[PDF] Analyse Numérique 1.5 Exercices du chapitre

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??? ?? ???????QR? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? x≈ ±mbp b 2 x-x∗ x

2bp-1=b1-N

2 x= 0.31415927 10-1-0.31415 10-1= 0.0000927 10-1= 0.927 10-4 A=XN XD =π-3,1415 10

4(π-3,1515)-0,927

XD= 104(0,927.10-4)-0,927 = 0,0

∗= 3,1415927

A=ERREUR

∗= 3,14159265

A=-0,18530...

∗= 3,141592653⌉

A=-0,197134...

∗= 3,141592654⌋

A=-0,201427...

∗= 3,1415926535⌉

A=-0,1992548...

∗= 3,1415926536⌋

A=-0,1996844...

∗= 3,14159265358⌉

A=-0,1995984...

∗= 3,14159265359⌋

A=-0,19964143...

∗= 3,141592653589

A=-0,19963713...

∗= 3,1415926535897⌉

A=-0,199640143...

∗= 3,1415926535898⌋

A=-0,1996405743...

∗= 3,14159265358979

A=-0,1996405312

∗= 3,1415927653589793

A=-0,1996405439...

a: = 0,23371258.10-4 b: = 0,33678429.102 c: =-0,33677811.102 a+b= 0,00000023(371258).102+ 0,33678429.102= 0,33678452.102. (a+b) +c= 0,33678452.102-0,33677811.102 = 0,00000641.102= 0,641.10-3. b+c= 0,33678429.102-0,33677811.102 = 0,00000618.102= 0,618.10-3 a+ (b+c) = 0,02337125(8).10-3+ 0,61800000.10-3= 0,64137126.10-3. ????a+b? ?? ? ? vf(a+b) = (a+b)(1 +ε1) 1 2

β1-n???? ???5.10-8? ??????η=vf(a+b)?

= [(a+b)(1 +ε1) +c](1 +ε2) =a+b+c+ (a+b)ε1(1 +ε2) + (a+b+c)ε2. vf((a+b) +c)-(a+b+c) a+b+c=a+b a+b+cε1(1 +ε2) +ε2. vf(a+ (b+c))-(a+b+c) a+b+c=b+c a+b+cε3(1 +ε4) +ε4. a+b a+b+c≃5.104,b+c a+b+c≃0,9. x∈R7-→f(x)∈R. f(x)-f(x∗) x f(x)-f(x∗) f(x)x-x∗ x ≃xf′(x) f(x) ?? ?? ?????x? ?? ?????? cond(f)x:=xf′(x) f(x) x xf′(x) f(x) =1 2 ??????? ????f(x) =a-x xf′(x) f(x) =x a-x f(x)? x. xf′(x) f(x) x+ 1) 2 x(x+ 1)-1 x =1 2 x x+ 1 1 2 ????x?????? ????? ??x??? ?????? ??

12345 = 111,113-111,108 = 0,500000.10-2.

?? ?? ?????? ?????? ????? ?f(12345) = 0,4500032....10-2.?? ? ???? ??? ?????? ?? x

0: = 12345

x

1: =x0+ 1

x x 1 x x 0 x

4: =x2-x3

x=1 x f(12345) =1

12346 +

12345
=1

222,221= 0,450002.10-2

e x=N∑ n=0x n n!(=SN)????N?????? N S

NN SNNSN

2-11,0...19 1629,87...36-0,001432...

3 61,0...20-996,45...37 0,000472...

4-227,0...21 579,34...38-0,0001454...

5 637,0...22-321,11...39 0,000049726...

6-1436,6...23 170,04...40-0,000010319...

7 2710,6...24-86,20...41 0,000007694...

8-4398,88...25 41,91...42 0,000002422...

9 6265,34...26-19,58...43 0,000003928...

10-7953,62...27 8,80...44 0,000003508...

11 9109,137...28-3,8130...45 0,000003623...

12-9504,78...29 1,5937...46 0,000003592...

13 9109,13...30-0,6435...47 0,000003600...

14-8072,94...31 0,2513...48 0,000003598...

15 6654,55...32-0,0950...49 0,000003599...

16-5127,44...33 0,0348...50 0,000003598...

17 3709,05...34-0,01238...

18-2528,47...35 0,004283...

?? ?????? ??e-12??? ?? ???? ??0,0000061442...? ?? ???????e-x=1 e x????? ???? ? ?? ?? ?????? ??? ?? ???????8? b 2-4ac 2a? ????x=-2c b sin(α+x)-sinα

0,1580 0,2653 0,2581.1010,4288.1010,6266.1020,7555.102

0,7889.1030,7767.1030,8999.104.

??? ???????1?1 6 ?1 6

2? ????1

6 x

0= 1?x1=1

6 ?xn+1=37 6 xn-xn-1???? ????n≥1. f(x) = 0 [ai,bi]? ??? ??????? ? f

1(x) =x-0,2sinx-0,5

f ′1(x) = 1-0,2cosx≥0???? ????x .

0, f1? ?? ?????? ???? ????[0,π].

f f

2excos(

x-π 4 4 +k 2 4 + (k+ 1) 2 4 +k 2

0, f(π)>0? ???? ?? ???? ????(0,π)? ?? ?? ??????? ?? ?????? ??f??(

2 ???f( 2 2 -0,7>0? ???? ?? ???? ??? ?? ???? ????[0, 2 4 )= 0,14>0? ??? ?????? ????? ??? ????[0, 4 ????n= 0,1,2,...,N,????? m:=(an+bn) 2 ?????an+1:=m, bn+1:=bn. ?? ? ?an+1-bn+1=1 2 2 n(a0-b0)? ?? ?? 1 2 ?????? ????? ???? ??? ?? ?????? ?? ?????? ??f?? ?? ?????? ??[0,π]???? ?????? ?? ??

Y(x) =f(x0) + (x-x0)f(x0)-f(x-1)

x

0-x-1,

Y(x1) = 0

x

1=x0-f(x0)x0-x-1

f(x0)-f(x-1). n+1 AB ????? ?? ?????? ?? ???????xn+1?????? ???? ?????? ?? ???? ??????? ???xn-1??xn? ????n= 0,1,2,... x n+1=xn-f(xn)xn-xn-1 f(xn)-f(xn-1). x n= 1 +1 2 +...+1 n |f(xn)|< ε.????? |f(xn)-f(xn-1)|< ε.

Y=f(xn) +f′(xn)(x-xn).

[????n= 0,1,2,... x n+1=xn-f(xn) f ′(xn). xn-xn-1 1 f (∗)f(x) = 0 (∗∗)g(x) =x n) x

2-x-2 = 0

g(x) =x2-2 2 +x g(x) = 1 +2 x g(x) =x-x2-x-2 m [????n= 0,1,2,... x n+1=g(xn). x x ∞=g(x∞). x? x+ 2 ????? ???0? ??xn? ?????? ???? ????x0∈[a,b]? ?? ????? ?????? ??? x n+1=g(xn)∀n∈N ∀x∈[a,b]g(x)∈[a,b]. g(x)-x?????? ? ?????h(a) =g(a)-a≥0???????g(a)∈[a,b] ????h(b) =g(b)-b≥0???????g(b)∈[a,b]. ????x∞?? ????? ???? ?? ? ? x lim n→∞|xn+1-x∞| |xn-x∞|p=C, x n+1-x∞=g(xn)-g(x∞) = (xn-x∞)g′(x∞) +1 2 x 2 |xn-x∞|2. g(x) =x-f(x) f ′(x)? ?? ??????? ??g??? ?????? ??? ? g ′(x) = 1-f′(x) f ′(x)+f(x)f′′(x) f ′2(x)=f(x)f′′(x) f ′2(x). ??f′(x∞)̸= 0? ?? ? ?????? ???????f(x∞) = 0? g ′(x∞) = 0. f(x∞) = 0, f′(x∞)̸= 0. ?????? ??x0??? ?????? ????? ???? ??x∞? ?? ??????? ?? ?????? ? x n+1=xn-f(xn) f ′(xn)∀n≥0 ???? ?????? ??????x0??? ????? ?????? ??x∞? x1x 2x0x1 f(xn) f ??????? ?f(x) =x2 x n+1=xn-f(xn) f ′(xn)=1 2 xn; ???f′(x∞)̸= 0∀x∈[a,b] |f(a)| |f′(a)|< b-a,|f(b)| |f′(b)|< b-a. g(x) =x-f(x) f ′(x) g ′(x) =f(x)f′′(x) f ′2(x) ???????f′(x∞)̸= 0? ?? ??????ε1??? ??? ? ∀x∈[x∞-ε1,x∞+ε1], f′(x)̸= 0. 2 <1. 2 x

0∈[x∞-ε2,x∞+ε2]

x n+1=g(xn) =xn-f(xn) f ′(xn) g? ??? ?? ?????? ??? g ′′(x∞) =f′′(x∞) f ′(x∞). x n+1:=xn-f(xn)xn-xn-1 f(xn)-f(xn-1)=xn-1f(xn)-xnf(xn-1) f(xn)-f(xn-1) x f(xn)-f(xn-1) x n+1-x∞= (xn-x∞)(xn-1-x∞)[ f(xn)-f(x∞) x n-x∞-f(xn-1)-f(x∞) x n-1-x∞] f(xn)-f(xn-1). f[x] : =f(x)quotesdbs_dbs27.pdfusesText_33

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