1  1j i=2 N     j=1 N   bij := (bij − ei−1 ∗ bi−1,j )/di  j=1 N  b := bN j /dN  Nj i=N −1 1 −1     j=1 N   bij := (bij − ei ∗ bi+1,j )/di

LX = B

LT C = X

ai i = 1, . . . , N !c

(N − 1)

i = 1, . . . , N − 1



     A=    

N ×N

  b2    .   bN −1   aN

aN −1 c1

...



cN −1

U 

     L=    



d1 d2

dN −1 f1

...

fN −1

 1           , U =         



  e2    .   eN −1   1

A = LU

ai A

k = 1, 2, . . . , N − 1

LU L

U A

d1 = a1

f 1 = c1 LU

LU A N −1

b1 = d1 e1 A

di = ai ,

f i = ci ,

A A

LU ei =

bi , di

i = 1, 2, . . . , N − 1. A

aN = f1 e1 + · · · + fN −1 eN −1 + dN , dN = aN −

N −1 ( j=1

fj ej .

LU ai 1 ≤ i ≤ N b i

ci 1 ≤ i ≤ N − 1 A

ai 1 ≤ i ≤ N c i 1 ≤ i ≤ N − 1 L d!

!a bi 1 ≤ i ≤ N − 1

 

i=1

U !b

(N − 1)

N −1

bi := bi /ai

aN := aN −

N' −1 j=1

1 L

A U

b j ∗ cj

LU U

N A!x = !b, A N

N ×N

(aij )1≤i,j≤N (bi )1≤i≤N

N !x

(xi )1≤i≤N LU

LLT

A N3 N N

!x0 , !x1 , !x2 , . . . , !xn , . . . lim 1!x − !xn 1 = 0.

n→∞

j !x

!xj

i A

xji N ×N

M A = K − M.

K K!x = M !x + !b !x = K −1 M !x + K −1!b. !x0

N n = 0, 1, 2, 3, . . .

!xn+1 = K −1 M !xn + K −1!b. !c = M !x + !b

!xn+1

!xn

K!xn+1 = !c K K!xn+1 = !c

K A A=D−E−F

D A i ≤ N eij = 0 1≤j≤i≤N

1 ≤ i ≤ j ≤ N A 

     A=     

−F D −E

aii $= 0

xn+1 i



     .     

i = 1, 2, . . . , N K=D

K −1 = D−1 =

A = K −M

fij

(aii )1≤i≤N −E eij = −aij 1 ≤ j < −F = −aij 1 ≤ i < j ≤ N fij = 0



K =D M = E+F (1/a11 , 1/a22 , 1/a33 , . . . , 1/aN N )

 N ( 1   = aij xnj + bi  , − aii j=1 j!=i

D (D − E) K =D−E

1 ≤ i ≤ N.

n+1

1≤i≤N !xn

xni

!x J

1≤i≤N

J = K −1 M = D−1 (E + F )

K = D−E

A=K −M

M =F

(D − E)!xn+1 = F !xn + !b !xn+1

xn+1 i

  ( 1  ( = − aij xn+1 − aij xnj + bi  , j aii j<i j>i (xni )1≤i≤N

xn+1 1

xn+1 2

xn+1 3

!xn

xn+1 , xn+1 , xn+1 ,... 1 2 3

  N 1  ( = − a1j xnj + b1  a11 j=2   N ( 1  = −a21 xn+1 − a2j xnj + b2  1 a22 j=3   N ( 1  = −a31 xn+1 − a32 xn+1 − a3j xnj + b3  1 2 a33 j=4

G G = K −1 M = (D − E)−1 F

xn+1 i

!b !x0

1 ≤ i ≤ N.

xn+1 i

N ×N

λ1 , λ2 , . . . , λN

ρ(B) = max |λj |, 1≤j≤N

|λj |

λj 1 ≤ j ≤ N

ρ(K −1 M )

K −1 M

A aii 1 ≤ i ≤ N

D−E

λ (D − E)−1 F ϕ !

|λ| < 1 N

ϕ1 , ϕ2 , . . . , ϕN (D − E)−1 F ϕ ! = λ! ϕ,

ϕ !

(D − E)−1 F

λ Fϕ ! = λ(D − E)! ϕ

A! ϕ = (D − E − F )! ϕ = (1 − λ)(D − E)! ϕ. λ

A! ϕ=0 A ϕ !∗

ϕ !=0 ϕ !∗ T

ϕ ! ∗

ϕ ! A! ϕ = (1 − λ)! ϕ∗ (D − E)! ϕ. ϕ ! ∗ A! ϕ

ϕ ! α = (1 − λ)! ϕ∗ (D − E)! ϕ λ)! ϕ∗ (D − E)! ϕ

ET = F ¯ ϕ∗ (D − F )! α = (1 − λ)! ϕ,

(1 −

¯ λ (1 − λ)

¯ (1 − λ)

λ ¯ = |1 − λ|2 ϕ (2 − λ − λ)α ! ∗ (2D − E − F )! ϕ

|1 − λ|

(1 − λ)

α=ϕ ! ∗ A! ϕ

A = D −E −F

¯ − |1 − λ|2 )α = |1 − λ|2 ϕ (2 − λ − λ ! ∗ Dϕ !. D ϕ ! ∗ Dϕ !

λ $= 1

α>0

¯ − |1 − λ|2 > 0. 2−λ−λ ¯ = 1−λ−λ ¯ + |λ|2 |1 − λ|2 = (1 − λ)(1 − λ) 1 − |λ|2 > 0, |λ| < 1 !

2D−A

N ×N

A 

  −1   A=    

−1 2



     .   −1 2 −1  −1 2

−1

A!x = !b λk k = 1, . . . , N D = 2I 4 − λk = 2 + 2 cos(kπ/(N + 1)) k = 1, . . . , N

A 2D − A

2D − A A!x = !b

D = 2I J = D−1 (E + F )

1 J = D−1 (E + F − D + D) = D −1 (D − A) = I − D −1 A = I − A. 2 J

1 − λk /2 = cos(kπ/(N + 1)) J

k = 1, . . . , N J ρ(J) = cos

π . N +1

ρ(J) < 1 A!x = !b cos x = 1 − x2 /2 + O(x4 ) N

|1 − ρ(J)| ≤ C

C 1 , N2

∀N > 1.

N ρ(J)

J 1

1 ρ(J) ρ(J)

ρ(J) N G = (D − E)−1 F A ρ(G) = ρ(J)2 N =2 |1 − ρ(G)| ≤ C/N 2 ∀N > 1

A 1≤i≤N

N ×N

aii A A = D − E − F.

A=K −M

. 1 1−ω A= D−E− D+F , ω ω K = ω −1 D − E M = ω −1 (1 − ω)D + F -

. . 1 1−ω n+1 D − E !x = D + F !xn + !b. ω ω ω=1

ω<1

ω>1

Gω =

1 D−E ω

.−1 -

1−ω D+F ω

. ρ(Gω ) ωopt

ω ρ(Gωopt ) < ρ(Gω )

ωopt

A 0<ω<2 ωopt ωopt = ρ(J) ρ(Gω )

2 6 , 1 + 1 − ρ(J)2 J

ω ωopt ω>1 ωopt

f (ω) = ρ(Gω ) lim f & (ω) = −∞

ω→ωopt ω<ωopt

lim f & (ω) = 1,

ω→ωopt ω>ωopt

F E

. . 1 1−ω n D − E y! = D + F !xn + !b, ω ω . . 1 1−ω n+1 D − F !x = D + E !y n + !b, ω ω

ρ(Gω ) 1 ρ(G) = ρ(J)2 ωopt − 1 ω 1

ωopt

ρ(Gω )

2 ω

!y n

!xn+1

!xn

A A!x = !b

N L(!y ) = R

1 T !y A!y − !bT !y . 2

L(!y ) A A!x = !b

L N ×N

!x N

L(!x) < L(!y ). !x

A!x = !b !z = !x − y!

!x L

1 (!x − !z )T A(!x − !z ) − !bT (!x − !z ) 2 1 1 1 1 = !xT A!x − !bT !x − !z T A!x − !xT A!z + !z T A!z + !bT !z . 2 2 2 2

L(!y ) = L(!x − !z ) =

!xT A!z = !z T A!x

!bT !z = !z T !b

A!x = !b

1 L(!y ) = L(!x) − !z T A!x + !z T !b + !z T A!z 2 3 4 1 = L(!x) − !z T A!x − !b + !z T A!z , 2 1 L(!y ) = L(!x) + !z T A!z . 2

A !z T A!z > 0

!z L(!y ) > L(!x) ! A!x = !b

!xn

!xn+1

L(!xn+1 ) < L(!xn )

w ! n+1 !xn+1 = !xn + αn+1 w ! n+1

αn+1

f (α) / 0 f (α) = L !xn + αw ! n+1 . !xn+1

w !

!xn

n+1

N =2

!xn %

w ! n+1

!xn+1 & !x (

N =2

αn+1 n+1

f (α

) ≤ f (α) ∀α ∈ R

αn+1 &

f (α)

N N ( 1 ( L(!y ) = aij yi yj − bi yi 2 i,j=1 i=1

A N

( ∂ L(!y ) = aij yj − bi . ∂yi j=1

N (

∂ L(!xn + αw ! n+1 ) ∂y i i=1   N N ( ( / 0 = win+1  aij xnj + αwjn+1 − bi 

f & (α) =

win+1

i=1

= (w !

j=1

3 / 4 0 ) A !xn + αw ! n+1 − !b .

n+1 T

αn+1

f & (αn+1 ) = 0

n+1

αn+1 =

3 4 (w ! n+1 )T !b − A!xn (w ! n+1 )T Aw ! n+1

!rn = !b − A!xn n !xn+1

!xn w ! n+1

αn+1 =

(w ! n+1 )T !rn ; (w ! n+1 )T Aw ! n+1

!xn+1 = !xn + αn+1 w ! n+1 w ! n+1 n+1 w ! n+1 !rn+1 = !b − A!xn+1 = !b − A(!xn + αn+1 w ! n+1 ) = !rn − αn+1 Aw ! n+1 . (w ! n+1 )T !rn+1 = 0

L n+1

w ! = ∂L(!y )/∂yi

−−−→

L(!x ) w ! n+1 =

!xn i

−−−→

L(!y )

L(!xn ) = A!xn − !b = −!rn . !rn

!x !rn $= 0

!rn

αn+1 = −

1!rn 12 . (!rn )T A!rn

!rn+1

!rn

/ 0 !rn+1 = !b − A!xn+1 = !b − A !xn + αn+1 w ! n+1 = !rn + αn+1 A!rn .

!z n+1 !r

n+1

!z n+1 = −A!rn

n+1 n+1

= !r − α

!x0 !xn+1

n = 0, 1, 2, . . .

!rn+1

!r0 = !b − A!x0

!z n+1 = −A!rn , αn+1 =

1!rn 12 , (!rn )T !z n+1

!xn+1 = !xn − αn+1!rn ,

!rn+1 = !rn − αn+1 !z n+1 ,

!rn+1 = 0 :

!rn

!xn n+1

!x −!rn

1 · 1A

w !n n+1

!x !x 1!y1A = (!y T A!y )1/2

w ! n+1 = −!rn + β n w ! n, βn

1!x − !xn+1 1A βn βn =

(!rn )T Aw !n . (w ! n )T Aw !n !z n

!z n = Aw !n

/ 0 !rn = !b − A!xn = !b − A !xn−1 + αn w ! n = !rn−1 − αn !z n . !x0

w ! 1 = −!r0 α1 =

(!r0 )T w !1 , (w ! 1 )T !z 1

!x1 = !x0 + α1 w ! 1.

!r0 = !b − A!x0

!z 1 = Aw ! 1,

!rn β n w ! n+1 !z n+1 αn+1 !xn+1

n = 1, 2, 3, . . .

!rn = !rn−1 − αn !z n , βn =

!rn = 0 :

n T n

(!r ) !z , (w ! n )T !z n

w ! n+1 = −!rn + β n w ! n, !z n+1 = Aw ! n+1 ,

αn+1 =

(!rn )T w ! n+1 , (w ! n+1 )T !z n+1

!xn+1 = !xn + αn+1 w ! n+1 . (w ! n )T !z n !rn

A N

ne !r0 , !r1 , . . . , !rn−1

N ×N

!rn = 0

n≤N

χ(A) C −1 A!x = C −1!b

A!x = !b

C χ(C −1 A)

χ(A) C −1 A!x = C −1!b

C −1 A C C = LLT ,

L C b −1!

−T

−1

A!x = L

A* = L−1 AL−T ,

−T

−1!

−1

!x* = LT !x C −1 A!x = C −1!b A* !x* = !b* .

−T

L !x = L

−1!

!b* = L−1!b.

C −1 A!x =



 −1  A= −1 2

Gω 

 ω 1 − ω   2 Gω =  ω ω2  . (1 − ω) 1 − ω + 2 4 J

ρ(J) =√1/2 ωopt = 8 − 4 3

ωopt

ρ(Gω ) = (1 − ω + ρ(Gω ) = ω − 1

ω2 ω6 2 )+ ω − 16ω + 16 8 8 ω → ρ(Gω )

lim f & (ω) = −∞

Gω = 

g11

g12

g21

g22



2 ω Gω =  −1 

2 ω  −1

ω ∈ [ωopt , 2].

f (ω) = ρ(Gω ) lim f & (ω) = 1.

ω→ωopt ω<ωopt



ω ∈ [0, ωopt ],

ω→ωopt ω>ωopt

 

−1  1−ω 0  2 ω 2  0 ω

    1−ω 0  g11  2 ω   2  =  g21 0 ω

  1  g11 1− ω =  2 g21 ω 

2 ω  −1

 g12  . g22

    0  g12   1  2  =  1− ω, g22 2 ω ω

g12 = ω/2 g22 = 1 − ω + ω 2 /4

g11 = 1 − ω g21 = ω(1 − ω)/2

Gω

 1 0 J = D−1 (E + F ) =  2 1 λ

 1 , 0

  1 −λ 1  2  det(J − λI) = det  1  = λ2 − . 4 −λ 2 det(J − λI) = 0 √λ = ±1/2 ωopt = 8 − 4 3

ρ(J) = 1/2

  ω 1−ω−λ   2 2 det(Gω − λI) = det  ω  ω (1 − ω) 1 − ω + −λ 2 4 . 2 ω = λ2 − λ 2(1 − ω) + + (1 − ω)2 . 4 Gω

λ1

λ2

ω .2 ω2 ∆ = 2(1 − ω) + − 4(1 − ω)2 4 . ω2 2 =ω 1−ω+ 16 2 3 √ 43 √ 4 ω = ω − (8 + 4 3) ω − (8 − 4 3) . 16 ω√ 0 ≤ ω ≤ 2 0 ≤ ω ≤ 8 − 4 3 = ωopt ∆≥0 P ω2 ω λ1 = (1 − ω) + + 1−ω+ 8 2P ω2 ω λ2 = (1 − ω) + − 1−ω+ 8 2

ω2 , 16 ω2 , 16

ρ(Gω ) = max{|λ1 |, |λ2 |} = (1 − ω) + ωopt < ω ≤ 2

∆<0

ω2 ω6 2 + ω − 16ω + 16. 8 8

P ω2 ω λ1 = (1 − ω) + +i −(1 − ω + 8 2P ω2 ω λ2 = (1 − ω) + −i −(1 − ω + 8 2

ω2 ), 16 ω2 ), 16

ρ(Gω ) = max{|λ1 |, |λ2 |} = ω − 1. ω → ρ(Gω ) 1

ρ(G) = ρ(J)2 = 0.25 ωopt − 1 0 0

1 √ ωopt = 8 − 4 3

ω → ρ(Gω )

f (ω) = ρ(Gω )

ω > ωopt

f & (ω) = 1

lim f & (ω) = 1.

ω→ωopt ω>ωopt

ω < ωopt f & (ω) = −1 +

ω 16 2 ω(2ω − 16) + ω − 16ω + 16 + √ 4 8 16 ω 2 − 16ω + 16 lim f & (ω) = −∞.

ω→ωopt ω<ωopt

A!x = !b 

A= −1

−1



 3/2  !x0 =  2 A!x = !b L



  1 !b =   . 1

,

!x2 = !x

!x0 !x1 !x2

L A

 2−λ det(A − λI) = det  −1

det(A − λI) = 0

λ=1





0 , !r0 = b − A!x0 =  −3/2   −3/2 , !z 1 = Aw !1 =  3   3/2 , !x1 = !x0 + α1 w !1 =  5/4 (!r1 )T !z 1 1 = , (w ! 1 )T !z 1 4   9/8 , !z 2 = Aw !2 =  0   1 !x2 = !x1 + α2 w !2 =   . 1 β1 =



−1

 = (2 − λ)2 − 1, 2−λ

λ=3 A

w ! 1 = −!r0 , (!r0 )T w !1 1 =− , 1 T (w ! ) !z 1 2   −3/4 , !r1 = !r0 − α1 !z 1 =  0   3/4 , w ! 2 = −!r1 + β 1 w !1 =  3/8 α1 =

α2 =

(!r1 )T w !2 2 =− , 2 T (w ! ) !z 2 3

A!x = !b

!x2 = !x 2×2

2   y1 !y =   y2

L 1 T y! A!y − !bT !y 2 = y12 + y22 − y1 y2 − y1 − y2 3 1 = (y1 − y2 )2 + (y1 + y2 )2 − y1 − y2 . 4 4

L(!y ) =

(1, 1)

2.5 !x0

% α1 w !1 $ !x2 ( !x1 (

1.5 1 0.5 0

0.5

1.5 L

2.5

N ×N λ1 λ2 . . . λN

ϕ !1 ϕ !2 . . . ϕ !N A! ϕj = λj ϕ !j ,

1 ≤ j ≤ N. N ×N

p(λ) = I

N ×N

(λI − A),

p(λ) = λN + aN −1 λN −1 + aN −2 λN −2 + · · · + a1 λ + a0 , aj 1 ≤ j ≤ N − 1 λ



      A=      

−aN −1

−aN −2

0 1

−aN −3

···

−a0

0 1

             

ϕj = λN −j 1 ≤ N N N ≥5

ϕ ! j≤N

A λ1 , λ2 , λ3 , . . . , λN

N ×N

|λ1 | ≥ |λ2 | ≥ |λ3 | ≥ . . . ≥ |λN |. RN

ϕ !1, ϕ ! 2, . . . , ϕ !N A! ϕj = λj ϕ !j , ϕ ! Tj ϕ ! k = δjk , δkj N ×N

1 ≤ j ≤ N, 1 ≤ j, k ≤ N, j =k Q ϕ ! 1, ϕ !2, . . . , ϕ !N Q ϕ ! Tk A! ϕj = λj δjk 1 ≤ j, k ≤ N

(QT = Q−1 ) QT AQ = D; D

(λ1 , λ2 , . . . , λN )

λ1 ϕ !1

!x(0) (!x(n) )∞ n=1

(µ(n) )∞ n=1

!x(n) = A!x(n−1) , µ(n) =

n = 1, 2, . . . ,

!x(n)T A!x(n) !x(n)T !x(n+1) = (n)T (n) , (n) 2 1!x 1 !x !x

n = 1, 2, . . . . !x(n) = An !x(0)

λ1 !x(0)

ϕ !1 ϕ !1 ϕ ! Tk !x(n) = 0, n→∞ 1! x(n) 1 lim

A |λ1 | > |λk | k = 2, 3, . . . , N ϕ ! T1 !x(0) $= 0 k = 2, 3, . . . , N,

lim µ(n) = λ1 .

n→∞

ϕ !1, ϕ ! 2, . . . , ϕ !N !x(0) !x(0) =

N (

αj ϕ !j

j=1

αj = ϕ ! Tj !x(0) ,

1 ≤ j ≤ N.

!x(1) !x(1) = A!x(0) =

N (

αj A! ϕj =

j=1

!x(n) =

N (

αj λj ϕ !j.

j=1

αj λnj ϕ !j .

j=1



1!x(n) 1 = 

N ( j=1

1/2

 α2j λ2n j

k = 2, 3, . . . , N -

λk λ1

αk ϕ ! Tk !x(n) αk λnk = = . ? @1/2 ? - .2n @1/2 1!x(n) 1 N N ' ' λ j α2j λ2n α2j j λ1 j=1 j=1 α1 = ϕ ! T1 !x(0) |λ1 | > |λk |

k = 2, . . . , N

(n)

!x(n)T !x(n+1) = (n)T (n) = !x !x

N '

j=1

α2j λ2n+1 j

N '

j=1

.2n+1 λj λ1 j=1 = λ1 - .2n . N ' 2 λj αj λ1 j=1 N '

α2j

α1

α2j λ2n j

|λ1 | > |λk |

k = 2, 3, . . . , N

.2n+1 λj λ1 α2 j=1 lim = 12 = 1, . 2n n→∞ ' N α1 λj α2j λ1 j=1 N '

α2j

µ(n) =

!x(n)T A!x(n) 1!x(n) 12

A !x(n) 1!x(n) 1 n ϕ !1

ϕ !2, ϕ ! 3, . . . , ϕ !N

ϕ ! T1 !x(0) $= 0 !x(0)

ϕ !1

ϕ !1 |λ1 | > |λk |

k = 2, 3, . . . , N

|λ1 | = |λ2 | > |λk |

k = 3, 4, . . . , N,

λ1

λ1 = λ2 λ1

λ1 = −λ2

ϕ ! Tk !x(n) = 0, n→∞ 1! x(n) 1 lim

k = 3, 4, . . . , N,

limn→∞ µ(n) = λ1 = λ2 A A + εI

N ×N

ε ε

A + εI ε µ(n)

λj + ε 1 ≤ j ≤ N |λ2 + ε| A

|λ1 + ε| A + εI λ1 + ε λ2 + ε

|λ1 − µ(n) | ≤ C µ(n)

λ2 λ1

.2n

λ1

λ2 /λ1

A λN

N ×N

λ1 , λ2 , . . . ϕ ! 1, ϕ !2, . . . , ϕ !N µ µ $= λj ,

1 ≤ j ≤ N.

A − µI ωj 1 ≤ j ≤ N

(A − µI)−1 ωj = (λj − µ)−1 ,

1 ≤ j ≤ N.

k |λk − µ| < |λj − µ|,

j = 1, 2, . . . , N ; j $= k. λk

µ A

2µ − λk |ωk | > |ωj |,

j = 1, 2, . . . , N ; j $= k.

ωk A n = 1, 2, . . .

λk (A − µI)−1 (A − µI)−1

(A − µI)−1 A !x(n)

!x(0)

!x(n) = (A − µI)−1 !x(n−1) . !x(0)

A !x(n) = (A − µI)−n !x(0) =

N (

αj ωjn ϕ !j ,

j=1

αj µ(n) =

!x(n)T A!x(n) , 1!x(n) 12

!x(n)T (A − µI)!x(n) 1!x(n) 12 - .2n−1 N ' ωj α2j ωk j=1 = (λk − µ) - .2n . N ' ωj 2 αj ωk j=1

µ(n) − µ =

αk

lim µ(n) = λk .

n→∞

ωj λk − µ = ωk λj − µ µ |ωk |

λk

µ !x(0)

ϕ !k µ(n−1) =

!x(n−1)T A!x(n−1) , 1!x(n−1) 12

!x(n) = (A − µ(n−1) I)−1 !x(n−1) , !x(n)

n = 1, 2, 3, . . .

(A − µ(n−1) I)!x(n) = !x(n−1) .

µ(n)

λk

n (A − µ(n−1) I)

(A − λk I) !x(n)

1rn !x(n) 1 = 1

QT AQ = D, D

λ1 , λ2 , . . . , λN {Q(k) }∞ k=1 lim Q(1) Q(2) Q(3) . . . Q(k) = Q.

k→∞

T (k) T (k) = Q(k)T Q(k−1)T . . . Q(1)T AQ(1) Q(2) . . . Q(k) .

lim T (k) = D.

k→∞

T (0) = A T (k) = Q(k)T T (k−1) Q(k) ,

(k−1)

(k) (k) tij qij (k−1) tmn m (k)

Q Q(k)

k = 1, 2, . . . ,

1 ≤ i, j ≤ N

(k)

T (k) Q(k) T (k−1)

$= n

(k) tmn

(k) (k) qmm = qnn = cos θk , (k) (k) qmn = −qnm = sin θk , (k)

qii = 1 (k) qij

i $= m

i $= n,

, (k)

θk

tmn mn t(k) mn =

N (

(k) (k−1) (k) qjn

qim tij

i,j=1 (k) (k−1) (k) (k) (k−1) (k) =qmm tmn qnn + qmm tmm qmn (k) (k−1) (k) (k) (k−1) (k) + qnm tnn qnn + qnm tnm qmn .

T (k) 4 / 2 0 3 (k−1) (k−1) t(k) cos θk − sin2 θk + t(k−1) (cos θk sin θk ) mn = tmn mm − tnn

(k−1) t(k) cos 2θk + mn = tmn

4 1 3 (k−1) tmm − t(k−1) sin 2θk . nn 2 (k)

θk

tmn = 0 (k−1)

2θk =

T (k)

tnn

(k−1)

− tmm

(k−1)

2tmn

T (k−1)

θk

(k)

(k−1)

(k) tnj

(k) tjn

(k−1) tmj

tmj = tjm = tmj

(k−1)

cos θk − tnj

sin θk ,

j $= m

j $= n,

(k−1) = = sin θk + tnj cos θk , j $= m j $= (k−1) 2 (k−1) t(k) sin2 θk − 2t(k−1) sin θk cos θk , mm = tmm cos θk + tnn mn 2 (k) (k−1) (k−1) 2 (k−1) tnn = tmm sin θk + tnn cos θk + 2tmn sin θk cos θk , (k) t(k) mn = tnm = 0, (k) (k−1) tij = tij , (i, j).

θk sin θk

cos θk sin θk

cos θk

α=

T =

θk

(k−1)

− tmm

tnn

(k−1)

2tmn

α=

1 = 2

1 −T T

2θk . .

1 − T2 , 2T

T T 2 + 2αT − 1 = 0. √ 1 + α2 − α √ T = − 1 + α2 − α T = T

1 cos θk = √ 1 + T2

h = T t(k−1) mn ,

α ≥ 0, α < 0.

sin θk = √

1 C= √ , 1 + T2

S = T C,

T . 1 + T2

τ=

S , 1+C

C2 + S2 = 1 (k)

(k)

3 4 (k−1) (k−1) − S tnj + τ tmj 3 4 (k−1) (k−1) (k−1) = tnj + S tmj − τ tnj (k−1)

tmj = tjm = tmj tnj = tjn

(k−1) t(k) mm = tmm − h, (k−1) t(k) + h, nn = tnn (k) t(k) mn = tnm = 0;

T (k−1) A

j $= m

j $= n,

j $= m

j $= n,

A anm α = (ann − amm )/2amn T h = T amn C = (1 + T 2 )−1/2 S = T C τ = S/(1 + C) m

amj = ajm := amj − S (anj + τ amj ) , anj = ajn := anj + S (amj − τ anj ) ,

j $= m, j $= n, j= $ m, j $= n,

amm := amm − h, ann := ann + h, amn = anm := 0.

A amn A

a21 , a31 , a41 , . . . , aN 1 , a32 , a42 , . . . , aN 2 , . . . ε amn

(k)

(1)

(2)

amn |amn | < ε

A . . . Q(k) (= R(k−1) Q(k) )

(3)

A 

A= −1

 −1 . 2

R(k)

amn =

!x(0) !x(0) A2 A3

  (0) x1 =  (0)  $= 0. x2 A4 µ(3)

!x(3)

!x(4) !x(0)

µ(3) A (0)

(0)

µ(3)

x1 = x2

n = 1, 2, . . . (n)

(n−1)

= A!x

!x(3) = A3 !x(0)

= A2 !x(n−2) = · · · = An !x(0) . !x(4) = A4 !x(0)



A2 =  −1 

    −1 2 −1 5 −4  = . 2 −1 2 −4 5

 −13  A3 = A2 A =  −13 14 !x(3) = A3 !x(0)

!x(4) = A4 !x(0)

µ(3) =



 −40 . A4 = A3 A =  −40 41



  (0) −13 x1  , = (0) −13 14 x2    (0) 41 −40 x1  . = (0) −40 41 x2 14

!x(3)T !x(4) !x(0)T A3 A4 !x(0) = (0)T 3 3 (0) . (3) 2 1!x 1 !x A A !x

    −13 14 −13 365 −364  = , A3 A3 =  −13 14 −13 14 −364 365      14 −13 41 −40 1094 −1093  = . A3 A4 =  −13 14 −40 41 −1093 1094 14

(0)

x1 = 0

x2 = 0 µ(3) =

1094 , 2.997. 365 A

µ(3)

3 A (0)

(0)

µ(3) = 1

x1 = x2

!x(0)

(0)

3 A  2 A= 0

 0 . 1 !x(0)

A !x(0) (0)

(0)

0 < x1 < x2 limn→∞ µ(n) = 1

  (0) x1 =  (0)  . x2 (0)

(0)

0 < x2 < x1 limn→∞ µ(n) = 2 µ(n−1) (n−1) x1

!x(n−1) (n−1)

(n−1)

$= 0

(n−1) x2

$= 0,

n=1

(n−1)

(n−1) 2

2(x1

(n−1) 2 )

(x1

 2 − µ(n−1)  0

(n−1) 2

) + (x2

)

(n−1) 2 )

+ (x2

0 1 − µ(n−1)

 

= ?

(n)



(n−1)



=

(n−1)



.

µ(n−1) 1

2 (n)

(n−1)

(n) x2

(n)

x2 (n)

x1 (n)

(n)

=−

(n−1)

(0)

x1 < x2 (0)

=−

x2 ?

(0)

1 − µ(n−1) . 2 − µ(n−1)

(n−1)

$= 0

(n−2)

@32

= · · · = (−1)n

(0)

@3n

(n)

lim

n→∞

x2 < x1

(n)

$= 0

(n−1) x2

(n)

lim µ(n) = 1,

n→∞

(n)

n→∞

(n) x2

lim µ(n) = 2.

= +∞

n→∞

QR A A = QR

A Q

A A = LR

1 LU

A2 = R1 Q1 Ak = Qk Rk Ak

LR LR QR QR LR Ak Ak+1

A1 = A A1 = Q1 R1 QR LR = Rk Q k A2 = R1 Q1 = QT1 AQ1

Ak+1 = Rk Qk = QTk Ak Qk = (Q1 . . . Qk )T A(Q1 . . . Qk ). Ak k A

f :Râ&#x2020;&#x2019;R

x Â¯

f (Â¯ x) = 0

xÂ¯ f x0 x0

x1 , x2 , x3 , . . . , xn , . . .

lim xn = x Â¯,

nâ&#x2020;&#x2019;â&#x2C6;&#x17E;

Î±

Î²

xÂ¯

f (Î±)f (Î²) < 0 f Î± Î± Î²

Î² f x1

f (Â¯ x) = 0.

f Î²

Î± x0 = (Î± + Î²)/2 f (x0 ) = 0 x0 f (x0 )

f (x0 )f (Î±) > 0 Î± := x0

x0 x1 = (Î± + Î²)/2

Î²

Î±

f (x0 )f (Î±) < 0 Î² := x0

x0 x1 = (Î± + Î²)/2

Î±

Î²

x0 x2

Î²

Î±

(xn )â&#x2C6;&#x17E; n=1

x1 x Â¯ M

f (Â¯ x) = 0 Îµ = |Î² â&#x2C6;&#x2019; Î±|

|¯ x − xM | ≤

ε 2M+1

x2 x0

p p

C |¯ x − xn+1 | ≤ C|¯ x − xn |p .

p=1

C <1

p=2 p=3 p=1

C = Cn

limn→∞ Cn = 0

f (x) = x−cos x x ¯ = cos x ¯ f x0 = 0.75

x ¯

x¯

f (¯ x) = 0

x¯ = cos x¯ xn+1 = cos xn ,

n = 0, 1, 2, . . . , x1 , x2 , . . .

x0 ∈ R

x¯

y = cos(x) y=x

1 y

0 -1 0

x Â¯1

x Â¯

x Â¯ = cos xÂ¯

x0 â&#x2C6;&#x2C6; [0, 1] x1 = cos x0 â&#x2C6;&#x2C6; [â&#x2C6;&#x2019;1, +1] x2 = cos x1 â&#x2C6;&#x2C6; [0, 1] x0 x0 â&#x2C6;&#x2C6; [0, 1] x Â¯ = cos x Â¯

52 5 |Â¯ x â&#x2C6;&#x2019; xn+1 | = | cos xÂ¯ â&#x2C6;&#x2019; cos xn | = 55

x Â¯

x0 â&#x2C6;&#x2C6; [0, 1] x1 = cos x0 â&#x2C6;&#x2C6; [0, 1] xn+1 = cos xn â&#x2C6;&#x2C6; [0, 1] n = 1, 2, . . .

5 5 sin tdt55 .

(xn )â&#x2C6;&#x17E; n=1

x Â¯

! x Â¯

x â&#x2020;&#x2019; cos x

x Â¯ = cos x Â¯

f (x) = 0 f (x) = 0 x = g(x).

g(x) = x − f (x), g(x) = x + αf (x) α ∈ R α $= 0 x

x ¯

x¯ ∈ R g

x ¯ = g(¯ x)

x ¯∈R

x ¯

x ¯ f x0

x ¯

xn+1 = g(xn ) n = 0, 1, 2, . . .

x g

(xn )∞ n=0 x

x = lim xn+1 = lim g(xn ) = g(x). n→∞

n→∞

g I

g : x ∈ I → g(x) ∈ R I

R g

χ<1 |g(x) − g(y)| ≤ χ|x − y|,

∀x, y ∈ I.

g (yn )∞ I n=1 |g(x) − g(yn )| ≤ χ|x − yn |

x I limn→∞ g(yn ) = g(x)

R g

g : I → R

g(I) ⊂ I g

x ¯ xn+1 = g(xn ), x ¯

x∈I I

g(x) ∈ I

x0 ∈ I

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

(xn )∞ n=0

g xn+1 = g(xn ) n = 0, 1, 2, . . . n = 1, 2, . . . I

ii) ii) i)

x0 ∈ I

|xn+1 − xn | = |g(xn ) − g(xn−1 )| ≤ χ|xn − xn−1 | χ<1 n |xn+1 − xn | ≤ χn |x1 − x0 |,

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

m |xn+m − xn | = |xn+m − xn+m−1 + xn+m−1 − xn+m−2 + xn+m−2 − · · · + xn+1 − xn | ≤ |xn+m − xn+m−1 | + |xn+m−1 − xn+m−2 | + · · · + |xn+1 − xn |.

/ 0 |xn+m − xn | ≤ χn+m−1 + χn+m−2 + . . . + χn |x1 − x0 | / 0 ≤ χn 1 + χ + χ2 + . . . + χm−1 |x1 − x0 |, |xn+m − xn | ≤ χn χ

1 − χm |x1 − x0 |. 1−χ

(xn )∞ n=0

x ¯ limn→∞ xn = x ¯ I g

x ¯

I g

x ¯ n

n |¯ x − xn+1 | = |g(¯ x) − g(xn )| ≤ χ|¯ x − xn |. x¯ I y¯ = x ¯ !

y¯ χ

g I |¯ x − y¯| = |g(¯ x) − g(¯ y)| ≤ χ|¯ x − y¯|

g(x) = cos x

I = [0, 1] x, y ∈ I

52 5 |g(x) − g(y)| = | cos x − cos y| = 55

≤ max | sin t| · |x − y|. t∈[0,1]

5 5 sin tdt55

xn+1

Ï&#x2021; = maxtâ&#x2C6;&#x2C6;[0,1] | sin t| I x Â¯ I = cos xn x0 â&#x2C6;&#x2C6; [0, 1]

Ï&#x2021;<1

g x Â¯

x0 â&#x2C6;&#x2C6; R

g:Râ&#x2020;&#x2019;R g x Â¯ = g(Â¯ x) |g & (Â¯ x)| < 1 |Â¯ x â&#x2C6;&#x2019; x0 | â&#x2030;¤ Îµ

x0 !

xn+1 = g(xn ),

x Â¯

x Â¯ Îµ>0

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

|g & (Â¯ x)| < 1

g& |g & (x)| â&#x2030;¤ Ï&#x2021;,

I = [Â¯ x â&#x2C6;&#x2019; Îµ, x Â¯ + Îµ]

Îµ>0

Ï&#x2021;<1

x â&#x2C6;&#x2C6; [Â¯ x â&#x2C6;&#x2019; Îµ, x Â¯ + Îµ]. x, y

52 5 |g(x) â&#x2C6;&#x2019; g(y)| = 55

y=x Â¯

I g

f xn

f :Râ&#x2020;&#x2019;R f (Â¯ x) = 0 x Â¯

x Â¯ f & (Â¯ x) $= 0 xn+1 f

Ox (xn , f (xn ))

f (xn ) x Â¯ xn+1

y = f (x)

x xnâ&#x2C6;&#x2019;1

f (xn )/(xn â&#x2C6;&#x2019; xn+1 ) = f & (xn )

x Â¯ xn+1 = xn â&#x2C6;&#x2019;

f (xn ) , f & (xn )

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

xÂ¯ g(x) = x â&#x2C6;&#x2019; f (x) = 0 f & (x) $= 0

f (x) , f & (x)

x = g(x)

x Â¯ xn+1 = g(xn ) g & (x) g & (Â¯ x)

f g & (x) = 1 â&#x2C6;&#x2019; f (Â¯ x) = 0

f & (x)2 â&#x2C6;&#x2019; f (x)f && (x) f & (x)2

f & (Â¯ x) $= 0 g & (Â¯ x) = 0.

xÂ¯ |Â¯ x â&#x2C6;&#x2019; x0 | â&#x2030;¤ Îµ xÂ¯

f f (Â¯ x) = 0 f & (Â¯ x) $= 0 (xn )â&#x2C6;&#x17E; n=0

g(x) = x â&#x2C6;&#x2019; f (x)/f & (x)

Îµ>0

x0 !

|g & (Â¯ x)| < 1

g & (¯ x) = 0 f

f (x) = f (xn ) + f & (xn )(x − xn ) + ξx

x f & (xn )

f && (ξx ) (x − xn )2 2 xn

x = x¯ f (¯ x) =

f (xn ) f && (ξx¯ ) 2 + x¯ − xn + & (¯ x − xn ) = 0. & f (xn ) 2f (xn )

|¯ x − xn+1 | =

|f && (ξx¯ )| |¯ x − xn |2 . 2|f & (xn )| max

x∈[¯ x−ε,¯ x+ε]

min

|f && (x)|

x∈[¯ x−ε,¯ x+ε]

|f & (x)|

|¯ x − xn+1 | ≤ C|¯ x − xn |2 .

! x0

x ¯

f f f & (¯ x) = 0 x0

f & (xn )

xn+1 = xn −

f (xn ) , f & (x0 )

f & (x0 )

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

(xn )∞ n=0 x ¯ = g(¯ x)

x¯

f & (x0 ) g(x) = x − f (x)/f & (x0 ) xn+1 = g(xn ) g

f (¯ x) = 0

x Â¯

x x2 x1

y = f (x)

xÂ¯ |Â¯ x â&#x2C6;&#x2019; x0 | â&#x2030;¤ Îµ xÂ¯

f f (Â¯ x) = 0 f & (Â¯ x) $= 0 (xn )â&#x2C6;&#x17E; n=0

Îµ>0

f Îµ >0

Ï&#x2021;<1

f & (Â¯ x) $= 0

I = [Â¯ x â&#x2C6;&#x2019; Îµ, x Â¯ + Îµ] 5 5 5 f & (x) 55 |g & (x)| = 551 â&#x2C6;&#x2019; & â&#x2030;¤ Ï&#x2021;, f (x0 ) 5

x â&#x2C6;&#x2C6; I.

! N

f : R â&#x2020;&#x2019; R f (Â¯ x) = 0

x Â¯ x1 , x2 , . . . , xN x

x ï£®

x â&#x2C6;&#x2C6; RN 1 â&#x2030;¤ j â&#x2030;¤ N

f (x) â&#x2C6;&#x2C6; RN f

N x â&#x2C6;&#x2C6; RN

ï£¹

ï£¯ ï£º ï£¯ ï£º ï£¯ x2 ï£º ï£º x=ï£¯ ï£¯ ï£º. ï£¯ ï£º ï£° ï£» xN f (x)

N RN

fj R



f1 (x)





f1 (x1 , x2 , . . . , xN )



         f2 (x)   f2 (x1 , x2 , . . . , xN )     . f (x) =  =          fN (x) fN (x1 , x2 , . . . , xN ) f (x) = 0 N

x2 . . . xN f1 (x1 , x2 , . . . , xN ) = 0, f2 (x1 , x2 , . . . , xN ) = 0,

fN (x1 , x2 , . . . , xN ) = 0. f 1 , f 2 , . . . , fN Df (x)

N ×N

x ∈ RN

f 

∂f1  ∂x1 (x)   ∂f2  (x)  Df (x) =  ∂x1     ∂fN (x) ∂x1

∂f1 (x) ∂x2 ∂f2 (x) ∂x2 ∂fN (x) ∂x2

... ...

...

Df (x)

Df (x)ij =

∂fi (x), ∂xj

 ∂f1 (x) ∂xN   ∂f2 (x)  ∂xN ;     ∂fN (x) ∂xN

1 ≤ i, j ≤ N.

xn+1 = xn − Df (xn )−1 f (xn ),

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

x ¯

f (¯ x) = 0

n xn

x x g : RN → RN g(x) = x − Df (x)−1 f (x), x ¯

x ¯

f (¯ x) = 0 x ¯ = g(¯ x)

Df (¯ x)

x¯

g |·|

g & (¯ x) x¯

I 1·1

Dg(¯ x)

f (¯ x) = 0

Df (¯ x)

N ×N

(xn )∞ n=0

f xj 1 ≤ j ≤ N x ¯

n x¯ 

1x0 − x ¯1 = 

N ( j=1

1/2

(x0j − x¯j )2 

lim 1xn − x ¯1 = 0.

n→∞

C 1 xn+1 − x ¯ 1≤ C 1 xn − x¯ 12 , xn+1

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

Df (xn )(xn − xn+1 ) = f (xn ),

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

!b = f (xn )

A = Df (xn ) A!y = !b LLT A

xn+1 = xn − !y Df (xn )

xn+1 = xn − Df (x0 )−1 f (xn ),

n = 0, 1, 2, . . . . Df (x0 )

xn+1 !b = f (xn ) L!z = !b

U!y = !z n+1

= x â&#x2C6;&#x2019; y!

Î» x Â¯ x Â¯ = Î»exÂ¯ . Î» < 1/e x Â¯1 < x Â¯2

Î» > 1/e

Î» = 1/e

Î» < 1/e

x Â¯1

xÂ¯2

x0 xn+1 = Î»exn ,

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

xn+1 = ln xn â&#x2C6;&#x2019; ln Î»,

n = 0, 1, 2, . . . . x Â¯1

x Â¯1

xÂ¯2

x Â¯2

x â&#x2020;&#x2019; Î»ex Î» = 1 > 1/e Î» = 1/e Î» = 1/e2 < 1/e Î» = 1/e x â&#x2020;&#x2019; Î»ex x â&#x2020;&#x2019; x x = 1 x Â¯ = Î»exÂ¯ Î» = 1/e x Â¯=1 Î» < 1/e xÂ¯ = Î»exÂ¯ 1 xÂ¯1 xÂ¯2 Î» < 1/e

Î» > 1/e xÂ¯ = Î»exÂ¯

xÂ¯1

x Â¯2 g(x) = Î»ex xÂ¯1

xn+1 = g(xn ) g g |g & (Â¯ x1 )| = Î»exÂ¯1 = x Â¯1 < 1.

g g & (x) = Î»ex

4 x ex e−1 ex e−2 ex

3 2 1 0 -1 -2 -4

-3

-2

-1

x → λex

xn+1 = g(xn ) x ¯1 x0 < x ¯2 x ¯2

(xn )∞ n=0

λ = 1 λ = 1/e λ = 1/e2

x ¯1

x0 g

g(x) = ln x − ln λ x ¯ = ln x ¯ − ln λ

x¯2 x ¯2 g g

xn+1 = g(xn ) x¯

ex¯ = eln x¯−ln λ =

x ¯ , λ

x¯

g & (x) = 1/x

x ¯2 > 1 |g & (¯ x2 )| =

1 < 1. x ¯2

xn+1 = g(xn ) x ¯2 x0 > x ¯1

x ¯2

x0 x0

x ¯1

x ¯

x0 = 1

x0 = 0

f (x) = x3 − x − 3

f (x) = x3 − x − 3 f (xn ) f & (xn ) x3 − xn − 3 = xn − n 2 3xn − 1 3 2x + 3 = n2 , 3xn − 1

xn+1 = xn −

n = 1, 2, . . . x0 x1 x2 x3 x0 = 1

f x0 = 0

x ¯

f (x) f (x)

x ¯ x0 = 1 xn n = 1, 2, . . .

x0 = 0

30 20 10 0

♦

-10

♦♦ ♦ x3 x2 x1

-20 -30 -3

-2

-1

f (x) = x3 − x − 3 x0 = 1

A x0 1 xn+1 = xn + (A − x2n ), 2

√ A

(xn )∞ n=0

√ − A |x0 −

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

√ A| ≤ ε

A ∈]0, 4[ (xn )∞ n=0

√ ε > 0 A

30 20 10 0♦ ♦ x1 x2 -10

♦ x3

♦ x0

-1

-20 -30 -3

-2

f (x) = x3 − x − 3 x0 = 0 √ − A

(xn )∞ n=0 x0 = 1

√ − √A − A

x2 − A = 0

A (xn )∞ n=0

x ¯

1 x¯ = x¯ + (A − x ¯2 ), 2 √ x ¯=± A

x¯2 = A

xn+1 = g(xn )

1 g(x) = x + (A − x2 ). 2

x ¯=

√ A

g & (x) = 1 − x g & (¯ x) = 1 − x ¯=1−

√ A.

0<A<4 |g & (¯ x)| = |1 − (xn )∞ n=0 A=2

√ A| < 1.

ε>0 √ x ¯= A √ x0 < − A x0

|¯ x − x0 | < ε

√ − A

√ x0 > − A (xn )∞ n=0

(xn )∞ n=0

√ − A √ A 3 2

x g(x)

1 0 -1 -2 -3 -3

-2

-1

g(x) = x + (A − x2 )/2

A=2

f (x) = x2 − A

xn+1 = xn −

f (xn ) x2n − A = x − , n f & (x0 ) 2x0

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

x0 = 1 xn+1 = xn −

x2n − A 1 = xn + (A − x2n ), 2 2

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

f (x) = x2 − A = 0

x0 = 1 f (x) = x2 − A = 0

xn+1 = xn − √

f (xn ) x2n − A x2n + A = x − = , n f & (xn ) 2xn 2xn

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

√ A x0 > 0

Df (xn )(xn â&#x2C6;&#x2019; xn+1 ) = f (xn ),

Df (xn )

x0 x0

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

f : (x, t) ∈ R × R t

R+ 3 f (x, t) ∈ R → x

x t t

u0 ∈ R u0 u : t ∈ R+ → u(t) ∈ R u(t) ˙ = f (u(t), t)

t > 0,

u(0) = u0 , u(t) ˙ = du(t)/dt

f (x, t) = 3x−3t u(t) ˙ = 3u(t) − 3t

u0 = α t > 0,

u(0) = α;

u(t) = (α − 1/3)e3t + t + 1/3

f (x, t) = u(t) ˙ =

√ 3 x u0 = 0

6 3 u(t)

t > 0,

u(0) = 0; u

u(t) = 0

t≥0

u(t) = ±

6 8t3 /27

f (x, t) = x3 u0 = 1 u(t) ˙ = u3 (t),

t > 0,

u(0) = 1; u

1/2

t ∈ [0, ∞[

√ u(t) = 1/ 1 − 2t

t ∈ [0, 1/2[ t

limt→1/2 u(t) = +∞ t<1/2

t=∞ f : R × R+ → R

t∈R

, : t ∈ R+ → ,(t) ∈ R

x, y ∈ R

(f (x, t) − f (y, t)) (x − y) ≤ ,(t)|x − y|2 .

f : R × R+ → R K ∂f (x, t) ≤ K, ∂x t ∈ R+

∀x ∈ R,

∀t ≥ 0.

ξ f (x, t) − f (y, t) =

x, y ∂f (ξ, t)(x − y). ∂x

x, y ∈ R

(f (x, t) − f (y, t)) (x − y) =

∂f (ξ, t)(x − y)2 ≤ K(x − y)2 . ∂x ,(t) = K ∀t ∈ R+

f 2

u(t) ˙ = −u3 (t) + e−t

t > 0,

u(0) = 1, 2

f (x, t) = −x3 + e−t

∂f ∂x (x, t)

f = −3x2 ≤ 0

f t ∈ R+

R × R+

x, y ∈ R

|f (x, t) − f (y, t)| ≤ L|x − y|.

f 2

(f (x, t) − f (y, t)) (x − y) ≤ L |x − y| , , !

∀x, y ∈ R, ∀t ∈ R+ .

,(t) = L ∀t ≥ 0

f (x, t) = |x| + sin x + e−t

u(t)

u0 = 1

5 5 |f (x, t) − f (y, t)| = 5|x| − |y| + sin x − sin y 5 5 5 2 x 5 5 5 = 5|x| − |y| + cos θdθ55 y 5 5 ≤ 5|x| − |y|5 + |x − y| ≤ 2|x − y|.

u(t) u(t)

t ∈ R+

R+ un+1

0 = t0 < t1 < t2 < t3 < . . . < tn < tn+1 < . . . un u t = tn n u , u(tn ) u t = tn+1 un un+1 − un = f (un , tn ), tn+1 − tn un+1 = un + (tn+1 − tn )f (un , tn ). u0 = u(0) = u0

|u(tn ) − un |

n = 1, 2, . . . ,

f (x, t) = 3x − 3t

u(t) =

u0 = α.

. 1 3t 1 α− e +t+ . 3 3 t = 10

α = 0.333333

u(10) = 10 + α=

1 3 1 3

u(10) = (0.333333 − 13 )e30 + 10 + = − 31 10−6 · e30 +

1 7 3 10

u(10)

α =

31 3

1 3

31 3 ,

1 −6 · e30 3 10

10−6 106 α=

α = 0.333333 u(10)

1 3

10−16 10

−4

u(10)

f (x, t) = −3x − 3t u0 = α

f (x, t) = 3x − 3t

1 1 u(t) = (α − )e−3t − t + . 3 3

α=

1 3

u(10) = −10 +

1 3

= − 29 3

u(10) = − 13 10−6 e−30 − 10 +

α = 0.333333

1 3

1 −19 , − 29 3 − 3 10

t 0 t1 t2

... 0 = t0 < t1 < t2 < . . . < tn < tn+1 < . . . . hn = tn+1 − tn u(t ˙ n) un

u(tn+1 ) − u(tn ) . hn

u(t ˙ n+1 ) u(tn )

un+1 − un = f (un , tn ), hn u0 = u0 .

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

un+1 − un = f (un+1 , tn+1 ), hn u0 = u0 .

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

un+1 u ,u ,u ,... 1

un u0 un+1

un+1 = un + hn f (un , tn ). un+1

un+1 − hn f (un+1 , tn+1 ) = un . un+1 g(x) = x − hn f (x, tn+1 ) − un g(x) x0 = un xm+1 = xm −g(xm )/g & (xm ) g (x) = 1 − hn ∂f (x, tn+1 )/∂x &

m = 0, 1, . . . x0 = un ,

xm+1 = xm −

xm − hn f (xm , tn+1 ) − un , ∂f 1 − hn (xm , tn+1 ) ∂x

m = 0, 1, . . . .

lim xm = un+1

m→∞

un+1

f (x, t) = −βx f (x, t) = −βx

u(t) ˙ = −βu(t), u(0) = u0 ,

t > 0,

u(t) = e−βt u0 u(t)

β t

Ot n = 0, 1, 2, . . .

tn = nh

h > 0

un+1 = (1 − βh)un ,

n = 0, 1, 2, . . .

un = (1 − βh)n u0 ,

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

u(t) u0 $= 0 n

t un

1 − βh < −1

−1 ≤ 1 − βh

h h≤

2 . β h

t (1 + βh)un+1 = un ,

u =

1 1 + βh

u0 ,

n = 0, 1, 2, . . .

n = 0, 1, 2, . . . . h>0

lim un = 0;

n→∞

h f (x, t) = −βx T |u(T ) − uN | u(t)

tn = nh

u(tn+1 ) = e−βh u(tn ),

/ 0n u(tn ) = e−βh u0 ,

u h = T /N N n = 0, 1, 2, . . . , N

n = 0, 1, 2, . . . , N − 1, n = 0, 1, 2, . . . , N.

β>0

e−βh = 1 − βh + O(h2 ), O(h2 )

|u(T )−u |

n=N

|u(T ) − uN | = |(e−βh )N − (1 − βh)N | · |u0 |. N ≥ βT

(1 − βh) ≥ 0

/ 0 aN − bN = (a − b) aN −1 + aN −2 b + aN −3 b2 + · · · + abN −2 + bN −1 , 1 − βx ≤ e−βx

∀x ∈ R,

1 − βh ≥ 0

e−β(N −1)h = eβh e−βT ≤ e · e−βT .

|e−βh − (1 − βh)| ≤

|u(T ) − uN | ≤

β 2 h2 β2T 2 = . 2 2N 2

eβ 2 T 2 e−βT 1 |u0 | · . 2 N u(T )

limN →∞ u

e−βh =

= u(T )

1 1 1 = = + O(h2 ). eβh 1 + βh + O(h2 ) 1 + βh

−βx

f : RÃ&#x2014;R+ â&#x2020;&#x2019; R x t u(t) T >0 h = T /N C T

N n = 0, 1, 2, . . . , N

tn = nh N

N |u(T ) â&#x2C6;&#x2019; uN | â&#x2030;¤

C C = h, N T

u 0 , u 1 , . . . , uN limN â&#x2020;&#x2019;â&#x2C6;&#x17E; |u(T ) â&#x2C6;&#x2019; uN | = 0

1/N

u(T ) uN 1/N = h/T = O(h) h h

h |u(T ) â&#x2C6;&#x2019; uN | = O(h)

t=T h

tn+1 u(tn+1 ) â&#x2C6;&#x2019; u(tn ) = un

f (u(t), t)dt. tn

un+1

u(tn )

un+1 â&#x2C6;&#x2019; un =

tn+1

u(tn+1 )

4 1 3 hn f (un , tn ) + f (un+1 , tn+1 ) , 2 hn = tn+1 â&#x2C6;&#x2019; tn

h2 un+1 un+1 uË&#x153;n+1 = un + hn f (un , tn ).

n = 0, 1, 2 . . . ,

un u

n+1

u ˜n+1 u

n+1

u ˜n+1

p1 = f (un , tn ), p2 = f (un + hn p1 , tn+1 ), hn un+1 = un + (p1 + p2 ). 2

tn+1/2 = (tn + tn+1 )/2 [tn , tn+1 ] un+1 − un = hn f (un+1/2 , tn+1/2 ), un+1/2

u(tn+1/2 ) un+1/2 un+1/2 = un +

hn f (un , tn ). 2 un

p1 = f (un , tn ), hn hn p2 = f (un + p1 , tn + ), 2 2 un+1 = un + hn p2 .

2 h

t = tn

n+1

un t = tn+1

p1 = f (un , tn ), hn hn p2 = f (un + p1 , tn + ), 2 2 hn hn p3 = f (un + p2 , tn + ), 2 2 n p4 = f (u + hn p3 , tn+1 ), hn un+1 = un + (p1 + 2p2 + 2p3 + p4 ). 6 h t=T

T N

tj = jh j = 0, 1, 2, . . . , N f |u(T ) − uN | ≤ Ch4 = C N

T4 , N4

C T h

f! : (!x, t) ∈ R

×R !u0

! x, t) ∈ R → f(! M M

!u : t ∈ R+ → !u(t) ∈ RM !u˙ (t) = f!(!u(t), t),

t > 0,

!u(0) = !u0 . M M u1 (t), u2 (t), . . . , uM (t) !u(t) !u˙ (t) u˙ 1 (t), u˙ 2 (t), . . . , u˙ M (t)

1·1 ! y , t))T (!x − !y ) (f!(!x, t) − f(!

(f (x, t) − f (y, t))(x − y)

|·|

!un+1 = !un + hn f!(!un , tn ). !un !un+1

!u(tn )

!u(tn+1 )

f : (x, y, t) ∈ R2 × R+ → f (x, y, t) ∈ R u0

u : t ∈ R+ → u(t) ∈ R, u¨(t) = f (u(t), u(t), ˙ t), t > 0, u(0) = u0 , u(0) ˙ = v0 , u¨(t) = d2 u(t)/dt2

t v(t) = u(t) ˙ u(t)

v(t)

u(t) ˙ = v(t), v(t) ˙ = f (u(t), v(t), t), t > 0, u(0) = u0 v(0) = v0 . 2 2

f f (x, y, t) = g(x, t) u¨(t) = g(u(t), t), u(0) = u0 ,

t > 0,

u(0) ˙ = v0 .

h>0

tn = nh n = 0, 1, 2, . . .

u(tn ) un+1 − 2un + un−1 = g(un , tn ), h2 u0 = u0 , 1 u1 = u0 + hv0 + h2 g(u0 , 0). 2

n = 1, 2, . . . ,

un+1

u1 = u(0) + hu(0) ˙ +

u(h)

h2 u¨(0); 2

t=0

g(x, t) = −λx λ > 0 u ¨(t) = −λu(t), u(0) = u0 ,

t > 0,

u(0) ˙ = v0 ,

u0 , v0 g(x, t) = −λx √ √ v0 u(t) = √ sin λt + u0 cos λt, λ √ P = 2π/ λ . λh2 α= 1− , 2

un+1 = 2αun − un−1 , 0

u = u0 ,

u1 = αu0 + hv0 .

n = 1, 2, . . . ,

un−1

r2 = 2αr − 1, r

un−1

1 un

r2 = α −

6 α2 − 1.

un+1

|α| r1 = α +

6 α2 − 1

un = a(r1 )n + b(r2 )n ,

a= un

1 h u0 + √ v0 , 2 2 α2 − 1

n = 0, 1, 2, . . .

1 h u0 − √ v0 , 2 2 α2 − 1

u(tn ) √ √ v0 u(tn ) = √ sin λtn + u0 cos λtn . λ

|u(tn )|

|r1 | ≤ 1 n

|r2 | ≤ 1

5 5 6 5 5 5α ± α2 − 15 ≤ 1.

|α| > 1√ √ 2 α ± i 1 − α2 √α ± α − 1 = |α± α2 − 1| = (α2 +(1−α2 ))1/2 = 1

|α| ≤ 1

2 h≤ √ . λ √ P = 2π/ λ

|un |

|α| ≤ 1

T >0

tn = nh

h n = 0, 1, 2, . . . , N N

h = T /N |u(T ) − uN | h2 N

9!10

u(t) ˙ = −(u(t))m + cos(t) u(0) = 0,

t > 0,

m h

tn = nh n = 0, 1, 2, . . . u(tn ) n = 0, 1, 2, . . . un+1 un

f (x, t) = −xm + cos t

u0 = 0

∂f (x, t) = −mxm−1 ≤ 0, ∂x f (f (x, t) − f (y, t))(x − y) ≤ 0

∀x, y ∈ R. ,(t) = 0

n+1

. n+1 m = u + h −(u ) + cos(tn+1 ) , n

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

u0 = 0.

un+1 u

u1 . u1 = u0 + h −(u1 )m + cos(t1 ) . = h −(u1 )m + cos(h) . u1

g g(x) = x + hxm − h cos(h).

g g(xk ) g & (xk ) xk + h(xk )m − h cos(h) = xk − . 1 + mh(xk )m−1

xk+1 = xk −

x0 = u0 = 0 x1 = h cos(h). u1

β >0 u(t) ˙ = −βu(t), u(0) = u0 , u0 un

t > 0,

h u(tn ) n = 0, 1, 2, . . .

tn = nh

lim un = 0

n→∞

p1 = −βun , p2 = −β(un + hp1 ) = βun (−1 + βh), . h β 2 h2 n+1 n u = u + (p1 + p2 ) = 1 − βh + un . 2 2

un =

.n β 2 h2 1 − βh + u0 , 2

limn→∞ un = 0 5 5 2 25 5 51 − βh + β h 5 < 1. 5 2 5

q(x) = 1−x+x2 /2 |q(x)| < 1

0<x<2

βh < 2 limn→∞ un = 0 h<

2 . β h ≤ 2/β

1 − x + x2 /2

−x

2 1 0 -1 -1

x → 1 − x + x2 /2

p1 = −βun , . . h βh n n p2 = −β u + p1 = βu −1 + , 2 2 . . h βh β 2 h2 p3 = −β un + p2 = βun −1 + − , 2 2 4 . β 2 h2 β 3 h3 p4 = −β (un + hp3 ) = βun −1 + βh − + , 2 4 h un+1 = un + (p1 + 2p2 + 2p3 + p4 ) 6 . β 2 h2 β 3 h3 β 4 h4 = 1 − βh + − + un . 2 6 24 .n β 2 h2 β 3 h3 β 4 h4 u = 1 − βh + − + u0 , 2 6 24 n

limn→∞ un = 0 5 5 2 2 3 3 4 45 5 51 − βh + β h − β h + β h 5 < 1. 5 2 6 24 5

r(x) = 1 − x + x2 /2 − x3 /6 + x4 /24 |r(x)| < 1 0 < x < x ¯ βh < 2.78

r x ¯ , 2.785 limn→∞ un = 0

2.78 . β

h ≤ x¯/β

r(x) e−x

2 1 0 -1 -1

x → 1 − x + x2 /2 − x3 /6 + x4 /24

u un+1

un+1 un−2 . . .

n−1

c [0, 1]

[0, 1] − u&& (x) + c(x)u(x) = f (x)

0 < x < 1,

u(0) = u(1) = 0.

x=0

x=1 f (x) u(x) c(x) = P/EI(x)

P x E

I(x) x −P

f (x)

x=0

x=1

u(x) x=0 f (x)

x=1 c(x) = 0 ∀x ∈ [0, 1]

f (x) 0

u(x)

u(0) = 0 u(1) = 0 câ&#x2030;¥0

[0, 1] u(x)

x â&#x2C6;&#x2C6; (0, 1)

N 0, 1, 2, . . . , N + 1

h = 1/(N + 1)

xj = jh j =

xN â&#x2C6;&#x2019;1 xN xN +1 1

u Î´h2 u(x) + O(h2 ) h2 u(x + h) â&#x2C6;&#x2019; 2u(x) + u(x â&#x2C6;&#x2019; h) = + O(h2 ), h2

u&& (x) =

O(h2 )

h h2 uj â&#x2C6;&#x2019;ujâ&#x2C6;&#x2019;1 + 2uj â&#x2C6;&#x2019; uj+1 + c(xj )uj = f (xj ) h2 u0 = uN +1 = 0.

u(xj ) 1 â&#x2030;¤ j â&#x2030;¤ N,

uj 1≤j≤N !u

u(xj )

uj , u(xj )

u 1 , u 2 , . . . , uN f! f (x1 ), f (x2 ), . . . , f (xN ) A N ×N



   1   A= 2 h    

2 + c1 h2



−1

2 + c 2 h2

−1

−1 −1

−1 2 + cN h2

ci = c(xi ) !u

     ,    

! A!u = f.

c(x) ≥ 0

x≥0

A !u A

c(x) ≥ 0 ∀x ∈ [0, 1] N

u C

h max | u(xj ) − uj |≤ Ch2 .

1≤j≤N

(uj )1≤j≤N lim

max | u(xj ) − uj |= 0.

N →∞ 1≤j≤N

v [0, 1] 2 −

[0, 1] 1

u&& (x)v(x)dx +

c(x)u(x)v(x)dx =

f (x)v(x)dx.

1 &

u (x)v (x)dx − u (1)v(1) + u (0)v(0) +

c(x)u(x)v(x)dx

= v 2

u (x)v (x)dx +

x=0

c(x)u(x)v(x)dx =

f (x)v(x)dx. 0

x=1

f (x)v(x)dx.

g g(0) = g(1) = 0

g& g

[0, 1]

V V

V u∈V

v∈V u

u u

u c(x) ≥ 0 ∀x ∈ [0, 1]

ϕ1 , ϕ2 , . . . , ϕN

V V

ϕi

g(x) =

g N (

gi ϕi (x),

i=1

N uh ∈ Vh

u&h (x)vh& (x)dx +

c(x)uh (x)vh (x)dx =

f (x)vh (x)dx

vh ∈ Vh

uh (x) =

N (

ui ϕi (x),

i=1

u 1 , u 2 , . . . , uN 1 ≤ j ≤ N u 1 , u 2 , . . . , uN N ( i=1

-2

ϕ&i (x)ϕ&j (x)dx

vh = ϕj

. c(x)ϕi (x)ϕj (x)dx =

f (x)ϕj (x)dx 0

j = 1, 2, . . . , N A N ×N 2 1 2 1 & & Aji = ϕi (x)ϕj (x)dx + c(x)ϕi (x)ϕj (x)dx, 0

c=0 N

A u 1 , u 2 , . . . , uN fj =

!u f!

f (x)ϕj (x)dx, 0

!u ! A!u = f. A

ϕ1 ϕ2 . . . ϕN Vh A A

uh u

N V

| · |1

|g|1 =

-2

.1/2 (g & (x))2 dx

g ∈ V.

c(x) ≥ 0 ∀x ∈ [0, 1]

|u − uh |1 ≤ C min |u − vh |1 , vh ∈Vh

C Vh

C = 1+maxx∈[0,1] |c(x)|

c(x) = 0 ∀x ∈ [0, 1]

uh 2

(u& (x) − u&h (x)) vh& (x)dx = 0

∀vh ∈ Vh .

e(x) = u(x) − uh (x) 2

|e|1 =

|e|1 ≤

e& (x)vh& (x)dx = 0, | · |1

-2

(e& (x)) dx =

∀vh ∈ Vh .

e& (x)(u& (x) − u&h (x))dx.

e& (x)u& (x)dx =

|e|1 = vh

e& (x)(u& (x) − vh& (x))dx,

.1/2 -2 1 .1/2 (e& (x))2 dx (u& (x) − vh& (x))2 dx , 0

|e|1 ≤ |e|1 |u − vh |1 . C=1 !

|e|1

vh ∈ Vh

[0, 1] N +1 N h = 1/(N + 1) xi = ih i = 0, 1, 2, . . . , N + 1 i = 1, 2, . . . , N  x−x i−1  xi−1 ≤ x ≤ xi ,   x − x  i i−1     x − xi+1 ϕi (x) = xi ≤ x ≤ xi+1 ,  x i − xi+1        0 x ≤ xi−1 x ≥ xi+1 . ϕi

ϕi ϕi (xj ) = δij , ϕi|[xj−1 ,xj ] ϕi

0 ≤ j ≤ N + 1, V

1 ≤ j ≤ N + 1. ϕ1 , ϕ2 , . . . , ϕN Vh

x0 , x1 , x2 , . . . , xN +1 [x0 , x1 ], [x1 , x2 ], . . . , [xN , xN +1 ] ϕ1 , ϕ2 , . . . , ϕN

Vh x1 , x2 , . . . , xN

x x1

...

xi−1

xi+1

. . . xN

xN +1 = 1

ϕi g ∈ Vh

ϕi g(x) =

N (

gi ϕi (x),

i=1

g g

g(xj ) = gj 1 ≤ j ≤ N

g(0) = g(1) = 0

g(x) g4 g3 g1 g2 0

x x1

...

xN g

xN +1 = 1 Vh

u∈V rh u =

N (

u(xi )ϕi

i=1

u rh u ∈ Vh min |u − vh |1 ≤ |u − rh u|1 .

vh ∈Vh

|u − uh |1 ≤ C|u − rh u|1 . u

uh u

c(x) ≥ 0 ∀x ∈ [0, 1] Vh |u − uh |1 ≤ Ch,

C˜

˜ |u − rh u|1 ≤ Ch, N w = u − rh u.

| · |1 rh u

rh u(xi ) = u(xi ) 0 ≤ i ≤ N + 1 w& (ξi ) = 0 0 ≤ i ≤ N w& (x) =

w(xi ) = 0 ξi ∈]xi , xi+1 [

rh u [xi , xi+1 ] 2

w&& (s)ds =

ξi

x ∈ [xi , xi+1 ] x

u&& (s)ds.

ξi

x ∈ [xi , xi+1 ] |w& (x)| ≤

xi+1

|u&& (s)|ds. x ∈ [xi , xi+1 ]

|w& (x)| ≤

-2

xi+1

12 ds

≤ h1/2

-2

xi+1

.1/2 -2

xi+1 xi

|u&& (s)|2 ds

.1/2

[xi , xi+1 ] 2

xi+1 xi

|w& (x)|2 dx ≤ h2

xi+1

|u&& (s)|2 ds.

i 2

|u − rh u|21 = |w|21 = =

N 2 ( i=0

= h2

|w& (x)|2 dx

xi+1

|w (x)| dx ≤ h

xi 1

N 2 ( i=0

xi+1

|u&& (s)|2 ds

|u&& (s)|2 ds. C˜

C˜ =

-2

|u (s)| ds

.1/2

A A!u = f!

Aji =

ϕ&i (x)ϕ&j (x)dx +

c(x)ϕi (x)ϕj (x)dx,

1 ≤ i, j ≤ N fj =

f (x)ϕj (x)dx

1 ≤ j ≤ N.

ϕ&i (x)ϕ&j (x)dx

   2/h  

i = j,

−1/h     0 11 0

|i − j| = 1, .

c(x)ϕi (x)ϕj (x)dx

11 0

f (x)ϕj (x)dx 11 0

Lh (,) = h

,(x)dx

. 1 1 ,(x0 ) + ,(x1 ) + ,(x2 ) + · · · + ,(xN ) + ,(xN +1 ) . 2 2   hc(x ) j Lh (cϕi ϕj ) =  0

i = j, i $= j,

Lh (f ϕj ) = hf (xj ). 11 c(x)ϕi (x)ϕj (x)dx 0

(xj )1≤j≤N ϕi ϕi

Vh u

h h = max |xi+1 − xi |. 0≤i≤N

Vh [xj , xj+1 ]

[0, 1] M +1 M h = 1/(M + 1) xi = ih i = 0, 1, . . . , M + 1 xi+1/2 = xi + h/2 i = 0, 1, . . . , M i = 1, 2, . . . , M  (x − xi−1 )(x − xi− 12 )    xi−1 ≤ x ≤ xi ,   (xi − xi−1 )(xi − xi− 12 )      (x − xi+1 )(x − xi+ 21 ) ψi (x) = xi ≤ x ≤ xi+1 ,   (xi − xi+1 )(xi − xi+ 12 )         0 x ≤ xi−1 x ≥ xi+1 ;

i = 0, 1, . . . , M  (x − xi )(x − xi+1 )    (x 1 − x )(x 1 − x ) i i+1 i+ 2 i+ 2 ψi+ 12 (x) =    0 ψi

ψi (x

x ≥ xi+1 .

) = 0,

0 ≤ j ≤ M + 1, 0 ≤ j ≤ M,

ψi|[xj−1 ,xj ] ψi+ 12 (xj+ 12 ) = δij , ψi+ 21 (xj ) = 0, ψi+ 21 |[x

x ≤ xi

ψi ψi+1/2 ψi+1/2

ψi (xj ) = δij , j+ 21

xi ≤ x ≤ xi+1 ,

j−1 ,xj ]

, 1 ≤ j ≤ M + 1; 0 ≤ j ≤ M, 0 ≤ j ≤ M + 1, , 1 ≤ j ≤ M + 1.

N = 2M + 1 ϕ1 = ψ1/2 ϕ2 = ψ1 ϕ3 = ψ3/2 ϕ4 = ψ2 ϕ5 = ψ5/2 ϕ6 = ψ3 . . . ϕ2M = ψM ϕ2M+1 = ψM+1/2 ϕ1 ϕ2 . . . ϕN V Vh x0 , x1 , x2 , . . . , xM+1

ψi (x)

# ψi− 12 (x)

ψi+ 12 (x)

! xi−1 xi− 12 xi

xi+ 12 xi+1 ψi−1/2 ψi

ψi+1/2

[x0 , x1 ], [x1 , x2 ], . . . , [xM , xM+1 ] x1/2 , x3/2 , x5/2 , . . . , xM+1/2 ϕ1 , ϕ2 , . . . , ϕN g ∈ Vh

ϕi g(x) =

N (

gi ϕi (x),

i=1

g(xj ) = g2j 1 ≤ j ≤ M g(0) = g(1) = 0 g

g(xj+1/2 ) = g2j+1 0 ≤ j ≤ M

#g(x) g3 g2 g1 x0

x1/2

x3/2

g u∈V rh u =

M ( j=1

u(xj )ϕ2j +

x5/2

x7/2

M ( j=0

u(xj+ 12 )ϕ2j+1

x M =3

u rh u ∈ Vh min |u − vh |1 ≤ |u − rh u|1 .

vh ∈Vh

u C

|u − rh u|1 ≤ Ch2 .

c(x) ≥ 0 ∀x ∈ [0, 1]

u uh

Vh |u − uh |1 ≤ Ch2 , C

c x f [0, 1] × R −→ c˜(x, v) ∈ R

u c˜ : (x, v) ∈

− u&& (x) + c˜(x, u(x)) = f (x),

0 < x < 1,

u(0) = u(1) = 0, u

c˜(x, v) = c(x)v

u(xj )

−uj−1 + 2uj − uj+1 + c˜(xj , uj ) = f (xj ) h2 u0 = uN +1 = 0.

1 ≤ j ≤ N,

c(x)

u 1 , u 2 , . . . , uN !u

N RN

F (!u)

2u1 − u2 h2



u 1 , u 2 , . . . , uN

     −u1 + 2u2 − u3   h2     F (!u) =       −u  N −2 + 2uN −1 − uN   h2    −uN −1 + 2uN h2



c˜(x1 , u1 ) − f (x1 )

      + c˜(x2 , u2 ) − f (x2 )      .       + c˜(xN −1 , uN −1 ) − f (xN −1 )     + c˜(xN , uN ) − f (xN ) !u

F (!u) = 0. !u0

!u !un+1 = !un − DF (!un )−1 F (!un ),

n = 0, 1, 2, . . .

c˜

d d(x, v) =

∂ c˜(x, v). ∂v

dnj = d(xj , unj ), def

1 ≤ j ≤ N. DF (!un )

 2 + dn1 h2   −1  1   n DF (!u ) = 2  h    4  !un+1

−1

2 + dn2 h2

−1

−1 −1

−1 2 + dnN h2 !un



     .    

F (!un ) ...

DF (!un ) y! DF (!un )!y = F (!un ) !un+1 = !un − !y

!u0

un1 un2

u 1 , u 2 , . . . , uN

unN

c˜(x, u) = c(x)u dnj = c(xj ) = cj

c(x)

d(x, u) = DF (!un ) n

A n=1

f : [0, 1] → R u : [0, 1] → R . d du − (1 + x) (x) = f (x), dx dx u(0) = u(1) = 0. N

0 < x < 1,

h = 1/(N + 1) xj = jh j = 0, 1, . . . , N + 1 Vh ϕ1 ϕ2 . . . ϕN

v [0, 1]

[0, 1]

−

d dx

. 2 1 du (1 + x) (x) v(x)dx = f (x)v(x)dx. dx 0

Kx=1 2 (1 + x)u (x)v (x)dx − (1 + x)u (x)v(x) = &

x=0

v 2

(1 + x)u& (x)v & (x)dx =

x=0 2

f (x)v(x)dx.

x=1

f (x)v(x)dx.

g g(0) = g(1) = 0 u∈V

v∈V Vh

ϕ1 ϕ2 . . . ϕN

uh ∈ Vh 2

(1 + x)u&h (x)vh& (x)dx =

f (x)vh (x)dx 0

vh ∈ Vh Aji =

(1 + x)ϕ&i (x)ϕ&j (x)dx,

1 ≤ i, j ≤ N.

ϕ&i 1 ≤ i ≤ N [xj−1 , xj ] 1 ≤ j ≤ N + 1 A Aii 1 ≤ i ≤ N Aii =

xi−1

(1 + x)ϕ&i (x)ϕ&i (x)dx +

xi+1 xi

Ai,i+1 1 ≤ i ≤ N −1

(1 + x)ϕ&i (x)ϕ&i (x)dx

h 1 h 1 (1 + xi−1 + 1 + xi ) 2 + (1 + xi + 1 + xi+1 ) 2 2 h 2 h 2 = (1 + ih). h =

xi+1

(1 + x)ϕ&i+1 (x)ϕ&i (x)dx . h 1 = (1 + xi + 1 + xi+1 ) − 2 2 h 1 = − (1 + ih + h/2). h

Ai,i+1 =

f : [0, 1] → R −u&& (x) = f (x)

u(0) = 0, u& (1) + αu(1) = 0.

α u : [0, 1] → R 0 < x < 1,

x=0 N

h = 1/(N + 1) xj = jh j = 0, 1, . . . , N + 1 Vh [xj−1 , xj ] j = 1, 2, . . . , N + 1

v [0, 1]

[0, 1]

− 2

u&& (x)v(x)dx =

f (x)v(x)dx.

u& (x)v & (x)dx − u& (1)v(1) + u& (0)v(0) = v 2

f (x)v(x)dx.

x=0

u& (x)v & (x)dx − u& (1)v(1) =

f (x)v(x)dx.

u (x)v (x)dx + αu(1)v(1) =

f (x)v(x)dx.

g g(0) = 0 u∈V

v∈V

x=1 ϕ1 ϕ2 ϕN +1

. . . ϕN x − xN xN +1 − xN ϕN +1 (x) =  0  

xN ≤ x ≤ xN +1 , .

Vh . . . ϕN ϕN +1 uh ∈ Vh 2 1 2 u&h (x)vh& (x)dx + αuh (1)vh (1) = 0

ϕ1 ϕ2

f (x)vh (x)dx

vh ∈ Vh Vh

uh =

N +1 (

uj ϕj ,

i=1

vh = ϕ1 vh = ϕ2 . . . vh = ϕN vh = ϕN +1 u1 , u2 , . . . , uN , uN +1 N +1 (

i=1

A!u = f!

-2

1 0

ϕ&i (x)ϕ&j (x)dx

. 2 + αϕi (1)ϕj (1) =

f (x)ϕj (x)dx

j = 1, 2, . . . , N, N + 1 A (N + 1) × (N + 1)

Aji =

ϕ&i (x)ϕ&j (x)dx + αϕi (1)ϕj (1),

1 ≤ i, j ≤ N + 1,

(N + 1) fj =

f (x)ϕj (x)dx,

1 ≤ j ≤ N + 1.

A ϕ1 ϕ2 . . . ϕN A

N 2 , h

Aii =

AN +1,N +1 =

1 Ai,i+1 = − , h

x=1

1 ≤ i ≤ N.

ϕ&N +1 (x)ϕ&N +1 (x)dx + αϕN +1 (1)ϕN +1 (1)

1 + α. h fj

fj =

f (x)ϕj (x)dx , hf (xj )

fN +1 =

f (x)ϕN +1 (x)dx ,

1 ≤ j ≤ N,

h f (xN +1 ). 2

A 

  −1   1 A=  h     

−1 2



         −1 2 −1   −1 2 −1   −1 1 + αh

−1

A!u = f! 

f (x1 )



    f (x2 )         . f! = h     f (xN −1 )       f (xN )    1/2 f (xN +1 )

Couv_3114_2016.qxp_Couverture.qxd 07.12.16 09:27 Page1

Presses polytechniques et universitaires romandes

Introduction à l’analyse numérique Jacques Rappaz Marco Picasso

Jacques Rappaz Marco Picasso

Introduction à l’analyse numérique

Presses polytechniques et universitaires romandes