.

y

x{f( )

x2 x1 x0

M0

P0

y=f(x)

f(b)

b

}

2008

.

2008

2

1 7

1.1 . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 . . . . . . . . . . . . . . . . . . . . 8

1.3 13

1.4 . . . . . . . 15

1.5 . . . . . . . . . . . . . . . . . . 17

2 24

2.1 . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Bolzano . 25

2.3 (Regula Falsi) . . 31

2.4 . . . . . . . . . . 33

2.4.1

. . . . . . . . . . . . . . . . . . . . . 41

2.5 Newton-Raphson . . . . . . . . . . . . . . 43

2.5.1 Newton-Raphson . . . 44

2.5.2 Newton 2 Bailey 54

2.6 . . . . . . . . . . . . . . . 55

2.7 - Aitken . . 58

2.8 . . . . . . . . . . . . . . . . 59

2.8.1 Horner . . . . . . . . . . . . . . 61

2.8.2 p(x) 63

3 -

66

3.1 Gauss . . . . . . . . . . . 66

3.1.1 Axk = bk, k = 1(1) . . . . . . . . 733.1.2 A1 . . . . . . . . . . . . . . 733.1.3 detA . . . . . . . . . . . . . 73

3

3.1.4 Gauss . . . . 74

3.1.5 Gauss 75

3.1.6 . . . . . . . . . . . . . . 79

3.1.7 Gauss

. . . . . . . . . . . . . . . . . . . . . 82

3.2 Jordan . . . . . . . . . . . 84

3.2.1 Jordan -

. . . . . . . . . . . . . . . . . . . 90

3.2.2 . . . . . . . . . 92

3.3 LU . . . . . . . . . . . . . . . . . . . . . 97

3.4 LU . . . . . . . . . . . . . 107

3.4.1 Choleski . . . . . . . . . . . 117

3.5 LU . . . . . . . . . . 118

3.5.1 LU - . . . . . . . . . . . . . . . . . . . . . . . 121

3.5.2 . . . . . . . . . 123

3.5.3 LU . 131

3.6 Norms . . . . . . . . . . . 135

3.7 . . . . . . . . . . . . . . . . . . . 143

4

146

4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.2 . . . . . . . . . . . 149

4.3 . . 164

4.4 Jacobi165

5 -

168

5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.2 . . . . . . . . . . . . . . . . 169

5.3 . . . . . . . 175

5.4 . . . . . . . 177

5.5 . . . 179

5.5.1 Aitken . . . . . . . . . . . . . . 179

5.5.2 Rayleigh . . . . . 180

5.5.3

Rayleigh . . . . . . . . . . . . . . . . . . . . . 182

4

5.5.4 (Shift of O-

rigin) . . . . . . . . . . . . . . . . . . . . . . . 183

5.6 . . . . . . . . . 184

6 187

6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.2 . . . . . . . . . . . . . . . . 187

6.3 . . . 191

6.3.1 -

Newton . . . . . . . . . . . . . . . . 191

6.3.2 -

Newton . . . . . . . . . . . . . . . . 193

6.4 . 195

6.4.1 Lagrange . . . . . . . . 195

6.4.2 . . . 197

6.4.3 . . . . . . . . . . . . . . 200

6.4.4 Newton -

. . . . . . . . . . . . . . . . . 202

7 207

7.1 . . . . . . . . . 207

7.2 . . . . . . . . . . . . . . . . . 210

8 218

8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8.2 -

. . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8.3 -

. . . . . . . . . . . . . . . . . . 222

8.4 . . . . . . 225

8.5

. . . . . . . . . . . . . . 228

8.6 . . . . . . . . . 229

9 231

9.1 . . . . . . . 231

9.2 . . . . . . . . 237

9.3 . . . . . . 239

9.4 . . . . . . 241

9.5 Romberg . . . . . . . . . . . . . 244

5

9.6 Gauss . . . . . . . . . . . . . . . 247

10 249

10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 249

10.2 Euler . . . . . . . . . . . . . . . . . . 250

10.3 Taylor . . . . . . . . . . . . . . 251

10.4

Euler Taylor . . . . . . . . . . . . . 260

10.5 -

. . . . . . . . . . . . . . . . . . . . . . . . . 263

10.6 Runge-Kutta 2 . . . . . . . . 264

10.7 Runge-Kutta 4 . . . . . . . . 266

10.8 . . . . . . . . . . 269

10.9 -

. . . . . 272

10.10

. . . . . . . . . . . . . . . . . . . . . 274

10.11

. . . . . . . . . . . . . . . . 276

6

1

1.1

-

.

1. -

.

,

.

2. .

.

3. . ,

, -

. -

.

4. .

.

,

.

7

1. x x,

|x x|

|x x||x| , x 6= 0.

1.2

-

. 432.52

4 102 + 3 101 + 2 100 + 5 101 + 2 102

101.11

1 22 + 0 21 + 1 20 + 1 21 + 1 22 = 5.75

(101.11)2 = (5.75)10

() - .

.

-

.

.

-

(bit) . -

n bits

8

,

:

n (111 . . . 1)2 = 1 2n1 + 1 2n2 + + 1 20 = 2n 1.

n = 15 215 1 = 32.767. , [(2n 1), 2n 1] .

.

-

-

.

-

. ,

15.546 = 0.15564 102.

, x ( 6= 0)

x = x 10e (1.1) e , (exponent)

x (mantissa). 0.1 x < 1. ,

x = x e, (1.2)

x = (0.a1a2 . . . an). (1.3)

ai, i = 1, 2, . . . n

0 ai 1, a1 6= 0. (1.4)

:

m e M (1.5)

9

m . m = M m = M 1. (1.3) n ,

n .

= 2. 1.1 64 32

bits .

{{{ {x x ee bits2-24 bits25-32bits2-12 bits13-64

64 bitCDC6000

1.1: . -

(bit) 0 1

.

(1.2) (floating point).

[s, L]

. = 2, x - n

|e| M. (1.6) x :

n (0.10 . . . 0)2 x

n (0.11 . . . 1)2

10

1

2 x 1 2n (1.7)

s =1

2 2e x 2e = |x| (1 2n) 2e = L < 2e

2e1 |x| < 2e. (1.8) x

x = (0.1a2a3 . . . an)2

e = M,M+1, . . . ,M1,M , 2n1 x. x 2n1 x [2e1, 2e) (2e,2e1] (. 1.2)

- L

( )

}

[]

}

)[ [ ]. . .[ ]. . . ( ]}} e = - M e = - M e = - M+1e = - M+1

0 s 2s 4s L- s- 2s- 4s

1.2: .

1

[2e1, 2e) (2e,2e1]. -

1.2 -

.

(1.6) (1.7), -

.

[L,s] [s, L]. x |x| > L

(overflow). |x| < s (underflow). -

x

11

,

e .

= 2, n = 3, m = 1, M = 2, .

(1.2) - (1.3)

x = x 2e, x = (0.1a2a3)2 0 ai 1, i = 2, 3. x (0.100)2, -

(0.001)2. e :0.100, 0.101, 0.110 0.111 e = 1, 0, 1, 2 :

(0.100)2 21 (0.101)2 21 (0.110)2 21 (0.111)2 21(14) ( 5

16) ( 6

16) ( 7

16)

(0.100)2 20 (0.101)2 20 (0.110)2 20 (0.111)2 20(12) (5

8) (6

8) (7

8)

(0.100)2 21 (0.101)2 21 (0.110)2 21 (0.111)2 21(1) (5

4) (6

4) (7

4)

(0.100)2 22 (0.101)2 22 (0.110)2 22 (0.111)2 22(2) (5

2) (6

2) (7

2)

,

. :

04

1

2

11

4

5

4

6

4

7

2

52

2

73

4

1-

2

1-

4

5-

4

6-

4

7-

2

5-

2

7- -1-2-3

(14, 0)

(0, 14). , -

.

.

12

1.3 -

x = (0.a1a2 . . . anan+1 . . .) e, a1 6= 0 (1.9)

(=2, 8, 10, 16). n, .

: x; -

x (- 1.3). 1.3 (),

x x an+1 . . . (. (1.9)),

x = (0.a1a2 . . . an) e. (1.10)

x

xx

x

xx

(a) (b)

1.3: x (x, x ).

1.3(b), x

x x (0.00 . . . 01) =

n,

x =((0.a1a2 . . . an) +

n) e. (1.11)

(rounding up). -

.

()

.

13

() 3.1, -

|x x| 12|x x| =

(1

2n

) e (1.12)

|x x||x|

12n ex e =

12n

x

12n

1=

1

2n+1. (1.13)

(1.12) (1.13)

(b) 1.3. -

x fl (x). (1.12) (1.13)

|x fl (x)| 12n e (1.14)

|x fl (x)|

|x| 1

2n+1, (1.15)

. (1.14) (1.15)

.

=fl (x) x

x,

fl (x) = (1 + )x (1.16)

(1.15)

|| 12n+1. (1.17)

(machineunit), , (1.16) fl (x) x.

(chopping). ,

x , x (. .

14

1.3). -

, x x,

|x fl (x)| en (1.18) |x fl (x)|

|x| n+1. (1.19)

(1.14), (1.15) (1.18) (1.19)

:

1.

.

2.

-

, .

Wil-

kinson.

. -

.

1.4 -

.

S =

ni=1

xi (1.20)

xi - . ,

15

S2 = fl (x1 + x2)

3 ...,

S3 = fl (x3 + S2)S4 = fl (x4 + S3)...

Sn = fl (xn + Sn1)

(1.21)

Sn S. (1.16),

S2 = (x1 + x2) (1 + 2)S3 = (x3 + S2) (1 + 3)...

Sn = (xn + Sn1) (1 + n)

(1.22)

|i| 12n+1, i = 2, 3, , n.

(1.22)

S2 = (x1 + x2) + (x1 + x2) 2S3 = [(x1 + x2 + x3) + (x1 + x2) 2] (1 + 3)

= (x1 + x2 + x3) + (x1 + x2) 2 + (x1 + x2 + x3) 3 + (x1 + x2) 23

, 23 2, 3, -

S3 (x1 + x2 + x3) + (x1 + x2) 2 + (x1 + x2 + x3) 3.

Sn ni=1

xi+(x1 + x2) 2+(x1 + x2 + x3) 3+. . .+(x1 + x2 + . . . + xn) n

(1.20)

Sn S x1 (2 + 3 + . . .+ n) + x2 (2 + 3 + . . .+ n)+x3 (3 + 4 + . . .+ n) + . . .+ xnn

16

|Sn S| . |x1| (|2|+ . . .+ |n|) + |x2| (|2|+ . . .+ |n|)+ |x3| (|3|+ . . .+ |n|) + . . .+ xn |n| . (1.23)

(1.23) -

|S Sn| - : ,

,

|x1| |x2| |x3| . . . |xn| .

, (1.23)

i xi.

1.5

x x (1.14) (1.15), .

|x| = |x x| 1210d

(1.14), x x d . x d ( x). x 6= 0,

.

,

().

|x| =xx

5 10s, (1.15), x x s . x s .

17

. -

.

() x = 28.254, x = 28.271, () x = 0.028254, x = 0.028271() x = e, x = 19/7 () x =

2, x = 1.414 () x = log2, x = 0.7.

()

|x| = |x x| = |28.254 28.271| = | 0.017|= 0.17 101 < 0.5 101.

, x .

|x| = |xx| = 0.17 10

1

28.254=

0.17 1010.28254 102 0.602 10

3 < 5 103

x .()

|x| = |x x| = |0.028254 0.028271| = 0.000017 0.17 104 < 0.5 104

|x| = |xx| = 0.17 10

4

0.28254 101 0.602 103 < 5 103

x .

10

,

[. () ()].

( ;)

() ,

|x| = |x x| = |e 19/7| = |2.718281 2.714286| 0.003995= 0.3995 102 < 0.5 102

18

|x| = |xx| = 0.3995 10

2

0.2714286 101 1.4718 103 < 5 103

x , .

()

|x| = |x x| = |2 1.414| = |1.414214 1.414| = 0.000214

= 0.214 103 < 0.5 103

|x| = |xx| = 0.214 10

3

0.1414214 101 1.51 104 < 5 104.

(+,,, /) , - . x y

x = x+ x y = y + y (1.24)

x y . xy

xy = xy xy= (xy xy) + (xy xy) . (1.25)

(1.25) -

xy.

fl (xy) = xy (1.26)

xy , (1.15) (1.26)

|xy xy| 12n+1 |xy| (1.27)

19

.

. = , (1.25)

xy = (x y) (x y)= (x x) (y y)

xy = x y

|xy| |x|+ |y| . (1.28) .

1.5.1. -

.

, x y - (, |x| , |x| < 12 104), xy xy 104. x y . , x y - , xy

|x| |y|. = ,

xy = xy xy = xy (x x)(y y)

xy = yx + xy + xy. (1.29)

= / y, y 6= 0,

x/y =x

y x

y=

x

y x x

y y =yx xyy2 + yy

. (1.30)

(1.29) (1.30)

xy xy + yx (1.31)

20

x/y yx xyy2

(1.32)

. (1.31) x y xy. , (1.32) x/y x / y.

(. /

-

/ ) -

.

1.5.2.

.

. (1.31) xy

xy =xyxy

=xx

+yy

+xx yy

(1.33)

xy = x + y + xy. (1.34)

|x|, |y|

x/y x y

|x/y| . |x|+ |y|. (1.37)

x y s , xy x/y s (;). , -

xy =xyx y =

xx y

yx y =

(x

x y)x

(y

x y)y.

(1.38)

(1.38) x y x y, x/(x y) y/(x y) xy.

.

x1 = 4.54, x2 = 3.00 x3 = 15.0 - . ,

() x1 x2 + x3() x1x2/x3() -

() ();

()

x1, x2 x3 , ,

|x1| 1

2102, |x2|

1

2103 |x3|

1

2101.

22

1.5.1

|x1x2+x3| |x1|+ |x2|+ |x3| 1

2102 +

1

2103 +

1

2101

=1

2(101 + 102 + 1)101 = 0.555 101 > 0.5 101.

, -

.

() -

|x1 |, |x2|, |x3| 5 103. 1.5.2

| x1x2/x3 | . |x1|+ |x2|+ |x3 | = 3 5 103 = 5 102.

.

23

2

2.1

-

f (x) = 0 (2.1)

f (x) - x. f (x) x. - f (x) .

( -

4). (2.1). -

(2.1) f () = 0. - [a, b] . , - .

,

.

24

2.1.1. f (x) C [a, b] 1 f (a) f (b) < 0, (a, b)

f () = 0.

(a, b) (2.1), -. , ,

() -

.

x0 (a, b) ()

x1, x2, . . . - {xn} , n = 0, 1, 2, . . . . - ,

.

-

{xn} , n = 0, 1, 2, . . ..

2.2 -

Bolzano

[a, b] .

, f (x) C [a, b] f (a) f (b) < 0, - (a, b) (2.1). - c0 = (a0 + b0) /2, a0 = a b0 = b. f (c0) = 0, = c0 f (a0) f (c0) < 0. , (a0, c0) -, a1 = a0 b1 = c0, [a1, b1] . f (a0) f (c0) > 0, f (c0) f (b0) < 0 (c0, b0), [a1, b1], a1 = c0

1 Cn [a, b] - [a, b] n-. C0 [a, b] C [a, b] [a, b].

25

b1 = b0. (a1, b1) , f (x) (. 2.1).

x

y

y = f(x)

(a, f(a))

a = a0 c0 c2 c1

(b, f(b))

b = b0

b0

b1

b2

a0

a1

a2

c0

c1

c2

2.1:

, -

[a0, b0], [a1, b1] , . . . , [an, bn]

a0 a1 a2 . . . b0b0 b1 b2 . . . a0 (2.2)

f (an) f (bn) 0, n = 0, 1, 2, . . . (2.3)

bn an = 12(bn1 an1) . (2.4)

{an} ,. , {bn} - . (2.4)

bn an = b0 a02n

. (2.5)

26

limn

bn limn

an = limn

2n (b0 a0) = 0.

= limn

an = limn

bn

(2.3)

f (x), [f ()]2 0, f (x).

[an, bn], [an, bn]

cn =an + bn

2. (2.6)

| cn| = an + bn2

bn an2 (2.5)

| cn| b0 a02n+1

. (2.7)

> 0,

| cn|

, (2.7), n

b0 a02n+1

n log (b0 a0) log 2

log 2

. (2.8)

- n -

. n [an, bn] ( (2.5)),

27

f (x) = 0. - [an, bn] , .

an bn, - (an, bn) f (x). (b0 a0) /2n+1 , .

-

.

f (x) ( ). 2.2 -

[a, b].

y

x

(a, f(a))

(b, f(b))

b

a

c

2.2:

[a, b].

f (x) C [a0, b0] f (a0) f (b0) < 0. n = 0, 1, 2, . . . :

28

1. cn =an+bn

2

2. f (cn) = 0 = cn

f (an) f (cn) < 0

an+1 = an, bn+1 = cn

an+1 = cn, bn+1 = bn.

-

|f (c)| < , 12 10k.

c f (x) (. 2). -

c < f (c) < , f (c) y = + y = (. 2.3).

y = +

y =

c

f(c)

x

y

2.3: |f(c)| < .

, |c | < , , c - < c < + , c

29

-

x = x = + (. 2.4). , -

c f (c) 2.3 2.4

, c f(c) - . ,

|c | < :

|cn cn1| < (2.9)

|cn cn1||cn| < , cn 6= 0. (2.10)

c

+

y = f(x)

x

y

2.4: |c | < .

. , k , 12 10k+2. , {cn}

30

|cn cn1| . (2.10)

( cn 6= 0).

2.3 (Re-

gula Falsi)

-

c . .

.

f (x) C [a, b] f (a) f (b) < 0, [a, b] f (x) = 0. , -, x0 (a, f (a)) (b, f (b)) x (. 2.5).

x1b1 = x0

y = f(x)

b = b0 = b1

(a, f(a))

(b, f(b))

x

y

2.5: -

.

a0 = a b0 = b (a0, f (a0)) (b0, f (b0))

31

y f (b0)x b0 =

f (a0) f (b0)a0 b0 . (2.11)

(2.11) y = 0 x = x0

x0 = b0 f (b0) b0 a0f (b0) f (a0) . (2.12)

f (a0) f (b0) < 0, (2.12) x0 . - . f (x0) = 0 = x0. f (a0) f (x0) < 0 (a0, x0) [a1, b1] a1 = a0 b1 = x0. - [a1, b1] x1 (2.12) a0, b0 - a1 b1, . , {xn} , n = 0, 1, 2, . . .

xn = bn f (bn) bn anf (bn) f (an) , n = 0, 1, 2, . . . (2.13)

f (an) f (bn) < 0 f (an) 6= f (bn) xn an < xn < bn bn < xn < an.

-

(2.13) , .

f (x) [a0, b0], f (a0) f (b0) < 0. n = 0, 1, 2, . . . :

1. xn (2.13)

2. f (xn) = 0 = xn

f (an) f (xn) < 0 an+1 = an, bn+1 = xn

an+1 = xn, bn+1 = bn.

32

2.4

f (x), x [a, b], f () = 0. g (x) - = g (). g (x) . , f (x) = x3 13x + 18, - g (x) : (a) g (x) = (x3 + 18) /13, (b) g (x) =

(13x 18)1/3 , (c) g (x) = (13x 18) /x2. , = g (), f () = 0.

= g () (fixed-point

problem) g (x).

g (x), x [a, b], f (x). g (x) I = [a, b] g (x) y = x (. 2.6).

a s1 s2 s3 b

a

g(a)

g(b)

b y = x

y = g(x)

2.6: s1, s2 s3 g(x).

I = [a, b], g (x) .

33

g (x) I, g (x). , x I, g (x) I. I, g () g (x) I.

2.4.1. x I g (x) I g (x) , g (x) I. - g (x) (a, b) L < 1

|g (x)| L < 1 x (a, b) , (2.14) [a, b] g (x).

. F (x) = g(x) x, F (x) . F (a) = g(a)a > 0 F (b) = g(b) b < 0. F (x) (a, b). F () = 0 g() = g(x). g(a) = a g(b) = b, .

-

I 1 I 2 I 1 6= 2.

g (1) g (2) = g () (1 2)

1 < < 2. , (2.14),

|1 2| = |g (1) g (2)| = |g() (1 2)| L |1 2| < |1 2|

. 1 = 2 [a, b] .

x I, g(x) I, .

2.4.2. ( ) f(x) C[a, b], - c1, c2 [a, b] f(c1) f(x) f(c2) x [a, b]. , f(x) (a, b), - c1 c2 [a, b] f (x).

34

2.4.1

g(x) [a, b]. - .

x0 [a, b] {xn}n=0

xn+1 = g(xn), n = 0, 1, 2, . . . (2.15)

{xn}n=0 g(x) ,

= limn

xn+1 = limn

g(xn) = g(limn

xn

)= g() (2.16)

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

method). f(x) = 0 g(x) - g(x) f(x). , (2.15),

.

-

(.

2.7). xn (2.15), xn+1 = g(xn) y- (xn, g(xn)) y = g(x). (xn+1, xn+1) = (xn+1, g(xn)) y = x. xn+1 - Ox. 2.7 . -

.

g(x) x0. n = 0, 1, 2, . . . :

xn+1 = g(xn)

2.4.1

.

35

y = x

x0x2x4x3x10

y

x

y = g(x)

2.7: -

.

2.4.3. g(x) C[a, b] g(x) [a, b] x [a, b]. g(x) (a, b) L < 1

|g(x)| L < 1 x (a, b). (2.17) x0 [a, b],

xn = g(xn1), n = 1, 2, . . . (2.18)

[a, b] g(x).. 2.4.1 -

[a, b]. g [a, b] , {xn}n=0 n 0 xn [a, b]. (2.17)

|xn | = |g(xn1) g()| = |g(n)| |xn1 | L |xn1 | n min(xn1, ) < n < max(xn1, ). -

|xn | L |xn1 | L2 |xn2 | . . . Ln |x0 | .(2.19)

36

0 L < 1, limnLn = 0,

limn

|xn | limn

Ln |x0 | = 0

{xn}n=0, (2.18), .

x3 2x2 1 = 0 [2, 3]. xn+1 = 2 + 1/x

2n

x0 [2, 3] .

g(x) = 2 + 1/x2 I = [2, 3]. g(x) I. g(x) = 2/x3, g(x) x [2, 3] m = g(3) = 2+1/9 M = g(2) = 2 + 1/4. , g(x) [2+1/9, 2+1/4] [2, 3] x [2, 3].

|g(x)| = 2x3

14 x [2, 3]. g(x) 2.4.3 xn+1 = 2+1/x

2n, n =

0, 1, 2, . . . x0 [2, 3] [2, 3].

f(x) = x2 x 2 2 -1. = 2 .

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

(a) g(x) = x2 2 (b) g(x) = 2 + x

(c) g(x) = 1 +2

x(d) g(x) = x x

2 x 2m

,m 6= 0.

37

(), g(x) > 1 x > 1/2 - (2.17)

= 2. (b),

g(x) =1

22 + x

.

x 0 g(x) 0 0 g(x) 1/8 < 1 (2.17). x k,

2 + x 2 + k 2 + k k k 2, g(x) = 2 + x k k 2. x [0, k], k 2 g(x) C[0, k]

2.4.3. x0 [0, k], k 2, = 2. , x0 = 0,

xn+1 =2 + xn, n = 0, 1, 2, . . .

x1 =2 = 1.41421

x2 =2 + x1 =

3.41421 1.84776

x3 =2 + x2 =

3.84776 1.96157

x4 =2 + x3 =

3.96157 1.99037

x5 =2 + x4 =

3.99037 1.99759

= 2. (c) (d) -

.

2.4.1.

2.4.3, n- n = xn ,

|n| Lnmax{x0 a, b x0}

|n| Ln

1 L |x0 x1| . (2.20)

. (2.19).

, (2.19)

|n| Ln |x0 | Lnmax{x0 a, b x0}

38

[a, b]. ,

|x0 | |x0 x1|+ |x1 | |x0 x1|+ L |x0 |

|x0 | 11 L |x0 x1| .

(2.19)

|n| Ln |x0 |

|x0 | - (2.20).

2.4.1

{xn}n=0 Ln/(1 L) L = maxx[a,b] |g(x)|. L, . L . (2.20)

-

.

L = maxaxb

|g(x)| < 1.

|g()| > 1. - xn+1 = g (xn) = g(),

| xn+1| = |g() g(xn)| = |g(n)| | xn| .

xn , |g(n)| > 1, | xn+1| | xn|. |g(n)| > 1. , ,

2.8.

, -

2.4.1.

-

.

39

y = x

y = g(x)

x0 x1 x2 x3

y

x

0 < g() < 1

y = x

y = g(x)

x3 x2 x1 x0

y

x

g() > 1

y = x

y = g(x)

x0 x2 x3 x1

y

x

1 < g() < 0

y = x

y = g(x)

x1 x0 x2

y

x

g() < 1

2.8:

2.4.4. g(x) C1[a, b]2 g(x) [a, b]. |g()| < 1, > 0, x0 (a, b) |x0 | < .

. g(x) (a, b) |g()| < 1, - K |g()| K < 1, > 0 x [, +] I, |g(x)| < K. K (2.17) 2.4.3 I. x I,

|g(x) | = |g(x) g()| = |g()| |x | K < 2 [a, b]

40

g(x) I x I. 2.4.3 I.

-

, x0, - .

> 0 g(x) = 12(x +

x) x > 0,

x0

.

g(x) = 12(1

x2), g(x) g(x)

x > 0. ,

g(x), = g(

). , g(

) = 0 < 1,

2.4.4

x0 .

2.4.1 -

-

. -

.

.

.

. {xn}n=0 , . c p

limn

|xn+1 ||xn |p = c, (2.21)

{xn}n=0 p, c ( p = 1 c < 1).

41

-

n |n+1| |n|p ( : ). p = 1 -, p = 2 , p = 3, ...

-

(2.21).

2.4.5. 2.4.4

1) g(x) p- I(g(x) Cp(I), 2)g(k)() = 0 k = 1, 2, . . . , p 1 gp() 6= 0 p 1, - I x I p .

. (1) g(x) Taylor x = ,

g(xn) = g() +(xn)

1!g() + (xn)

2

2!g() + . . .

+ (xn)p1

(p1)! g(p1)() + (xn)

p

p!g(p)(n)

n xn . 2)

n+1 = xn+1 = g(xn) g() = (xn )p

p!g(p)(n)

limn xn =

limn

|n+1||n|p =

1

p!

g(p)() . (2.22) (2.22)

p.

2)

g(x) x = .

p = 1, |g()| < 1 2.4.4

. p > 1, |g()| = 0 < 1, .

42

2.5 Newton-Raphson

Newton-Raphson

f(x) = 0. f(x) - [a, b], f(x) C2[a, b]. xn [a, b] f (xn) 6= 0 |xn | . Taylor

f(x) xn

f(x) = f(xn) + (x xn)f (xn) + (x xn)2

2!f ((x)) (2.23)

(x) x xn. x = (2.23)

f() = 0 = f(xn) + ( xn)f (xn) + ( xn)2

2!f (()) . (2.24)

(2.24),

|xn | |xn |2 ,

0 f(xn) + ( xn)f (xn)

xn f(xn)f (xn)

.

-

{xn}n=0

xn+1 = xn f(xn)f (xn)

, n = 0, 1, 2, . . . (2.25)

Newton-Raphson.

f (xn) 6= 0.

Newton-Raphson

f(x) C2[a, b] - x0 [a, b].

n = 0, 1, 2, . . . - :

xn+1 (2.25)

43

2.5.1 Newton-Raphson

x0 Newton-Raphson.

2.5.1. f(x) C2[ , + ], > 0. f() = 0 f () 6= 0, 0 Newton-Raphson x0 I0 =[0, + 0]. .

. 2.4.4 (-

)

g(x) = x f(x)f (x)

.

I0 = [ 0, + 0] :

1. g(x) C1 (I0),

2. g() =

3. |g()| L < 1 L L (0, 1).

f () 6= 0 f (x) I = [ , + ], 1 f (x) 6= 0 x [1, +1]., g(x) [ 1, + 1].,

g(x) =f(x)f (x)

[f (x)]2

f(x) C2[ , + ] g(x) C1[ 1, + 1].

g() =f()f ()

[f ()]2= 0 < 1

g() = . 2.4.4 -

g(x) = x f(x)f (x)

o I1., Newton-Raphson x0 I0 0 1. g() = 0, 2.4.5, .

44

2.5.1

Newton-Raphson x0 . , (2.22) p = 2

limn

|n+1||n|2

=1

2|g()| .

g() =f ()f ()

limn

|n+1||n|2

= M (2.26)

M =1

2

f ()f () . (2.27)

(2.26)

|n+1| M2n

|Mn+1| (Mn)21 (Mn1)2

2 . . . (M0)2n

limn |n+1| = 0

|M0| < 1

| x0| < 1M

= 2

f ()f () . (2.28)

(2.28) x0 M . M , x0 Newton-Raphson .

Newton-Raphson.

Newton-Raphson

f () 6= 0 f(xn)

45

f (xn) . .

. f(x) k f(x) f(x) = (x )kh(x), x 6= ,

limx

h(x) 6= 0.

2.5.2. f(x) Ck[a, b], k

f() = f () = f () = . . . = f (k1)() = 0

f (k)() 6= 0. , f () 6= 0, -

: f(x) = 0. - , Newton-Raphson ,

.

2.5.3. f(x) C2[a, b] [a, b] f(x) = 0, k > 1, Newton-Raphson , .

. , Ne-

wton -Raphson,

g(x) =f(x)f (x)

[f (x)]2.

f () = 0 - x = . . k > 1, f(x)

f(x) = (x )kh(x), h() 6= 0. (2.29)

f (x) = (x )k1 [kh(x) + (x )h(x)] (2.30)

46

f (x) = (x)k2 [k(k 1)h(x) + 2k(x )h(x) + (x )2h(x)] .(2.31)

g() = limx g(x) = limxf(x)f (x)

[f (x)]2

= limx(x)kh(x)(x)k2[k(k1)h(x)+2k(x)h(x)+(x)2h(x)]

(x)2k2[kh(x)+(x)h(x)]2

= k(k1)[h()]2

k2[h()]2= k1

k< 1.

, .

k = 2

g(x) = x 2 f(x)f (x)

g(x) =2f(x)f (x) [f (x)]2

[f (x)]2.

L Hospital

g() = 0.

,

k,

xn+1 = xn k f(xn)f (xn)

, n = 0, 1, 2, . . . (2.32)

x0. ,

. -

(x)

(x) =f(x)

f (x). (2.33)

47

k 1,

f(x) = (x )kh(x) h() 6= 0

(x) =(x )kh(x)

k(x )k1h(x) + (x )kh(x) =(x )h(x)

kh(x) + (x )h(x) , 1. Newton-Raphson (x),

g(x) = x (x)(x)

= xf(x)f (x)

[f (x)]2[f(x)][f (x)][f (x)]2

g(x) = x f(x)f(x)

[f (x)]2 f(x)f (x) . (2.34)

g(x) ,

xn+1 = g(xn), n = 0, 1, 2, . . . (2.35)

. -

f (x) .

Newton-Raphson x0 [a, b]. .

2.5.4. ( ).

1. f(x) C2[a, b]2. f(a)f(b) < 0

3. f (x) 6= 0, x [a, b]4. f (x) [a, b]

5. | f(c)f (c)

| b a, c [a, b] |f (x)|

48

x0 [a, b] Newton-Raphson - f(x) [a, b].

(2)

f(x) [a, b]. (3) f(x) - (f > 0) (f < 0) [a, b]. [a, b] f(x). (4) f(x) (f (x) 0) (f (x) 0). (5), -

.

y

x{f( )

x2 x1 x0

M0

P0

y=f(x)

f(b)

b

}

2.9: Newton-Rapshon

2.9

tan = f (x0) =M0P0x0 x1 =

f(x0)

x0 x1

x0 x1 = f(x0)f (x0)

x1 = x0 f(x0)f (x0)

x0 Ox Newton-Raphson.

(5) : -

x0 [a, b], x1 [a, b].

49

x0 a b, x1 [a, b].

f(x) = x3 2x 1, x [1, 2].

Newton-Raphson -

.

2.5.4 -

f(x) C2[1, 2] f(1)f(2) < 0. f (x) = 3x22 6= 0 f (x) = 6x > 0 x [1, 2]. , 5) - |f(1)/f (1)| = 2 > 1, Newton-Raphson [1, 2]. [3

2, 2],

f(32)f(2) < 0

f(32)/f (3

2) = 5

30< 1. Newton-

Raphson x0 [32 , 2].

1. f(x) = x2 c, x > 0, c > 0 . NR

c x0 > 0.

[a, b] 0 < a 0. NR

xn+1 = xn x2n c2xn

50

xn+1 =1

2(xn +

c

xn), n = 0, 1, 2, ...

.

2. 1/c . c > 0, NR 1/c.

1/c

f(x) =1

x c = 0

NR

xn+1 = xn 1xn c

1x2n

xn+1 = xn(2 cxn), n = 0, 1, 2, ... (2.36) (2.36)

.

f (x) = 1x2

< 0, f (x) =2

x3> 0

x > 0 4.8 [a, b] 0 < a < c1 < b

f(b)

f (b)= b(bc 1) b a

b1 b b2,

b1,2 =11 ac

c

x > 0 0 < b < 2c1. NR x0

0 < x0 < 2c1 (2.37)

51

3. NR - e1. c = e (2.37) 0 < x0 < 2e1 0.735776 x0 = 0.3 (2.36) :

x0 = 0.3 2 ex0 = 1.1845157x1 = 0.355355 2 ex1 = 1.0340461x2 = 0.36745345 2 ex2 = 1.00111583x3 = 0.36787907 2 ex3 = 1.0000014x4 = 0.36787958 2 ex4 = 1.0000000

, .

4. f(x) C1[a, b], f (x) 6= 0 x [a, b] f(x) = 0 [a, b]. g(x) = x + h(x)f(x), h(x) .

g(x) = g(). -

g() = 0.

g(x) = 1 + h(x)f(x) + h(x)f (x)

x =

g() = 0 = 1 + h()f ()

f() = 0.

h() = 1f ()

h(x) ( )

h(x) = 1f (x)

.

xn+1 = g(xn)

xn+1 = xn f(xn)f (xn)

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

52

Newton-Raphson.

5. f(x) = xk c, x > 0, c > 0 k , NR k

c

x0 > 0.

2.5.4

[a, b] 0 < a < kc b ,

b > 1k[(1 k)a+ c/ak1].

NR

xn+1 = xn xkn ckxk1n

xn+1 = (1 1k)xn +

1

kcx1kn , n = 0, 1, 2, ...

k = 2

xn+1 =1

2(xn + c/xn).

17 x0 = 4

:

x1 = 4, 12x2 = 4, 123 106x3 = 4, 123 1056 2561 77x4 = 4.123 1056 2561 7660 5498 2140 9856

x4 28 . -

.

-

. -

(100 ..-100 ..)

53

2.5.2 Newton 2

Bailey

, Newton 2 -

( )

. -

( 2 ).

f(x) = 0.

f(x) Taylor x = xn

f (xn+1) = f (xn) + hf (xn) +

h2

2!f (xn) +

h3

3!f (xn) + . . .

xn+1 = xn + h.

f (xn) = 0 ,

f (xn) + h

[f (xn) +

hf (xn)2

]= 0.

h , h, h = f (xn) /f (xn)(. Newton-1 ) ,

f (xn) + h

[f (xn) f (xn) f

(xn)2f (xn)

]= 0

h

h = f (xn)f (xn)

(f (xn)f(xn)

2f (xn)

) .

xn+1 = xn + h

xn+1 = xn f (xn)f (xn) f (xn)f(xn)2f (xn)

, n = 0, 1, 2, . . . (2.38)

Newton 2 .

54

2.6

Newton -

,

x0 - f (xn). f(x) ,

. -

Newton-Raphson.

.

-

.

Newton-Raphson f (xn)

f (xn) f (xn1)xn xn1

.

y

x

x0=bx2x1 x3x-1 =

2.10: .

55

f(x) C[a, b] x1, x0 [a, b]. n = 0, 1, 2, . . . xn+1

xn+1 = xn f(xn)(xn xn1)f(xn) f(xn1) =

f(xn)xn1 f(xn1)f(xn)f(xn) f(xn1)

(2.39)

f(xn) 6= f(xn1). ,

f(a)f(b) > 0, .

f (x) , 6= xn1 6= xn n = x xn. (2.39) :

n+1 = xn+1 = f(xn)n1 f(xn1)nf(xn) f(xn1) =

=f(xn)n1 f(xn1)n

xn1 xn xn1 xn

f(xn) f(xn1) .(2.40)

f(xn)n1 f(xn1)nxn1 xn = nn1

f(xn)n

f(xn1)n1

xn1 xn =

= nn1

f(xn)f()xn

f(xn1)f()xn1

xn xn1= nn1f [xn, , xn1] (2.41)

f [xn, , xn1] =f(xn)f()

xn f(xn1)f()

xn1xn xn1 . (2.42)

G(x) = f(x)f()x , -

f [xn, , xn1] =G(xn)G(xn1)

xn xn1 = G(n). (2.43)

56

G(x) =f (x)(x ) + f() f(x)

(x )2 (2.44)

Taylor f (x)

f() = f(x) + f (x)( x) + f()2

( x)2. (2.45)

(2.44) (2.45)

G(x) =f ()2

(2.43)

f [xn, , xn1] = G(n) =

f ()2

. (2.46)

, (2.44), (2.42), (2.43), (2.46)

n+1 = nn1f [xn, , xn1](xn1 xn)

f(xn) f(xn1)= nn1

(f (n)

2

)( 1f (n)

), (2.47)

n n xn1, xn . (2.47) -

.

2.6.1. f() = 0, f () 6= 0 f (x) , x1, x0 I = [ , + ], > 0.

. Ma f (x)f (x)

x x [ a, + a]. f (x) f (x) f () 6= 0, > 0,

f (x)f (x) M x x [ , + ]. > 0

57

M = K < 1, < M M. |x1 | |x0 | , |1| |10|M 2M = K < .

, |2| |01|M = |1|(|0|M) < (K)(M) < K2. i < K

i i1 < Ki1, - |i+1| < Ki+1 < , xi+1 I i. limi i = 0 I.

-

m > 0 n+1 = Kmn .

n = Kmn1 n1 = K

1m

1mn . ,

(2.47)

n+1 = Mnn1 = Mn(K1m

1mn ) Kmn (2.48)

1 +1

m= m (2.49)

m =15

2(2.50)

n+1 = Kmn m = 1.618.

2.7 - -

Aitken

Aitken.

{xn}, n = 0, 1, 2, . . . :

n+1n

= k, n = 0, 1, 2, . . . (2.51)

n = xn , |k| < 1. - xn, xn+1, xn+2 ., (2.51) :

xn+1 xn =

xn+2 xn+1

58

=xnxn+2 x2n+1

xn+2 2xn+1 + xn = xn (xn+1 xn)2

xn+2 2xn+1 + xn (2.52)

= xn (xn)2

2xn(2.53)

xn = xn+1xn 2xn = xn+1xn = xn+22xn+1+xn. {xn} n = 0, 1, 2, . . . ,

{xn} n = 0, 1, 2, . . . (2.53)

xn = xn (xn)

2

2xn, n = 0, 1, 2, . . . (2.54)

Aitken. -

limn

xn xn = 0

limn

xn+1 xn = k < 1

.

,

Aitken, .

2.8

p(x) = anxn + an1xn1 + a1x+ a0 = 0

p(x) n (an 6= 0) , -

.

59

.

, -

.

.

2.8.1. p(x) n 1, p(x) = 0 .

.

2.8.1. p(x) n 1, 1, 2, . . . , k m1, m2, . . . , mk

ki=0mi p(x) =

an(x 1)m1(x 2)m2 (x k)mk .

i 1 (m1 =m2 = mk = 1), n n.

2.8.2. p(x) q(x) n. 1, 2, . . . , k, k > n p(i) = q(i) i = 1, 2, . . . k, p(x) = q(x) x.

2.8.3. p(x) n, q(x) p(x) (x )q(x) + p(). n 1, q(x) n 1, q(x) 0. 2.8.2. p(x) n 1 , n 1 p(x) = (x )q(x). 2.8.4. p(x) n 1. p(x) m p() = p() = = pm1() = 0 p(m)() 6= 0.

-

p(x), q(x) 2.8.2.

2.8.5. p(x) n 1 p(x). q(x) n1 p(x) = (x )q(x) (. 2.8.2 ), :

60

) m q(x) m 1. ( m = 1, q(x)).

) 6= , p(x) q(x).

2.8.1 Horner

p(x) = anxn+an1xn1+ a1x+a0.

:

) xk, k = 2(1)n, n 1 -,

) akxk, k = 1(1)n, n ,

) n

k=0 akxk, n .

2n 1 n . p(x) = anx

n + an1xn1 + a1x+ a0, an 6= 0 (2.55) p(x0) x0,

p(x) (x x0)q(x) + r. (2.56)

p(x0) = r. (2.57)

q(x) = nxn1 + n1xn2 + + 2x+ 1, (2.58)

p(x) = anxn + an1xn1 + a1x+ a0

(x x0)(nxn1 + n1xn2 + + 2x+ 1) + r= nx

n + n1xn1 + + 2x2 + 1x(nx0xn1 + n1x0xn2 + + 2x0x+ 1x0) + r

= nxn + (n1 nx0)xn1 + + (1 2x0)x+ r 1x0

(2.59)

61

an = n

an1 = n1 nx0... (2.60)

a1 = 1 2x0a0 = r 1x0

n = an

n1 = an1 + nx0... (2.61)

1 = a1 + 2x0

r = a0 + 1x0

i = ai + i+1x0, i = n(1)0 (2.62)

n+1 = 0 0 = r = p(x0), n n p(x0). i :

an an1 an2 a1 a0x0 bnx0 bn1x0 b2x0 b1x0

bn bn1 bn2 b1 b0 = p(x0)

Horner p(x)

p(x) = (. . . ((

n1 anx0 + an1)x0 + an2

n2

)x0 + + a1)x0 + a0 (2.63)

62

2.8.2

p(x)

Horner -

p(x). ,

p(x) = (x x0)q(x) + r (2.64)

p(x) = q(x) + (x x0)q(x) (2.65)

p(x0) = q(x0) (2.66)

p(x0) Horner,

q(x) q(x) x x0. p(k)(x) k + 1 Horner., Horner :

p(x) = (x x0)q1(x) + r0q1(x) = (x x0)q2(x) + r1

... (2.67)

qn1(x) = (x x0)qn(x) + rn1qn(x) = (x x0) 0 + rn.

p(x) = (x x0)q1(x) + r0= (x x0)[(x x0)q2(x) + r1] + r0 (2.68)= (x x0)2q2(x) + (x x0)r1 + r0

p(x)

p(x) = rn(xx0)n+ rn1(xx0)n1+ + r1(xx0)+ r0. (2.69)

63

p(x) Taylor x0

p(x) = p(x0)+(x x0)

1!p(x0)+

(x x0)22!

p(x0)+ +(x x0)n

n!p(n)(x0).

(2.70)

p(x)

p(k)(x0) = k!rk, k = 0(1)n (2.71)

rk = qk(x0), k = 0(1)n. (2.72)

Horner

p(x) x = x0 .. p(x)

an an1 a2 a1 a0x0 nx0 3x0 2x0 1x0

n n1 2 1 0 = r0x0 nx0 3x0 2x0

n n1 2 1 = r1x0 nx0 3x0

n n1 2 = r2 p(x0) = 2!

2

1. p(x) = 6x4 53x3 + 184x2 295x +186. Horner p(2), p(2), p(2), p(2), p(4)(2).

6 -53 184 -295 186

2 12 -82 204 -182

6 -41 102 -91 4 = r02 12 -58 88

6 -29 44 3 = r12 12 -34

6 -17 10 = r22 12

6 = r4 5 = r3

64

p(k)(x0) = k!rk k = 0(1)4 :

p(2) = 0!r0 = 1 4 = 4p(2) = 1!r1 = 1 (3) = 3p(2) = 2!r2 = 2 10 = 20p(2) = 3!r3 = 6 (5) = 30p(4)(2) = 4!r4 = 24 6 = 144

2. Horner -

p(x) = 3x2 4x + 5 p(x) = a + (x 2) + (x 2)2.

Taylor x0 = 2 :

p(x) = p(2) +(x 2)

1!p(2) +

(x 2)22!

p(2),

rk =p(k)(x0)

k!, k = 0, 1, 2,

p(x) = r0 + r1(x 2) + r2(x 2)2.

r0, r1, r2, Horner :

3 -4 5

2 6 4

3 2 9 = r02 6

3 = r2 8 = r1

p(x) = 9 + 8(x 2) + 3(x 2)2.

65

3

3.1 Gauss

-

.

-

. 20 Gramer ,

2 ,

2

! -

,

. -

Gauss.

a(1)11 x1 + a

(1)12 x2 + a

(1)13 x3 + . . .+ a

(1)1nxn = b

(1)1

a(1)21 x1 + a

(1)22 x2 + a

(1)23 x3 + . . .+ a

(1)2nxn = b

(1)2

a(1)31 x1 + a

(1)32 x2 + a

(1)33 x3 + . . .+ a

(1)3nxn = b

(1)3

a(1)n1 x1 + a

(1)n2x2 + a

(1)n3x3 + . . .+ a

(1)nnxn = b

(1)n

(3.1)

66

a(1)11 a

(1)12 a

(1)13 a(1)1n

a(1)21 a

(1)22 a

(1)23 a(1)2n

a(1)31 a

(1)32 a

(1)33 a(1)3n

......

......

a(1)n1 a

(1)n2 a

(1)n3 a(1)nn

x1x2x3...

xn

=

b(1)1

b(1)2

b(1)3...

b(1)n

(3.2)

A(1)x = b(1). (3.3)

detA(1) 6= 0 . b(1) 6= . (3.1)

, (3.1).

a(1)11 x1 + a

(1)12 x2 + a

(1)13 x3 + . . .+ a

(1)1n xn = b

(1)1

a(2)22 x2 + a

(2)23 x3 + . . .+ a

(2)2n xn = b

(2)2

a(2)32 x2 + a

(2)33 x3 + . . .+ a

(2)3n xn = b

(2)3

...a(2)n2 x2 + a

(2)n3x3 + . . .+ a

(2)nnxn = b

(2)n

(3.4)

(3.1) x1 . (3.4)

a(1)11 a

(1)12 a

(1)13 . . . a

(1)1n

a(2)22 a

(2)23 . . . a

(2)2n

0 a(2)32 a(2)33 . . . a(2)3n...

... ...a(2)n2 a

(2)n3 . . . a

(2)nn

x1x2x3...

xn

=

b(1)1

b(2)2

b(2)3...

b(2)n

A(2)x = b(2).

(.(3.4) )

67

x2.

a(1)11 x1 + a

(1)12 x2 + a

(1)13 x3 + . . .+ a

(1)1nxn = b

(1)1

a(2)22 x2 + a

(2)23 x3 + . . .+ a

(2)2nxn = b

(2)2

a(3)33 x3 + . . .+ a

(3)3nxn = b

(3)3

... ...a(3)n3 x3 + . . .+ a

(3)nnxn = b

(3)n

a(1)11 a

(1)12 a

(1)13 . . . a

(1)1n

a(2)22 a

(2)23 . . . a

(2)2n

a(3)33 . . . a

(3)3n

0... ...

a(3)n3 . . . a

(3)nn

x1x2x3...

xn

=

b(1)1

b(2)2

b(3)3...

b(3)n

A(3)x = b(3).

r 1

a(1)11 x1 + a

(1)12 x2 + . . . + a

(1)1,r1xr1 + a

(1)1,rxr + . . . + a

(1)1nxn = b

(1)1

a(2)22 x2 + . . . + a

(2)2,r1xr1 + a

(2)2,rxr + . . . + a

(2)2nxn = b

(2)2

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

a(r1)r1,r1xr1 + a

(r1)r1,rxr + . . . + a

(r1)r1,nxn = b

(r1)r1

a(r)r,rxr + . . . + a

(r)rn xn = b

(r1)r

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

a(r)n,rxr + . . . + a

(r)nnxn = b

(r)n

a(1)11 a

(1)12 . . . a

(1)1,r1 a

(1)1r . . . a

(1)1n

a(2)22 . . . a

(2)2,r1 a

(2)2r . . . a

(2)2n

0 . . . ... ... . . . ...a(r1)r1,r1 a

(r1)r1,r . . . a

(r1)r1,n

a(r)rr . . . a

(r)rn

0 ... . . . ...a(r)nr . . . a

(r)nn

x1x2...

xr1xr...

xn

=

b(1)1

b(2)2...

b(r1)r1b(r)r

...

b(r)n

68

A(r)x = b(r).

, n 1 -

a(1)11 x1 + a

(1)12 x2 + . . .+ a

(1)1,r1xr1 + a

(1)1,rxr + . . .+ a

(1)1nxn = b

(1)1

a(2)22 x2 . . .+ a

(2)2,r1xr1 + a

(2)2,rxr + . . .+ a

(2)2nxn = b

(2)2

...a(n1)n1,n1xn1 + a

(n1)n1,nxn = b

(n1)n1

a(n)nnxn = b

(n)n

(3.5)

A(n)x = b(n), (3.6)

A(n) . (3.5)

xn =b(n)n

a(n)nn

xi =b(i)i

nj=i+1 a

(i)ij xj

a(i)ii

, i = n 1(1)1. (3.7)

Gauss

{A(k)

}, k = 1(1)n, A(1) = A

{b(k)}, k = 1(1)n

A(n) U .

x1 n1 .

a(1)ij = aijb(1)i = bi

, i = 1(1)n, j = 1(1)n,

x1 i- (3.1) i = 2(1)n, mi1 ,

69

, i- ,

mi1 = a(1)i1

a(1)11

, i = 2(1)n (3.8)

a(1)11 6= 0. a(1)11 .

[a(1)i2 +mi1a

(1)12

]x2 + . . .+

[a(1)in +mi1a

(1)1n

]xn = b

(1)1 +mi1b

(1)1

a(2)i2 x2 + . . .+ a

(2)in xn = b

(2)i , i = 2(1)n

a(2)ij = a

(1)ij +mi1a

(1)1j , i = 2(1)n, j = 2(1)n (3.9)

b(2)i = b

(1)i +mi1b

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

x1 i-

a(2)i1 = a

(1)i1 +mi1a

(1)11 = 0.

, n 1

a(2)22 x2 + . . .+ a

(2)2nxn = b

(2)2

......

...

a(2)n2 x2 + . . .+ a

(2)nnxn = b

(2)n .

a(2)22 6= 0, mi2

x2 n2 ... (3.6). -

M (1)

M (1) =

1 0m21m31 In1...

mn1

(3.10)

70

Gauss -

M (1). :

M (1)A(1)x = M (1)b(1) (3.11)

A(2)x = b(2) (3.12)

A(2) = M (1)A(1) b(2) = M (1)b(1).

3.1.1. A(1)x = b(1) A(2)x = b(2).

. (3.1) ,

detA(1) 6= 0. (3.11) (3.12)

A(2) = M (1)A(1)

detA(2) =[detM (1)

] [detA(1)

]= detA(1) 6= 0

detM (1) = 1. A(2) (3.4) . (3.11)

(3.1) (3.4), (3.1)

(3.4) .

r1 , a(r)rr 6= 0,

a(r)rr xr + . . .+ a(r)rnxn = b

(r)r .

xr n r mir i- i =r + 1(1)n.

mir = a(r)ir

a(r)rr

, i = r + 1(1)n

[a(r)i,r+1 +mira

(r)r,r+1

]xr+1+. . .+

[a(r)in +mira

(r)rn

]xn =

[b(r)i +mirb

(r)r

]71

a(r+1)i,r+1xr+1 + . . .+ a

(r+1)in xn = b

(r+1)i , i = r + 1(1)n

a(r+1)ij = a

(r)ij +mira

(r)rj , i = r + 1(1)n, j = r + 1(1)n (3.13)

b(r+1)i = b

(r)i +mirb

(r)r , i = r + 1(1)n.

A(r+1)x = b(r+1). (3.14)

M (r) =

Ir1 0 00 1

mr+1,r 1 0... mr+2,r 0 1

.... . .

0 mn,r 0 1

(3.15)

r

M (r)A(r)x = M (r)b(n)

A(r+1)x = b(r+1).

-

-

M = M (n1) . . .M (2)M (1). (3.16)

MA(1)x = Mb(1) (3.17)

A(n)x = b(n). (3.18)

3.1.2. A(n)x = b(n) A(1)x = b(1).

. 3.1.1

.

72

3.1.1 Axk = bk, k = 1(1)

Axk = bk, k = 1(1), (3.19)

xk = [x1k, x2k, . . . , xnk]T

bk = [b1k, b2k, . . . , bnk]T

AX = B

X B n . detA 6= 0, Gauss

b B.

3.1.2 A1

AX = I

(3.19).

3.1.3 detA

A(n)

A(n) = MA(1) (3.20)

detA(n) = [detM ][detA(1)

].

(3.16)

detM =

n1r=1

detM (r)

73

M (r), r = 1(1)n 1

detM = 1 (3.21)

A(n)

detA(1) = detA(n) = a(1)11 a

(2)22 . . . a

(n)nn (3.22)

Gauss.

3.1.4 Gauss

Gauss Ax = b.

1. A = (aij) , b = (bi)

2. i = 1(1)n ai,n+1 = bi

3. r = 1(1)n 1 3.1-3.33.1 p ap,r 6= 0, p =

r(1)n. p ( - ). .

3.2 p 6= r ( p r ). q = r(1)n

bq = arqarq = apqapq = bq

3.3 i = r + 1(1)n 3.3.1-3.3.2

3.3.1

mir = airarr

3.3.2 j = r + 1(1)n+ 1

aij = aij +mirarj

74

4. ann = 0 . - .

5. ( )

xn = an,n+1/ann

6. i = n 1(1)1

xi =

[ai,n+1

nj=i+1 aijxj

]aii

7. xi, i = 1(1)n. .

3.1.5

Gauss

,

Gauss .

r- -

, , a(r)ir .

i r .

-

.

a(r)ir ;

3.1.3. A(1) ,

a(r)ir .

. -

3.1.1 detA(r) = detA(1) detA(r) 6= 0.

detA(1) = det

a(1)11 a

(1)12 a(1)1n

0 a(2)22 a(2)2n

......

...

0 a(2)n2 a(2)nn

= a(1)11 det

a

(2)22 a(2)2n...

...

a(2)n2 a(2)nn

= a(1)11 detA

(2)n1

75

detA(r) =[a(1)11 a

(2)22 . . . a

(r1)r1,r1

]detA

(r)nr+1,

A(r)nr+1 =

a

(r)rr a(r)rn...

...

a(r)nr a(n)nn

.

detA(r)nr+1 6= 0 Anr+1

.

6= 0 A(r)nr+1.

a(r)rr = 0, r

, , a(r)ir 6= 0 (. 3.1.3).

r i . - A(r) (r) A(r) b(r), .

A(r)x = (r) (3.23)

(r < i)

r i

Iri =

r

i

1 | |. . . | 0 | 0

1 | | 0 1

| 1 |0 | . . . | 0

| 1 | 1 0

| | 10 | 0 | . . .

| | 1

,

(3.24)

76

-

I2ri = I. (3.25)

r (3.23) Iri - M (r). [

M (r)Iri]A(r)x =

[M (r)Iri

](r) (3.26)

A(r+1)x = (r+1). (3.27)

i = r Iri = Irr = I.

A(1)x = (1) (3.28)

M = [M (n1)In1,in1] [M (n2)In2,in2] . . . [M (2)I2,i2] [M (1)I1,i1](3.29)

MA(1)x =M(1) (3.30)

A(n)x = (n). (3.31)

-

(3.31) (3.28).

3.1.4. A(1)x = (1) A(n)x = (n).. detA(1) 6= 0. (3.30) (3.31)

A(n) =MA(1) (3.32)

77

detA(n) = [detM] [detA(1)] . (3.33) detA(n) 6= 0 detM 6= 0. (3.29)

detM =[n1r=1

detM (r)

][n1r=1

det Ir,ir

](3.34)

M (r), r = 1(1)n 1

n1r=1

detM (r) = 1. (3.35)

n1r=1

det Ir,ir 6= 0 (3.36)

r = 1(1)n 1 r 6= irdet Ir,ir = 1

det Ir,ir I - r i ( r = ir, det Irr = 1). (3.34), (3.35) (3.36)

detA(n) 6= 0

(3.31) .

(3.28)

x =[A(1)

]1(1)

(3.31) (3.30),

A(n)x =MA(1){[A(1)]1 (1)} =M(1) = (n)

(3.28) (3.31)

.

78

3.1.6

Gauss -

. -

, ,

-

, .

-

,

Gauss

.

.

.

a(r)rr

a(r)rr (r < i). a

(r)rr , -

mir -

,

. Wilkinson

.

-

. -

r A(r) r , |mir| 1, :

a(r)pr = maxi

a(r)ir , r i n r p A(r) r-. - r

79

n r + 1 A(r)

a(r)pr = maxij

a(r)ij , r i, j n p r . ,

= 103. -

,

.

x1 + 2x2 x3 = 02x1 x2 = 1x1 + 7x2 3x3 = 5

-

Gauss i) ii) .

(i) [A...b],

1 2 1 02 1 0 1

1 7 3 5

Gauss

. -

21

1 2 1 02 1 0 1

1 7 3 5

3

1 2 1 00 3 2 1

0 9 4 5

1

80

1 2 1 00 3 2 1

0 0 2 2

2

x1 + 2x2 x3 = 03x2 2x3 = 1

2x3 = 2

x3 = 1, x2 = 1 x1 = 1.

(ii) Gauss -

. 1 2 1 02 1 0 1

1 7 3 5

(1)(2)

(3)

1/21/2

2 1 0 11 2 1 0

1 7 3 5

(2)(1)

(3)

2 1 0 10 3/2 1 1/2

0 15/2 3 9/2

(2)(1)

(3)1

1/5

2 1 0 10 15/2 3 9/2

0 3/2 1 1/2

(2)(3)

(1)

2 1 0 10 15/2 3 9/2

0 0 2/5 2/5

(2)(3)

(1)2

2x1 x2 = 1152x2 3x3 = 92

25x3 = 25

81

x3 = 1, x2 = 1 x1 = 1.

3.1.7 Gauss -

Gauss

1. A = (aij), b = (bi) n.

2. i = 1(1)n ai,n+1 = bi.

3. i = 1(1)n

h(i)= i

4. r = 1(1)n 1 4.1-4.4 (- ).

4.1. p

r p n

|a(h(p),r)| = maxrjn

|a(h(j),r)|

4.2. a(h(p), r) = 0 ( ). .

4.3. h(r) 6=h(p) ( h(p) h(r))

q = h(r)h(r) = h(p)h(p) = q

( ).

4.4. i = r + 1(1)n i ii

i.

m(h(i),r) = a(h(i),r)a(h(r),r)

82

ii. j = r + 1(1)n+ 1

a(h(i),j) = a(h(i),j)+m(h(i),r)a(h(r),j)

a(h(n),n) = 0 ( ). .

5. ( )

xn = a(h(n),n+ 1)/a(h(n),n)

6. i = n 1(1)1

xi =a(h(i),n+ 1)nj=i+1 a(h(i),j)xj

a(h(i),i)

7. xi, i = 1(1)n. .

-

,

.

30, 00x1 + 591.400x2 = 591.7005, 291 6, 130x2 = 46, 78.

-

m21 =5, 291

30, 00= 0, 1764

30, 00x1 + 591.400x2 = 591.700 104.300x2 = 104.400

x2 = 1, 001 x1 = 10, 00.

x1 = 10, 00 x2 = 1, 000. ,

83

(scaled)

.

.

.

30, 00

591.400= 0, 00005073

5, 291

6, 130= 0, 8631

-

.

3-4.1,

:

3. i = 1(1)n 3.1-3.3

3.1. si = max1jn |aij |3.2. si = 0

3.3. h(i) = i

4. r = 1(1)n 1 4.1-4.44.1. p r p n

|a(h(p),r)|s(h(p))

= maxrjn

|a(h(j),r)|s(h(j))

.

-

Ax = b D1 i- (si)

1.

3.2 Jordan

Jor-

dan

.

Gauss

xi .

84

(3.1), -

Jordan

Gauss, (3.4) :

a(1)11 x1 + a

(1)12 x2 + a

(1)13 x3 + . . . + a

(1)1n xn = b

(1)1

a(2)22 x2 + a

(2)23 x3 + . . . + a

(2)2n xn = b

(2)2

a(2)32 x2 + a

(2)33 x3 + . . . + a

(2)3n xn = b

(2)3

...

a(2)n2 x2 + a

(2)n3 x3 + . . . + a

(2)nnxn = b

(2)n

(3.37)

x1 n1

A(2)x = b(2).

Jordan

x2 n 2 , .

a(1)11 x1 a

(3)13 x3 + . . . + a

(3)1nxn = b

(3)1

a(2)22 x2 + a

(3)23 x3 + . . . + a

(3)2nxn = b

(3)2

a(3)33 x3 + . . . + a

(3)3nxn = b

(3)3

...

a(3)n3x3 + . . . + a

(3)nnxn = b

(3)n

(3.38)

A(3)x = b(3) (3.39)

. r 1

a(1)11 x1 +a

(r)1r xr + . . .+ a

(r)1nxn = b

(r)1

a(2)22 x2 +a

(r)2r xr + . . .+ a

(r)2nxn = b

(r)2

. . ....

a(r1)r1,r1xr1 +a

(r)r1,rxr . . .+ a

(r)r1,nxn = b

(r)r1

a(r)rr xr + . . .+ a

(r)rnxn = b

(r)r

...

a(r)nr xr + . . .+ a

(r)nnxn = b

(r)n

(3.40)

85

A(r)x = b(r)

r 1 . , n :

a(1)11 x1 = b

(n+1)1

a(2)22 x2 = b

(n+1)2

. . ....

a(n)nnxn = b

(n+1)n

(3.41)

A(n)x = b(n+1) (3.42)

A(n) . (3.42)

xi =1

a(i)ii

b(n+1)i

a(i)ii 6= 0, i = 1(1)n.

a(1)ij = aij i, j = 1(1)n

b(1)i = bi i = 1(1)n

a(1)11 6= 0,

mi1 = a(1)i1

a(1)11

, i = 2(1)n.

x1 i- , - mi1 i-, - (3.37),

a(2)ij = a

(1)ij +mija

(1)1j , i = 1(1)n, i 6= 1

j = 2(1)n

b(2)i = b

(1)i +mi1b

(1)1 , i = 1(1)n, i 6= 1.

86

-

a(2)1j = a

(1)1j , j = 2(1)n

b(2)1 = b

(1)1 .

x2 n2 , (3.38)

a(3)ij = a

(2)ij +mi2a

(2)2j , i = 1(1)n, i 6= 2, j = 3(1)n

b(3)i = b

(2)i +mi2b

(2)2 , i = 1(1)n, i 6= 2

mi2 = a(2)i2

a(2)22

, i = 1(1)n, i 6= 2.

a(3)2j = a

(2)2j , j = 3(1)n

b(3)2 = b

(2)2 .

(3.41). -

A - n n . Jordan

n n M MAx = Mb (3.43)

MA = I. (3.44)

M = A1 (3.45)

x = Mb. (3.46)

A(1)x = b(1) (3.47)

87

M (1) -

M (1) =

11 02131 In1...

n1

(3.48)

i1 =

{1/a

(1)11 , i = 1

a(1)i1 /a(1)11 , i 6= 1 a

(1)11 6= 0. (3.49)

Jordan

(3.47) M (1), :

M (1)A(1)x = M (1)b (3.50)

A(2)x = b(2). (3.51)

r :

M (r)A(r)x = M (r)b (3.52)

M (r) =

1r

Ir1... 0

r1,r0 0 rr 0 0

r+1,r

0 ... Inrnr

(3.53)

ir

i1 =

{1/a

(r)rr , i = r

a(r)ir /a(r)rr , i 6= r, a(r)rr 6= 0, r = 1(1)n.

(3.52)

A(r+1)x = b(r+1) (3.54)

88

A(r+1) =

[Ir 0

](3.55)

. -

M = M (n)M (n1) . . .M (2)M (1) (3.56)

M (1) M (i), i = 2(1)n 1 (3.48) (3.53),

M (n) =

1,n

In1...

n1,n0 0 nn

. (3.57)

MA(1)x = Mb(1)

x = Mb(1). (3.58)

Gauss, -

Jordan

. ( -

).

Gauss -

. Jordan

.

Jordan -

.

89

1 2 1 02 1 0 1

1 7 3 5

(1)(2)

(3)

1/21/2

2 1 0 11 2 1 0

1 7 3 5

(2)(1)

(3)

2 1 0 10 3/2 1 1/2

0 15/2 3 9/2

(2)(1)

(3)1

2/15

1/5

2 1 0 10 15/2 3 9/2

0 3/2 1 1/2

(2)(3)

(1)

115/2

2 0 2/5 8/50 15/2 3 9/2

0 0 2/5 2/5

(2)(3)

(1)2

2 0 0 20 15/2 0 15/2

0 0 2/5 2/5

(2)(3)

(1)3

x1 = 1, x2 = 1 x3 = 1.

3.2.1 Jordan -

Jordan

Ax = b :

1. A = (aij) , b = (bi) n.

2. i = 1(1)n

ai,n+1 = bi

90

3. i = 1(1)n

h(i)= i

4. r = 1(1)n 4.1-4.4 ().

4.1. p

r p n

|a(h(p), r))| = maxrjn

|a(h(j), r)|

4.2. a(h(p),r)= 0 -. .

4.3. h(r) 6=h(p)

q=h(r)h(r)=h(p)h(p)=q

( )

4.4. i = 1(1)n i 6= r i ii

i.

m(h(i),r) = a(h(i),r)a(h(r),r)

ii. j = r + 1(1)n+ 1

a(h(i),j) = a(h(i),j)+m(h(i),r)a(h(r),j)

5. i = 1(1)n

a(h(i),i) = 0 . .

xi = a(h(i),n + 1)/a(h(i),i)

6. xi, i = 1(1)n.

91

3.2.2

-

, ,

Gauss Jordan.

k Gauss :

n k

a11 a12 a13 a1k a1,k+1 a1na22 a23 a2k a2,k+1 a2n

a33 a3k a3,k+1 a3n0

. . ....

......

akk ak,k+1 aknak+1,k ak+1,k+1 ak+1,n

0...

......

ank an,k+1 ann

x(1)1 x

(2)1 x()1

x(1)2 x

(2)2 x()2

x(1)3 x

(2)3 x()3

......

...

x(1)k x

(2)k x()k

x(1)k+1 x

(2)k+1 x()k+1

......

...

x(1)n x

(2)n x()n

=

nk

l

n k

=

b(1)1 b

(2)1 b()1

b(1)2 b

(2)2 b()2

b(1)3 b

(2)3 b()3

......

...

b(1)k b

(2)k b()k

......

...

b(1)n b

(2)n b()n

(3.59)

l

- A. -

,

(nk) (nk+ ). (

k ). (n k)(n k+ ) - . n k - mik, i = k + 1(1)n n k . , -

n k + k A

92

. n k (n k)(n k + ) -. k = 1(1)n 1

n1k=1

(n k)

n1k=1

(n k)(n k + ) (3.60)

n1k=1

(n k)(n k + ) .

mk=1

k =m(m+ 1)

2

mk=1

k2 =m(m+ 1)(2m+ 1)

6

(3.60)

n(n 1)2

n(n 1)(2n 1 + 3)6

(3.61)

n(n 1)(2n 1 + 3)

6.

x(i)k , k = 1(1)n, i = 1(1) -

(3.7)

i n k aijxj , j = k + 1(1)n, n k k = 1(1)n 1, x(i)n - . x

(i)k

nk=1

1

93

n1k=1

(n k)

n1k=1

(n k)

n

n(n 1)2

(3.62)

n(n 1)

2.

(3.59) :

n(n 1 + 2)2

n(n 1)(2n 1 + 6)6

(3.63)

n(n 1)(2n 1 + 6)

6.

( = 1) Gauss

n2

2+n

2

n3

3+n2

2 5n

6 (3.64)

n3

3+n2

2 5n

6.

A1 Gauss ( = n)

3n2

2 n

2

4n3

3 3n

2

2 n

6 (3.65)

94

4n3

3 3n

2

2 n

6.

Gauss

O(n3/3) . -

(

) A1. Jordan

a11a22

. . .

a1k a1,k+1 a1n...

......

......

...

akk ak,k+1 aknak+1,k ak+1,k+1 ak+1,n

......

...

ank an,k+1 ann

X

=

B

nk

n 1 (n 1)(n k + ). -

Jordan.

(n 1)(n k + ) . A

nk=1

(n 1)

nk=1

(n 1)(n k + )

nk=1

(n 1)(n k + )

95

n(n 1) n(n 1)(n 1 + 2)

2

n(n 1)(n 1 + 2)2

.

n n . Jordan

n(n 1 + ) n(n 1)(n 1 + 2)

2 (3.66)

n(n 1)(n 1 + 2)2

.

-

( = 1)n2

n3

2 n

2 (3.67)

n3

2 n

2,

( = n)

2n2 n 3n3

2 2n2 + n

2 (3.68)

3n3

2 2n2 + n

2.

(3.67) -

Jordan

O(n3/2). Jordan Gauss

. -

Jordan .

96

A1 , -

. Jordan, ,

.

Gauss

.

Wilkinson.

Gauss

.

3.3 LU

-

Gauss , A(1)x = b(1)

MA(1)x = Mb(1)

A(n)x = b(n)

A(1) = M1A(n) (3.69)

(3.56)

M1 =[M (1)

]1 [M (2)

]1. . .[M (n1)

]1(3.70)

[M (r)

]1=

Ir1 01

mr+1,r 1 00 mr+2,r 0 1

......

. . .

mnr 0 0 1

. (3.71)

97

M1 =

1m21 1 0m31 m32 1m41 m42 m43 1

......

......

. . .

mn1 mn2 mn3 mn4 1

(3.72)

M1 . A(n) , (3.69) , -

A(n) A(1), A(1) -

M1 A(n). .

3.3.1. n n A = (aij) LU , L U ,

det [a11] 6= 0, det[a11 a12a21 a22

]6= 0, . . . , detA 6= 0.

,

L U .

. A

A = LU (3.73)

L ,

A =

121 1 031 32 1...

......

. . .

n1 n2 n3 1

u11 u12 u13 u1n

u22 u23 u2nu33 u3n

0. . .

...

unn

.(3.74)

98

, n2 - n2 . L ( - =1), n . -

L U . LU A

u11 = a11u12 = a12

...

u1n = a1n

(3.75)

A U -.

LU ,

21u11 = a2131u11 = a31

...

n1u11 = an1.

(3.76)

u11 = a11, a11 6= 0, (3.76) L. , r U r L, k = 1(1)r 1 U L, , .

1

21. . . 0

...

r1 1r+1,1 r+1,r 1...

......

. . .

n1 nr 1

u11 u1r u1,r+1 u1n. . .

......

...

ur1,r ur1,r+1 ur1,nurr ur,r+1 urn

0 . . ....

unn

99

r L r, r + 1, . . . , n U A,

r1j=1

rjujr + urr = arr

r1j=1

rjuj,r+1 + ur,r+1 = ar,r+1

r1j=1

rjujn + urn = arn.

urp, p = r(1)n

urp = arp r1j=1

rjujp, p = r(1)n. (3.77)

r U r + 1, r + 2, . . . , n L A

r1j=1

r+1,jujr + r+1,rurr = ar+1,r

r1j=1

r+2,jujr + r+2,rurr = ar+2,r

r1j=1

njujr + nrurr = anr.

pr, r + 1(1)n , urr 6= 0, r L

pr =1

urr

(apr

r1j=1

pjujr

), p = r + 1(1)n. (3.78)

100

, (3.77) p = r

urr = arr r1j=1

rjujr, r = 1(1)n

r = 2

u22 = a22 21u12= a22 a21a11a12

(= a

(2)22

)

u22 =1

a11det

[a11 a12a21 a22

].

a11 6= 0 det[a11 a12a21 a22

]6= 0

u22 6= 0. , urr 6= 0, r = 1(1)n.

.

L :

A(1) = LU

M1 = L A(n) = U.

Gauss

LU L - . Wilkinson ([1965] . 223)

-

,

-

, .

,

Gauss

. -

Gauss

-

n

j=1 ajbj

101

.

,

.

Gauss

, -

.

.

1. ,

-

, ,

. ,

,

|aii| >j 6=i

|aij | , i = 1(1)n. (3.79)

2. A

A = LU

Ax = b LUx = b (3.80)

Ly = b (3.81)

Ux = y. (3.82)

(3.81)

(ii = 1, i = 1(1)n)

y1 = b1

yi = bi i1j=1

ijyj , i = 2(1)n (3.83)

(3.82)

xn = yn/unn

102

xi =

(yi

nj=i+1

uijxj

)/uii, i = n 1(1)1.

3. LU L U , - -

Gauss. , L U A .

, A = (aij), n = 4u11 u12 u13 u1421 u22 u23 u2431 32 u33 u3441 42 43 44

.

, L -

Doolittle U Crout.

LU

A =

1 1 11 2 22 1 1

A = LU

1 1 11 2 22 1 1

=

1 0 02 21 1 0

314 32 1

u11 u1

12 u13

0 u3

22 u23

0 0 u5

33

103

u11 = 1 u12 = 1 u13 = 121u11 = 1 21 = 1/u11 = 131u11 = 2 31 = 2/u11 = 221u12 + u22 = 2 u22 = 2 21u12 = 121u13 + u23 = 2 u23 = 2 21u13 = 131u12 + 32u22 = 1 32 = (1 31u12)/u22 = 331u13 + 32u23 + u33 = 1 u33 = 1 31u13 32u23 = 2.

A =

1 0 01 1 02 3 1

1 1 10 1 1

0 0 2

.

LU

n n A LU .

1. : n, aij , i, j = 1(1)n A lii, i = 1(1)n L uii, i = 1(1)n U .

2. l11 u11 l11u11 =a11.

l11u11 = 0 ( ). -.

3. j = 2(1)n

u1j = a1j/l11( U)

lj1 = aj1/u11( L)

4. r = 2(1)n 1 4.1-4.2

4.1 lrr urr

104

lrr urr = arr r1j=1

lrj ujr

lrr urr = 0 ( ).

4.2 p = r + 1(1)n

urp =

(arp

r1j=1

lrj ujp

)/ lrr( r U)

lpr =

(apr

r1j=1

lpj ujr

)/ urr( r L)

5. unn lnn

unn lnn = ann r1j=1

lnj ujn

( lnn unn = 0, A = LU A )

6. (lij, j = 1(1)i, i = 1(1)n)

(uij, j = 1(1)n, i = 1(1)n)

.

L U . :

u11 u12 u13 . . . u1nl21 u22 u23 . . . u2nl31 l32 u33 . . . u3n...

......

...

ln1 ln2 ln3 . . . unn

105

-

, , , ...

,

u1j = a1j , j = 1(1)n.

r - ,

1

l21 1 0l31 l32 1...

.

.

....

. . .

.

.

....

.

.

.. . .

lr1 lr2 lr3 . . . lr,r1 1...

.

.

....

.

.

.. . .

ln1 ln2 ln3 . . . ln,r1 . . . ln,n1 1

u11 u12 . . . u1r . . . u1nu22 . . . u2r . . . u2n

0 . . ....

.

.

.

urr . . . urn

0 . . ....

unn

-

, r L

lr1u11 = ar1lr1u12 + lr2u22 = ar2lr1u13 + lr2u23 + lr3u33 = ar3

......

......

lr1u1r + lr2u2r + lr3u3r + . . . + lr,r1ur1,r1 = ar,r1

u11u12 u22 0u13 u23 u33...

......

. . .

u1,r1 u2,r1 u3,r1 . . . ur1,r1

lr1lr2lr3...

lr,r1

=

ar1ar2ar3...

ar,r1

r U

urp = arp r1j=1

lrj ujp, p = r(1)n.

106

L U .

1l21 1

l31 l32 1 0...

......

. . .

......

.... . .

lr1 lr2 lr3 . . . lr,r1 1lr+1,1 lr+1,2 lr+1,3 . . . lr+1,r1 lr+1,r1...

......

......

. . .

ln1 ln2 ln3 ln,r1 ln,r . . . 1

u11 u12 u1r u1nu22 u2r u2n

. . ....

......

urr urn0 . . .

...

unn

u1r = a1rl21u1r + u2r = a2r

......

...

lr1u1r + lr2u2r + . . . + lr,r1ur1,r + urr = arr

1

l21 1 0l31 l32 1...

. . ....

. . .

lr1 lr2 lr3 . . . lr,r1 1

u1ru2ru3r...

urr

=

a1ra2ra3r...

arr

r L

lpr =

(apr

r1j=1

lpj ujr

)/ urr, p = r + 1(1)n.

3.4 LU

LU A . A = LU LU A D L = LD U = D1U ,

107

A = LU = LDD1U = LU

LU LU A. - LU

.

:

A = LDU

LDU A L , D U .

3.4.1. A LDU A[r], r = 1(1)n 1

A[r] =

a11 . . . a1r... ...ar1 . . . arr

.

. A LDU - . A = L1D1U1 A =L2D2U2. A , D1 D2.

L1D1U1 = L2D2U2

L12 L1 = D2U2U11 D

11 (3.84)

(3.84) -

.

,

L12 L1 = I

108

L1 = L2.

U1 = U2.

L1 U1 , L1D1U1 =L2D2U2

D1 = D2.

A LDU - A[1] . . . A[n1] .

A = LDU LDU - A. L, D U . L U D L[k], D[k] U [k] . A[k] = L[k]D[k]U [k]

A[k] . , A[r], r = 1(1) 1 . 3.3.1 -

Gauss A = LA(n) L A(n) (3.72) (3.20), . A(n)

a(k)kk , k = 1(1)n

. D = diag (a(1)11 , a

(2)22 , . . . , a

(n)nn ) U = D1A(n),

A = LDU

LDU A.

LDU A

lij = mij , i = 2(1)n, j = 1(1)i 1dii = a

(i)ii , i = 1(1)n

uij =a(i)ij

a(i)ii

, i = 1(1)n, j = i+ 1(1)n.

109

LDU .

A = (LD)U = LU

l11 0l21 l

22

......

. . .

ln1 ln2 . . . l

nn

1 u11 . . . unn

1 . . . u2n. . .

...

0 1

= LU

lii = dii, i = 1(1)n, lij = lijdii, i = 2(1)n, j = 1(1)i 1

Crout . , -

A = LDU , - 3.3.1

A =

1l21 1

l31 l32 1 0...

......

. . ....

......

. . .

lr1 lr2 lr3 . . . lr,r1 1lr+1,1 lr+1,2 lr+1,3 . . . lr+1,r1 lr+1,r1...

......

......

. . .

ln1 ln2 ln3 ln,r1 ln,r . . . 1

d11 0

d22. . .

0 dnn

1 u12 . . . u1r u1,r+1 . . . u1n. . .

......

.... . .

......

...

1 ur,r+1 . . . urn1 . . . ur+1,n

0 . . ....

1

110

A =

1l21 1

l31 l32 1 0...

......

. . ....

......

. . .

lr1 lr2 lr3 . . . lr,r1 1lr+1,1 lr+1,2 lr+1,3 . . . lr+1,r1 lr+1,r 1...

......

......

. . .

ln1 ln2 ln3 ln,r1 ln,r . . . 1

d11 d11u12 . . . d11u1r d11u1,r1 . . . d11u1n. . .

......

.... . .

......

...

drr drrur,r+1 . . . drrurn. . .

...

0 . . ....

dnn

d11 = a11d11u12 = a12

...

d11u1n = a1n

d11 = a11

uij =aijd11

, j = 2(1)n.

111

l21d11 = a21...

ln1d11 = an1

li1 =ai1d11

, i = 2(1)n.

arr =r1j=1

lrj djj ujr + drr, r = 2(1)n

arp =r1j=1

lrj djj ujp + drr urp, p = r + 1(1)n

apr =

r1j=1

lpr djj ujr + lpr drr, p = r + 1(1)n

, L,D U

drr = arr r1j=1

lrj djj ujr

urp =

(arp

r1j=1

lrj djj ujp

)/ drr

lrp =

(arp

r1j=1

lpj djj ujr

)/ drr

p = r + 1(1)n, r = 1(1)n

A = L(DU) = LU

112

1 0l21 1...

. . .

ln1 . . . 1

u11 . . . . . . u

1n

. . ....

. . ....

0 unn

= LU

uii = dii, i = 1(1)n uij = di uij = a

(i)ij , i = 1(1)n, j = 1(1)n

Doolittle .

A LU - (L 6= UT ).

A = LDLT (3.85)

L . - D

D1/2 = diag(d1/211 , . . . , d

1/2nn ) (3.86)

A

A = LDLT = (LD1/2)(LD1/2)T = LLT

(3.87)

L = LD1/2. (3.88)

A

A = LLT .

Cho-

leski

Cho-

leski . -

Choleski.

113

. A x 6= 0

xTAx > 0

xTAx 0

A mn , B = ATA ,

BT = (ATA)T = ATA = B.

B . , x 6= 0 y = Ax,

xTBx = xTATAx = yTy =

ni=1

y2i 0.

(rank) A n, x 6= 0 y = Ax 6= 0, xTBx > 0 B .

3.4.2. A n n A .

. x 6= 0 Ax = 0

xTAx = 0

A . Ax = 0 A .

3.4.3. 1

.

1 A

A, i1 0, i =1(1)n ( ;). 3.3.1

l11 = u11 =a11

A

lj1 =aj1u11

=a1jl11

= u1j , j = 2(1)n (3.90)

U - L.

115

k < n k U - k L.

k + 1 U k+1 L. L U

lj,k+1 = 0 = uk+1,j, j = 1, 2, . . . , k.

A

lk+1,k+1 uk+1,k+1 = ak+1,k+1 k

j=1

lk+1,j uj,k+1

lk+1,k+1 uk+1,k+1 = ak+1,k+1 k

j=1

l2k+1,j

(. 4.1 3.3.1)

lk+1,k+1 = uk+1,k+1 =

(ak+1,k+1

kj=1

l2k+1,j

)1/2. (3.91)

(3.91) . -

Ak+1 3.4.3. -

.

0 < det (Ak+1) = det (Ak)lk+1,k+1 uk+1,k+1

det (Ak) > 0,

lk+1,k+1 uk+1,k+1 > 0

lk+1,k+1 (3.91) - . (.

4.2 3.3.1)

116

lp,k+1 =

(ap,k+1

kj=1

lpj uj,k+1

)/ uk+1,k+1

=

(ak+1,p

kj=1

ujp lk+1,j

)/ lk+1,k+1

= uk+1,p, p = k + 2(1)n.

3.4.1 Choleski

n n A :

1. n aij , i, j = 1(1)n A.

2. l11 =a11

3. j = 2(1)n lj1 = aj1/l11

4. r = 2(1)n 1 4.1-4.24.1

lrr =

[arr

r1j=1

l2rj

]1/2. (3.92)

4.2 p = r + 1(1)n

lpr =1

lrr

[apr

r1j=1

lpj lrj

](3.93)

5.

lnn =

[ann

n1j=1

l2nj

]1/2(3.94)

6. lrp, p = 1(1)r, r = 1(1)n. .

. Choleski

-

.

117

3.5 LU

3.3 -

Gauss

.

;

.

3.5.1. n n A P , L U PA

PA = LU (3.95)

. (. [Young and Gregory] . 127-129).

-

Gauss PA A .

PA. detP 6= 0 Ax = b

PAx = Pb (3.96)

A - PA. Wilkinson [1967] PA - P .

Gauss -

, A a11 . -

(3.77) (3.78)

U L, . A

118

u11 u12 u13 . . . u1n b1 0l21 a22 a23 . . . a2n b2 0l31 a32 a33 . . . a3n b3 0. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .ln1 an2 an3 . . . ann bn 0

.

,

. r

sr = arr r1j=1

lrj ujr

sr+1 = ar+1,r r1j=1

lr+1,j ujr

. . . . . . . . . . . . . . . . . . . . . . . . . . . (3.97)

sn = anr r1j=1

lnj ujr

.

r = 2

u11 u12 u13 . . . u1n b1 0l21 a22 a23 . . . a2n b2 s2l31 a32 a33 . . . a3n b3 s3. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .ln1 an2 an3 . . . ann bn sn

(3.97) r = 1

s1 = a(1)11 s2 = a

(1)21 . . . sn = a

(1)n1 .

119

r = 2

s2 = a(1)22 l21 u12

= a(1)22 +m21 a

(1)12

= a(2)22

s3 = a(2)32 , . . . , sn = a

(2)n2

sr = a(r)rr sr+1 = a

(r)r+1,r, . . . , sn = a

(r)nr (3.98)

si. p

|sp| = maxrin

|si| . sr r p (r < p) sp sr.

urr = sr

r u

urp = ar,p r1j=1

lrj ujp, p = r + 1(1)n

r L

lpr = sp/urr, p = r + 1(1)n

.

, -

L. , lij - / Gauss

.

120

3.5.1 LU -

A LU - Lz = b Ux =z L U .

1. n, aij , i = 1(1)n, j = 1(1)n+ 1 A lii, i = 1(1)n L uii, i = 1(1)n U .

2. p 1 p n

|ap1| = max1jn

|aj1|

( )

|ap1| = 0 ( ). .3. p 6= 1 p 1 A.4. l11 u11

l11 u11 = a11

5. j = 2(1)n

u1j = a1j/l11 ( U)lj1 = aj1/u11 ( L)

6. r = 2(1)n 1 6.1-6.4.6.1 p r p n

apr r1j=1

lpj ujr

= maxrknakr

r1j=1

lkj ujr

( )

( -

)

121

6.2 p 6= r p r A L.

6.3 lrr urr

lrr urr = arr r1j=1

lrj ujr

6.4 p = r + 1(1)n

urp =

(arp

r1j=1

lrj ujp

)/lrr (r U)

lpr =

(apr

r1j=1

lpj ujr

)/urr (r L)

7.

hold = ann r1j=1

lnj ujn

hold = 0 ( ). .

lnn unn

lnn unn = ann r1j=1

lnj ujn

( 8 9

Lz = b).

8.

z1 = a1,n+1/l11

9. i = 2(1)n

zi =

(ai,n+1

i1j=1

lij zj

)/ lii

122

( 10 11

Ux = z)

10.

xn = zn/unn

11. i = n 1(1)1

xi =

(zi

nj=i+1

uij xj

)/ uii

12. xi, i = 1(1)n. .

3.5.2

b1x1 +c1x2 = d1a2x1 +b2x2 +c2x3 = d2

a3x2 +b3x3 +c3x4 = d3. . . . . .

. . . . . .an1xn2 +bn1xn1 +cn1xn = dn1

anxn1 +bnxn = dn(3.99)

Gauss .

.

Gauss

.

ai, . ci . b1 6= 0 x1 ,

123

m1 = a2b1

b2x2 + c2x3 = d2

b2 = b2 +m1c1

d2 = d2 +m1d1

b2 6= 0, x2 , /.

m2 = a3b2

b3x3 + c3x4 = d3

b3 = b3 +m2c2

d3 = d3 +m2d2

i- , xi i + 1, /

mi = ai+1bi

(3.100)

i+ 1

bi+1xi+1 + ci+1xi+2 = di+1 (3.101)

bi+1 = bi+1 +mici (3.102)

124

di+1 = di+1 +midi (3.103)

i = 1(1)n 1

b1x1+ c1x2 = d1

b2x2+ c2x3 = d2

. . .. . .bn1xn1+ cn1xn = d

n1

bnxn = dn

(3.104)

b1 = b1 d1 = d1.

xn =dnbn, bn 6= 0 (3.105)

xi = (di cixi+1)/ bi, i = n 1(1)1

bi 6= 0. .

1. n A, bi, i = 1(1)n, ci, i = 1(1)n 1, ai, i = 2(1)n di, i = 1(1)n

2. i = 1(1)n 1 2.1-2.32.1

mi = ai+1bi

2.2

bi+1 = bi+1 +mici

2.3

di+1 = di+1 +midi

( )

125

3. bn = 0 ( ). .

4.

xn =dnbn

5. n 1(1)1

xi = (di cixi+1)/bi6. xi, i = 1(1)n. .

LU . A (3n 2) , L U (3n 2).

L =

l1k2 l2 0

k3 l3

0 . . . . . .kn ln

U =

1 u11 u2 0

1 u3. . .

. . .

0 1 un11

(3.106)

(2n1) L (n1) U , (3n 2) A = LU ,

l1 = b1, l1u1 = c1 k2 = a2

k2 u1 + l2 = b2, l2u2 = c2 k3 = a3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.107)

kn un1 + ln = bn kn = an

126

l1 = b1

u1 = c1/l1

ki = ai, i = 2(1)n

li = bi ki ui1, i = 2(1)nui = ci/li, i = 2(1)n 1 (3.108)

l1 = b1

u1 = c1/l1

li = bi ai ui1, i = 2(1)n (3.109)ui = ci/li, i = 2(1)n 1

(3.99)

LUx = d

Lz = d (3.110)

Ux = z. (3.111)

(3.110)

z1 = d1/l1 (3.112)

zi = (di ai zi1)/li, i = 2(1)n (3.111)

xn = zn (3.113)

xi = zi ui xi+1, i = n 1(1)1. Grout

(3.99)

127

1. n A, bi, i = 1(1)n, ai, i = 2(1)n, ci, i = 1(1)n 1 di, i = 1(1)n

2.

l1 = b1

u1 = c1/l1

3. i = 2(1)n 1

li = bi ai ui1ui = ci/li

4.

ln = bn an un1( 5 6 Lz = d).

5.

z1 = d1/l1

6. i = 2(1)n

zi = (di ai zi1) / li( 7 8 Ux = z).

7.

xn = zn

8. i = n 1(1)1

xi = zi ui xi+1

128

9. xi, i = 1(1)n. .

2 1 0 01 2 1 00 1 2 10 0 1 2

x1x2x3x4

=

1001

Grout

1. n = 4, b1 = b2 = b3 = b4 = 2, a2 = a3 = a4 = 1c1 = c2 = c3 = 1 d1 = 1, d2 = 0, d3 = 0 d4 = 1

2. l1 = 2, u1 = 1/23. i = 2

l2 = b2 a2 u1 = 2 (1)(1

2

)= 3

2

u2 = c2/l2 = 1/(32

)= 2

3

i = 3

l3 = b3 a3 u2 = 2 (1)(2

3

)= 4

3

u3 = c3/l3 = 1/(43

)= 3

4

4. l4 = b4 a4 u3 = 2 (1)(3

4

)= 5

4

5. z1 = d1/l1 = 1/2

6. i = 2

z2 = (d2 a2 z1)/l2 = 0(1)(12)

( 32)= 1

3

i = 3

x3 = (d3 a3 z2)/l3 = 0(1)(13)

( 43)= 1

4

i = 4

z4 = (d4 a4 z3)/l4 = 1(1)(14)

( 54)= 1

7. x4 = z4 = 1

129

8. i = 3

x3 = z3 u3 x4 = 14 (3

4

) 1 = 1i = 2

x2 = z2 u2 x3 = 13 (2

3

) 1 = 1i = 1

x1 = z1 u1 x2 = 12 (1

2

) 1 = 1 x1 = x2 = x3 =x4 = 1

3.5.2. A (3.110) ai, ci 6= 0 i = 2(1)n1. |b1| > |c1|, |bi| |ai|+ |ci| i = 2(1)n1 |bn| > |an|, detA 6= 0 i)|ui| < 1, ii)|ci| 0, A = 0 A = 0ii)cA = |c| A c iii)A+B A+ B (3.123)iv)AB A B.

137

norms norms -

A = maxx=1

{Ax}

Ax A x. (3.124)

3.6.4. norms -

norms

A1 = max

ni=1

|ai|

A2 = [S(AHA)]1/2 (3.125)

A = maxi

n=1

|ai|

S(A) = max1in

|i| i A AH A.

. .

norms -

.

{x(k)}, k = 0, 1, 2, . . . {x(k)i }, i = 1(1)n {x(k)i }, k =0, 1, 2, . . . i = 1(1)n ,

limk

x(k) = 0 x(k) k

0

3.6.5.

limk

x(k) = x

138

limk

x(k) x = 0.

. .

{A(k)}, k = 0, 1, 2, . . . - {a(k)ij }, i, j = 1(1)n - n2

{a(k)ij }, k = 0, 1, 2, . . . i, j = 1(1)n , -

limk

A(k) = 0 A(k) k

0.

3.6.6.

limk

A(k) = A

limk

A(k) A = 0.

. .

3.6.7.

A n, , lim

kA(k) = 0

S(A) < 1. (3.126)

. (. [Golub and Van Loan])

3.6.8. n

Ak Ak, k = 0, 1, 2, . . .. k = 0, 1 . k > 1

Ak = Ak1A Ak1 A = Ak2A A Ak2 A2 Ak.

139

3.6.9.

A n -, lim

kA(k) = 0

A < 1.

. 3.6.6 limk0

Ak = 0 limk

Ak = 0 3.6.8 lim

kAk = 0,

A < 1.

3.6.10. A n

S(A) A. (3.127)

. A x -

Ax = x

norms

Ax = x norms

|| x A x

|| A

S(A) = max || A.

3.6.11.

m=0

Am = I + A+ A2 + + Am + . . .

140

limm

Am = 0.

m=0

Am = (I A)1.

. (i) limm

Am = 0

3.6.7 S(A) < 1 = 1 A. det(I A) 6= 0 (I A)1 .

(I A)(I + A+ A2 + + Am) I Am+1

I + A+ A2 + + Am (I A)1 (I A)1Am+1.

I + A+ A2 + + Am m

(I A)1.(ii)

S(m) = I + A+ A2 + + Am m

(I A)1

S(m)ij (I A)1ij , m

n2 S(m)ij -

.

m- . i j (Am)ij 0 m lim

mAm = 0.

3.6.1.

m=0

Am = (I A)1

S(A) < 1.

141

.

3.6.7 3.6.11.

3.6.2. norm

A < 1

m=0

Am = (I A)1.

.

3.6.9 3.6.11.

3.6.12. A < 1 norm I A I + A

1

1 + A (I A)1 1

1 A (3.128)

1

1 + A (I + A)1 1

1 A . (3.129)

. 3.6.10 S(A) A < 1 = 1 A det(I A) 6= 0 I A . (3.128)

I = (I A)(I A)1. norms

1 = I = (I A)(I A)1 I A (I A)1 (I+ A) (I A)1 = (1 + A) (I A)1.

(3.128).

(I A)1 = (I A+ A)(I A)1 = I + A(I A)1

norms

(I A)1 = I + A(I A)1 I+ A(I A)1 1 + A (I A)1.

,

A < 1, (3.128). (3.128) A A (3.129).

142

3.7

, -

-

.

Ax = b

,

B = [A, b].

, -

-

.

. -

A b. A b A b,. , -

,

x, x.

(A+ A)(x+ x) = b+ b (3.130)

.

3.7.1. A

A1 A < 1 (3.131)

xx

1 A A1[bb +

AA

](3.132)

= (A) = A1 A. (3.133)

143

. (3.130)

(A+ A) x = b Ax x A + A = A(I + A1 A) .

S(A1A) A1A < 1 (3.131), I + A1A .

x = (I + A1A)1A1(b Ax)

norms

x = (I + A1A)1A1(b Ax) (I + A1A)1 A1 (b Ax).

3.6.12 , A1A A1 A < 1,

(I + A1A)1 11 A1A

11 A1 A .

x A1

1 A1 A [b+ A x]

xx

A A11 A1 A

[ bA x +

AA

].

Ax = b b A x

xx

A A11 A1 A

[bb +

AA

].

144

7.2

A = 0

xx A A

1bb .

7.3 b = 0

xx =

A A11 A1 A

AA .

(i) A , 7.2 ( A = 0), (A) = A1 A.

(ii) (A) , A b A b x.

(iii) = A1 A A1 A = I = 1.

A ,

(A) = A A1 (3.134) Ax = b. (A) - , A b x -.

(ill-conditioned).

3.7.2. A ,

(A) =

1n , (3.135)

1 n A, .

. .

145

4

4.1

Au = b (4.1)

A u, b - . A ( 106), .

-

. ,

.

.

u(n+1) = Gu(n) + k, n = 0, 1, 2, . . . (4.2)

G k ., (4.2)

:

146

1. (4.1), -

().

2. (4.2)

, (4.1) ( -

).

(4.2)

u(0).

4.1.1. (4.2)

S(G) < 1. (4.3)

. u u(n), n =0, 1, 2, . . . e(n) = u(n) u,

e(n+1) = Ge(n) (4.4)

u

u = Gu+ k. (4.5)

(4.4)

e(n) = Gne(0). (4.6)

limn

u(n) = u limn

e(n) = 0 (4.6)

limn

(G(n)e(0)) = 0 e(0). -

3.6.7 limn

Gn = 0

(4.3).

S(G) < 1, I G (I G)u = k . S(G) < 1 lim

nGn = 0 lim

n Gn = 0.

Gne(0) Gn e(0) limn

Gne(0) = 0

(4.6) limn

e(n) = 0 limn

u(n) = u,

(4.2) .

147

G

G < 1 (4.7) = 1 S(G) G . -

(4.2) ,

u(0)( ) - (4.2) n = 0, 1, 2, . . .. ,

:

u(n+1) u(n)

u(n+1) u(n) u(n+1)

= 1 2 . (4.2), -

.

e(n) 0 n . (4.6) u(0) 6= u

e(n) / e(0) Gn . (4.8)

Gn norm e(0) . n

Gn . (4.9)

n

Gn 1

n log /(1nlog Gn) (4.10)

x x. (4.10)

148

(4.2). -

( 1nlogGn).

Rn(G) = 1nlog Gn . (4.11)

,

R(G) = limn

Rn(G) = logS(G) (4.12)

[34]

S(G) = limn

( Gn 1n ).

( )

(4.2) -

n logR(G)

. (4.13)

(4.12) -

G . -

-

-

.

4.2

(4.2). -

(4.1) A1, . -

A1 , (4.1)

R1, R1Au = R1b, (4.14)

149

R - (

Rs = t ). (4.14)

u(n+1) = u(n) + R1(bAu(n)), n = 0, 1, . . . (4.15) 6= 0

(4.15). -

(4.15)