.

13 2008

1

- . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 (BOLZANO) . . . . . . . . . . . . . . . . . . . . 1.2 . . . . . . . . . . . . . . 1.3 MULLER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 x = g(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Aitken . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 NEWTON - RAPHSON . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Newton-Raphson (Halley) . . . . . . . . . . . . . 1.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 - . . . . . . . . . . . . . . . . 1.6.1 Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 x = g(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 GAUSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 GAUSS-JORDAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 L-U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 JACOBI() . . . . . . . . . . . . . . . . . . . . . . . . 2.5 GAUSS SEIDEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 . . . . . . . . . . 2.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Aitken . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 . . . . . . . . . . . . . . . . . . . 2.7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 4 10 11 14 15 18 19 22 23 25 28 31 32 35 36 39 40 42 42 42 43 44 46 46 47 48

2

VI

3

& . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 LAGRANGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Lagrange . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Lagrange . . . . . . . . . . . . . . . . . . . . . . . . 3.2 xi . . . . . . . . . . . 3.2.1 xi . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 (HERMITE) . . . . . . . . . . . . . . . . . . 3.4 TAYLOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 SPLINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 50 51 52 54 56 57 58 59 65

4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 NEWTON-COTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Simpson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Simpson(3/8) . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Romberg . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Splines . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 . . . . . . . . . . . . . . 5.2.1 Filon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 GAUSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Gauss-Legendre . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Gauss . . . . . . . . . . . . . . . . . . . . . . . 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 75 78 80 80 80 82 84 86 88 89 91 93

5

6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.1.1 Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.1.2 Euler & Euler - Heun . . . . . . . . . . . . . . . . . . . . . . 97 6.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.1.4 Runge-Kutta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2 . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.1 Adams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.2 Milne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.3 - . . . . . . . . . . . . . . . . . . . . . 105 6.4 . . . . . . . . . . . . . . . . . . . . 107 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

VII

7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 7.1 . . . . . . . . . . . . . . . . . . . . . 111 7.1.1 . . . . . . . . . . . . . . . . . . . . . . . 111 7.1.2 . . . . . . . . . . . . . . . . . . . . . 114 7.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 . . . . . . . . . . . . . . . . . . . . 117 7.3 . . . . . . . . . . . . . . . . . . . . 119 7.3.1 . . . . . 121 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 . . . . . . . . . .125 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 8.1.1 Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.2.1 . . . . . . . . . . 131 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.3.1 : . . . . . . . . . . . . . . . . . 136 8.3.2 : Crank-Nicholson . . . . . . . . . . 138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141 .0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 .0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 .0.3 . . . . . . . . . . . . . . . . . . . . . . . . 143 .0.4 O(hn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 .0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 .0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147

1 -

, f (x) , x (a, b) :

f () = 0

(1.1)

- . - , . , ,

xn+1 = (xn )

(1.2)

x0 , x1 , . . . , x , . . . (1.1). : , x0 .

: K , , F x. :

F =K

12 x

1+

x 12

N

1

(1.3)

2

1 -

N . , (1.3). , N , , K NK ,

f (NK ) = F

12K x

1+

x 12

NK

1 =0

(1.4)

,

f (N ) = 0 .

(1.5)

, , . , , . ,

f (x) = 0 .

(1.6)

, , .

1.1 (BOLZANO) , . 1000 50 65000. ( ) , x1 = 0.10 x2 = 0.15, :

f (0.10) = 3286.4 f (0.15) = 3881.8 0.10 ( 10% ), 3286.4, 0.15 3881.8. , 0.125, (0.1, 0.15). x3 = 0.125 : 174.5 , (0.10, 0.125). , , x4 = 0.1125,

f (0.125) = 174.5

1.1 (BOLZANO)

3

f (0.1125) = 1585.6 . (0.1125, 0.125), , x5 = (0.1125 + 0.125)/2, . , 50 . Bolzano. : [a0 , b0 ], f (a0 ) f (b0 ) < 0. 0 = (a0 + b0 )/2, : () f (0 ) f (a0 ) < 0 () f (0 ) f (b0 ) < 0 () f (0 ) = 0 .

(), , [0 , b0 ] (II) [a1 , b1 ] = (1.7) [a0 , 0 ] (I)

1.1.

4

1 - REPEAT SET x3 = (x1 + x2 )/2 IF f (x3 ) f (x1 ) < 0 SET x2 = x3 ELSE SET x1 = x3 ENDIF UNTIL (|x1 x2 | < E) OR f (x3 ) = 0

1.1. - .

: ,

n = | xn | xn .

n

1 |an bn | 2

(1.8)

n+1 =

n 0 n1 = 2 = = n+1 2 2 2

(1.9)

. E :

n = log2

0 E

(1.10)

, 0 , E (1.10) , n, .

1.2 , . ,

1.2

5

. . : [x1 , x2 ] f (x), f (x1 ) f (x2 ) < 0, (x1 , f (x1 )) (x2 , f (x2 )) :

y (x) = f (x1 ) +

f (x1 ) f (x2 ) (x x1 ) x1 x2

(1.11)

Ox x3 :

x3 =

x2 f (x1 ) x1 f (x2 ) f (x1 ) f (x2 ) f (x2 ) = x2 (x2 x1 ) f (x2 ) f (x1 )

(1.12)

[x1 , x2 ] f (x) Ox . (1.12) . (1.12) . ( ) , : () f (x1 ) f (x3 ) < 0 x2 = x3 () f (x2 ) f (x3 ) < 0 x1 = x3 () f (x3 ) = 0

(1.12) :

xn+2 = xn+1

f (xn+1 ) (xn+1 xn ) . f (xn+1 ) f (xn )

(1.13)

xn xn+1 xn+2 - . ( ) , x1 = 0.1 x2 = 0.15

6

1 -

1.2.

x3 = 0.1229 f (x3 ) = 122.1 , , , , : x4 = 0.12375 f (x4 ) = 4.4. 1 2 3 4

x10.10 0.1229 0.12375 0.12378

x20.15 0.15 0.15 0.15

x30.1229 0.12375 0.1237787 0.1237798

f (x3 )122.114 4.361 0.156 0.00571

1.2.

, , , , . :

1.2

7

.

: () (x1 , x2 ) REPEAT SET x3 = x2 f (x2 ) f (xx2 x1 ) )f (x IF f (x3 ) f (x1 ) < 0 SET x2 = x3 ELSE SET x1 = x3 ENDIF UNTIL |f (x3 )| < E 1.3. .2 1

( 1.3) , , , x2 . , , , . 1.2 . , , x1 x2 (1.13) x3 , x3 x1 , x2 , |f (x)| . , : () (x1 , x2 ) REPEAT x x SET x3 = x2 f (x2 ) f (x 2 1 ) )f (x2 1

(x1 ) | < |f (x2 ) | SET x2 = x3 ELSE SET x1 = x3ENDIF UNTIL |f (x3 )| < E 1.4. .

IF |f

f (x) x2

8

1 -

f (x2 )/2 . : () ( )

F1 = f (x1 ) F2 = f (x2 ) S = f (x1 )REPEAT x SET x3 = x2 F2 F2 x1 2 F1 IF f (x3 ) F1 < 0 SET x2 = x3 SET F2 = f (x3 ) IF f (x3 ) S > 0 SET F1 = F1 /2 ENDIF ELSE SET x1 = x3 SET F1 = f (x3 ) IF f (x3 ) S > 0 SET F2 = F2 /2 ENDIF ENDIF SET save = f (x3 ) UNTIL |f (x3 )| < E 1.5.

. n = | xn | f (x) = 0 x = xn ,

n+1 = k 1.618 n

(1.14)

, .

xn+2 = xn+1

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

(1.15)

(1.16)

1.2

9

f () = 0. , n ( ) xn . Taylor x1 , x2 x3

f (xi ) = f ( + i ) = f () + i f () +

2 i f () 2

(1.17)

+ n+2 = + n+1

n+1 f () + 2 f ()/2 n+1 (n+1 n ) 1 f () (n+1 n ) + 2 f () 2 2 n n+1 n f () 2 f () f () 2f ()

:

n+2 = n+1 1 1

= n+1 n

n + 2 n + 1 n. n+1 = k m k n m . , n + 1 n + 2 :

n+1 = k m n n+2 = k m n+1 1 m1

k m = n+1 n n+1

f () 2f ()

n+1 =

1 k

1

m1 n A m1

1

A =

f () 2f ()

(1.18)

n+1 n k m.

k

m n

=

A k

1 m1

n

1 m1

k= m=

A 1/(m1) k 1 m1

m2 m 1 = 0

m=

f () 1 5 = 1.618 k m = A = 2 2f ()

:

n+1 = k 1.618 n

(1.19)

.

10

1 -

1.3 MULLER Muller . xi2 , xi1 , xi , f (x) 2- P (x) . xi : f (xi ) P (xi ) = Ax2 + Bxi + (1.20) i

3

B = (2q + 1) P (xi ) (1 + q) P (xi1 ) + q 2 P (xi2 ) = (1 + q) P (xi )

A = qP (xi ) q (1 + q) P (xi1 ) + q 2 P (xi2 )2

(1.21)

xi xi1 . (1.22) xi1 xi2 xi+1 q= Ax2 + Bx + = 0 . (1.23)

xi+1 = xi (xi xi1 )

. :

2 B B 2 4A

(1.24)

n+1 = k1.84 n

(1.25)

. : .

1.4 x = g(x)

11

1.4 x = g(x) , g (x) , p = g (p). pn+1 = g (pn ) n = 0, 1, ... . : g (x) pn , . lim pn = p,n

p g (x). f (x) f () = 0 x = g (x) , , f () = 0 = g(), xn = g(xn ) . ,

xn+1 = g (xn ) n

(1.26) (1.27)

lim xn = .

: I = ( , + ) f (x) = 0 , x = g(x) |g (x)| < 1, x0 I n . , f (x) I . xn I ,

xn+1 = g(xn ) g() = |g (xn )| < 1,

g(xn ) g() (xn ) = g (n )(xn ) xn

|xn+1 | |xn | < |xn | n

|xn | n |x0 | lim |xn | = 0n

, 1 ,

1 =

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

|g ()| < 1

| 1 | | 1 | < | 1 | . I .

12

1 -

1.3. x = g(x) g (x) < 1.

1.3 1.4 . 1.3 |g (x)| < 1 1.4 |g (x)| > 1 (1.4), , :

xn+1 = g (xn ) =

12K F

1+

xn 12

N

1

, , |g (x)| > 1 x [0.05, 0.15] , . ,

xn+1 = g1 (xn ) = 12

xn F 1 12k

1/N

1

( ) |g1 (x)| < 1 (.. x = 0.1, g1 (x) 0.108)

n n = xn , :

1.4 x = g(x)

13

1.4. x = g(x) g (x) > 1. 1 2 3 4 5 ... 29

xn0.1 0.1043 0.1080 0.1111 0.1136 ... 0.12376 1.6.

(xn )/-0.157 -0.1275 -0.1028 -0.0825 ... -0.000135

14

1 -

n+1 = xn+1 = g () g(xn ) = g ()( xn ) = g () n g () , y () = g (). :

n+1 = g ()n

(1.28)

, g (x) . f (x) = x+ln(x) [0.1, 1]. : |g (x)| =xn

xn+1 = ln(xn )

1 [0.1, 1] . xn+1 = e |y (x)| = |ex | e0.1 0.9 < 1 . : x = (x + ex )/2 1 |g (x)| = 1 |1 ex | 2 |1 e1 | = 0.316 . 2 x 1 1 x = x+2e |g (x)| = 3 |12ex| 3 |12e1 | = 0.03 . 3 xn + 2exn . 3

1 x

:

xn+1 =

1.4.1 Aitken , , en+1 = g ()en , , n , , . n xn+1 = g (xn ) :

xn+1 g ()( xn ) n + 1 :

xn+2 g ()( xn+1 )

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

1.5 NEWTON - RAPHSON

15

:

= xn

(xn xn1 )2 xn 2xn1 + xn2

(1.29)

Aitken. 1.6 x = g(x) x0 = 0.1080, x1 = 0.1111 x2 = 0.1136, Aitken r = 0.1240 = 0.00024. 25 x = g(x).

1.5 NEWTON - RAPHSON f (x) f (x) f (x) , , f (x) = 0 . Newton-Raphson , 1 f (x). x0 , . (x0 , f (x0 )) Ox x1 f (x1 ) = 0. , x1 x0 A (. 1.5) :

tan = f (x0 ) =

f (x0 ) x1 x0

(1.30)

x1 :

x1 = x0

f (x0 ) . f (x0 )

(1.31)

x1 x2 . 1 2 3

x00.15 0.1247 0.12378

x10.1247 0.1237810 0.1237798

f (x1 )-132.475 -0.01681 -0.00101

1.7. Newton-Raphson .

16

1 -

1.5. Newton-Raphson.

(1.31) . xn+1 xn xn+1 xn+1 = xn + n . :

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

2 n f (xn ) + 2

, xn+1 f (x), f (xn+1 ) = 0,

0 = f (xn ) + n f (xn )

1.5 NEWTON - RAPHSON

17

n =

f (xn ) f (xn )

(1.32)

:

xn+1 = xn

f (xn ) f (xn )

(1.33)

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

+ n+1 = + n

1 f () + n f () + 2 2 f () f ( + n ) n = + n f ( + n ) f () + n f ()

f () = 0

1 f () 1 n 1 + f ()/f () f ()

n+1 =

f () 2 2f () n

(1.34)

, , . Newton-Raphson . : 1

Newton-Raphson a. a :

f (x) = x2 a f (x) = 2x

18

1 -

(1.33)

xn+1 = xn :

x2 a n 2xn

xn+1 =

1 2

xn +

a xn

.

(1.35)

COMPUTE f (x1 ), f (x1 ) SET x2 = x1 IF (f (x1 ) = 0) END (f (x1 ) = 0) REPEAT SET x1 = x1 SET x2 = x1 f (x1 )/f (x1 ) UNTIL (|x1 x2 | < E) OR (|f (x2 )| < E ) ENDIF 1.8. Newton-Raphson.

1.5.1 Newton-Raphson (Halley) Newton-Raphson . Taylor . n = xn+1 xn ,

f (xn+1 ) = f (xn + n ) = f (xn ) + n f (xn ) +, f (xn+1 ) 0,

2 n f (xn ) + . . . 2

f (xn ) + n f (xn ) +

n f (xn ) = 0 2

n

n =

f (xn )

f (xn ) n + 2 f (xn )

n (1.32) Newton - Raphson , :

1.5 NEWTON - RAPHSON

19

n :

f (xn ) f (xn )

xn+1 = xn

f (xn ) f (x1 ) f (xn )f (xn ) 2f (xn )

= xn

2f 2 (xn )

2f (xn ) f (xn ) (1.36) f (xn ) f (xn )

f (xn ) = 0 NewtonRaphson, (1.33). Halley. Halley ( ).

n+1 =

1 f () 1 6 f () 4

f () f ()

2

3 n

(1.37)

Halley. a :

xn+1 =

x3 + 3xn a n 3x2 + a n

(1.38)

f (x) = x2 a f (x) = 2x f (x) = 2 Halley . 9. f (x) = x2 9 = 0. x0 = 15 ( ). . 1.5.2 f (x) = 0 , f (x) = (x)m q(x) m . , f (x) = (x )m1 [mq(x) + (x )q (x)] f (x) f (x) x = f (x)/f (x) .

20

1 - Newton-Raphson x0 =15 x1 =7.8 x2 =4.477 x3 =3.2436 x4 =3.0092 x5 =3.00001 Halley x0 =15 x1 =5.526 x2 =3.16024 x3 =3.00011 x4 =3.0000000... x5 = 3.0000000...

0 1 2 3 4 5

= 12 = 4.8 = 1.477 = 0.243 = 9.15 103 = 1.39 105

0 1 2 3 4 5

= 12 = 2.5 = 0.16 = 1.05 104 = 3.24 1014 = 0.0

1.9. Newton-Raphson Halley 9.

(x) =

m = 1 :

(x )q(x) f (x) = (x) f mq(x) + (x )q (x)

xn+1 = xn

f (xn )/f (xn ) (xn ) = xn (x ) (x )]2 f (x )f (x )} /[f (x )]2 n {[f n n n n f (xn ) f (xn ) = xn (1.39) [f (xn )]2 f (xn )f (xn )

, Newton-Raphson (x) = 0 . f (x) = x4 4x2 + 4 = 0 = 2 = 1.414213.... . Newton-Raphson

xn+1 = xn

x2 2 n 4xn

(A)

, (1.39),

xn+1 = xn

(x2 2)xn n x2 + 2 n

(B).

x () () () () x0 1.5 8.6102 1.5 8.6102 2 x1 1.458333333 4.410 1.411764706 -2.4103 2 x2 1.436607143 2.210 1.414211439 -2.1106 2 x3 1.425497619 1.110 1.414213562 -1.61012

1.5 NEWTON - RAPHSON

21

. ( ) . Newton-Raphson. :

|n+1 | =a n |n | lim :

|n | a|n1 | a2 |n2 | ... an |0 | n |n |:

n

:

log10 |n | log10 |0 | log10 |a| |n+1 | =b |n |2

(1.40)

n

lim

:

|n | b|n1 |2 b3 |n2 |4 b7 |n3 |8 ... b2

n+1

1

|0 |2

n+1

n |n |:

2n+1

n 106 n 106 0 = 0.5 a = b = 0.7 (1.40)

log10 |n | + log10 |b| log10 |0 | + log10 |b|

(1.41)

n

log10 |106 | log10 |0.5| 37 log10 |0.7|

37 .

22

1 -

2n+1

log10 |106 | + log10 |0.7| 13.5 log10 |0.5| + log10 |0.7|

3 . , n 1014 89 4 !

3

2

(1.56,1.25)

1

0

(-1.98,-0.29)-1

-2

-4

-2

0

2

4

1.6. - (1.42).

1.6 - - - . Newton-Raphson x = g(x). - . :

f (x, y) = ex 3y 1 g(x, y) = x + y 42 2

(1.42)

1.6 -

23

f (x, y) = 0 g (x, y) = 0 f (x, y) = 0 g (x, y) = 0 xy (. 1.6). 1.6.1 Newton Newton NewtonRaphson . - N N .

f (x, y) = 0 g (x, y) = 0 , n + 1 (xn+1 , yn+1 ) , f (xn+1 , yn+1 ) 0 g(xn+1 , yn+1 ) 0. = = (xn , yn ) xn+1 = xn + n yn+1 = yn + n , Taylor , :

f f 0 f (xn+1 , yn+1 ) = f (xn + n , yn + n ) f (xn , yn ) + n + n = = x y g g + n 0 = g(xn+1 , yn+1 ) = g(xn + n , yn + n ) = g(xn , yn ) + n x y n n

n = n =

f g g f x y x y g f g x + f x f g g f x y x y

f g + g f y y

(1.43)

(1.44)

n n , NewtonRaphson 2 - :

24

1 -

xn+1 = xn yn+1

f gy g fy fx gy gx fy g fx f gx = yn fx gy gx fy

(1.45) (1.46)

fx = f /x. N N :

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

N (x1 , x2 , ..., xN ). (x1 , x2 , ..., xN ) n+1 n+1 n+1 , N - (x1 , x2 , ..., xN ) n n n

x1 = x1 + x1 n+1 n n.. .. ..

xN = xN + xN n+1 n n :

0 f1 (x1 , ..., xN ) = f1 (x1 + x1 , ..., xN + xN ) = n+1 n+1 n n n n f1 f1 f1 + 1 x1 + ... + N xN n n x x.. .. .. (1.47)

0 fN (x1 , ..., xN ) = fN (x1 + x1 , ..., xN + xN ) = n+1 n+1 n n n n fN fN x1 + ... + N xN fN + n n x1 x, xi ( i = 1...N ) n

f1 . . .

x1

f1 x2

f1 xN

. . .

fN fN x1 x2

fN xN

N xi . n

1 xn . . . xN n

f1 . = . . fN

(1.48)

1.6 -

25

, N - (x1 , x2 , ..., xN ), 0 0 0 xi n N - (x1 , x2 , ..., xN ) : 1 1 1

x1 = x1 + x1 1 0 0.. .. .. (1.49)

xN 1

=

xN 0

+

xN 0

, , max |xi | < E E . n 1.6.2 x = g(x) 1.4, . N

f1 (x1 , x2 , . . . , xN ) = 0. . . . . . (1.50)

fN (x1 , x2 , . . . , xN ) = 0 :

x1 = F1 (x1 , x2 , . . . , xN ) ... xN = FN (x1 , x2 , . . . , xN )(1.51)

x = g(x). : , N (x1 , . . . , xN )0 (1.51) N - (x1 , . . . , xN )1 . N - ...

26

1 -

N (x1 , . . . , xN )0 (1.51) ( ) (x1 )1 . , (x1 )1 , (x2 , . . . , xN )0 (x2 )1 . (x1 , x2 )1 , (x3 , . . . , xN )0 ... xi = fi (x1 , x2 , . . . , xN ) i = 1, . . . , N ,

f1 f1 f1 + + +

j=1 j=i

|aij |

i = 1, 2, . . . , N

(2.27)

x(k+1) = D1 B Lx(k+1) Ux(k) A =lower

(2.28)

L +

diagonal

D

+ U . L upper

A , D A U A . Gauss-Seidel .

42

2

x(k+1) = y (k+1) z (k+1) :

7 + y (k) z (k) 4 21 + 4x(k) + z (k) = 8 15 + 2x(k) y (k) = 5

(1, 2, 2) (1.75, 3.75, 2, 95)

(1.95, 3.97, 2.99) (1.996, 3.996, 2.999)

, , (2, 4, 3). Jacobi 5 , , Gauss-Seidel , .

2.6 2.6.1 , Gauss . . :

a11 a12 a13 a a 0 a22 a23 = a11 12 23 = a12 a22 a33 0 a33 0 0 a332.6.2 N N , N . , :

ab cd xy z w

xy z w 10 01

.

ab cd

=

x, y, z w :

2.7

43

ab cd

x y

=

1 0

ab cd

y w

=

0 1

Gauss-Jordan , . . , , . () :

a11 a12 ... a1N a21 a22 ... a2N aN 1 aN 2 ... aN N A

1 0 ... 0 0 1 ... 0(2.29)

0 0 ... 1 I

I Gauss Jordan A.

1 0 ... 0 0 1 ... 0 0 0 ... 1

a11 a12 ... a1N a21 a22 ... a2N aN 1 aN 2 ... aN N

(2.30)

aij = A1 .

2.7 N N ( ) ( ) , A x = x (2.31) (, x = 0). A. x, (2.31), A, . ,

1 23 2 2 1 3 1 1 = 1 1 2 01 2 2 A u1 1 u1 u1 = (2, 1, 2) 1 = 1 A. (2.31)

44

2

det (A I) = 0

(2.32)

, (2.32), , A. , (2.7),

3 52 + 3 + 9 = 0 i = 1, 3 3 A. , , . , , . , , , , . , . 2.7.1 /. 1 ,

1 2 3 = 3 52 + 3 + 9 = 0 det 1 3 1 2 0 1

(2.33)

|1 | > |2 | |3 | . . . |N |

(2.34)

, x N u(1) , u(2) , . . . , u(N ) . , u(i) :

Au(i) = i u(i)

(1 i N )

(2.35) (2.36)

(2.36) A, :

x = a1 u(1) + a2 u(2) + + aN u(N )

x(1) Ax = a1 1 u(1) + a2 2 u(2) + + aN N u(N )

(2.37)

k (2.37) A,

x(k) Ak x = a1 k u(1) + a2 k u(2) + + aN k u(N ) 1 2 N = k 1 a1 u(1) + a2 2 1k

u(2) + + aN

N 1

k

u(N )

(2.38)

2.7

45

1 ( 2.34), k (j /1 ) , k . , k :

x(k) = Ak x k a1 u(1) 1 ,

(2.39)

rk

k+1 a1 u(1) x(k+1) Ak+1 x 1k = 1 Ak x x(k) 1 a1 u(1)

(2.40)

1 , k . (2.39) u(1) x(k) .

x(5) = A5 x = (232, 628, 560) .

1 01 A = 1 2 2 1 = 3.4142, 2 = 2 1 03 3 = 0.5858 u(1) = (0.3694, 1, 0.8918) , u(2) = (0, 1, 0) u(3) = (0.7735, 1, 0.3204) . , x = (1, 2, 1) , , , : 68 0 164 A5 = 136 32 428 164 0 396

x(6) = A6 x = (792, 2144, 1912) , (2.40)

232 0 560 A6 = 532 64 1484 560 0 1352

1

2144 x(6) 3.4140 = (5) 628 x

792/232 1912/560. , u(1) x(6) , ( ) (0.3694, 1, 0.8918). , .

46

2

2.7.2 Aitken rk 1 (2.40) 1 k = |rk 1 |. , rk+1 k+1 = |rk+1 1 |. ,

k+1 = Ak

(2.41)

Aitken . , , 1.5 1 , rk , rk+1 , rk+2 . :

1

2 rk rk+2 rk+1 rk+2 2rk+1 + rk

(2.42)

. , r5 = 3.414, r4 = 3.413 r3 = 3.407. Aitken 1 = 3.4142, r5 . 2.7.3 , , . : : A, 1 A1 .

1

: Ax = x. x = A1 (x) = A1 x. A1 x = x, , 1 A1 .

:

, A N |1 | > |2 | |N 1 | > |N | > 0, , A1 1 , i

1 > 1 1 > 0 1 N N 1

(2.43)

, 1 N A1 . 1

rk 1 ,

2.7

47

A1 , 2.6, /. , A1 x Gauss A x(k+1) = x(k) , x(k) k - A x, , x(k) = Ak x. , x(0 , x(1) = A1 x(0) . , Ax(1) = x(0) . , x(1) . , x(k+1) :(k+1)

x(k+1) = A1 x(k)

Ax(k+1) = x(k)

(2.44)

1 01 A = 1 2 2 1 03 (0) x = (1, 2, 1) , Ax(1) = x(0) Gauss. : x(1) = 1, 3 , 0 . 2 3 1 Ax(2) = x(1) x(2) = 2 , 2, 2 ... ,x r6 = x(6) 1.70707, A 3 = 1/1.7070707 = 0.5858.(7)

2.7.4 , . , . : : N i (i = 1, . . . , N) N N A, A I (I: ) (i ) (i = 1, . . . N ). , (N , 1 ) , k 0 < |k | < |i | > . , , |k | A I, , . x(k+1) (A I) x(k+1) = x(k) . , , rk , i :

48

2

i =

1 + rk

(2.45)

, i , . , rk = j , j = rk + .

2.8 1.

det (A), A1 A x = b. A.

0.002 4.000 4.000 A = 2.000 2.906 5.387 3.000 4.031 3.112

7.998 b = 4.481 4.413

3 &

() (). (.. 12:15) 20:00 19:00. ( ) . .. . , , . . . , : LAGRANGE: n- (xi , yi ), xi = xi+1 xi ,

50

3 &

n 1 , n . . xi : n . : n- (xi , yi , yi ). 2n 1. TAYLOR: Taylor n, , n . SPLINES: . , n- (xi , yi ) n 1 ( spline), (xi , yi ) (xi+1 , yi+1 ) (n 1)- n .

3.1 LAGRANGE , n n + 1 . 3 :

x f (x)

x0 3.2 22.0 f0

x1 2.7 17.8 f1

x2 1.0 14.2 f2

x3 4.8 38.3 f3

x4 5.6 51.7 f4

3.1.

, , . ax3 + bx2 + cx+ d = P (x), 4 4 , a, b, c d. : a = 0.5275, b = 6.4952, c = 16.117 d = 24.3499. , :

P (x) = 0.5275x3 + 6.4952x2 16.117x + 24.3499, , , f (3) f (3) = 20.21. Lagrange , . , :

3.1 LAGRANGE

51

P3 (x) =

xi f (xi ). 3.1.1 Lagrange , : . 0 () : P0 (x) = a0 = f1 x, Ox (x1 , f1 ). . 1 : P1 (x) = a0 + a1 x, :

(x x0 )(x x2 )(x x3 ) (x x1 )(x x2 )(x x3 ) f0 + f1 (x0 x1 )(x0 x2 )(x0 x3 ) (x1 x0 )(x1 x2 )(x1 x3 ) (x x0 )(x x1 )(x x2 ) (x x0 )(x x1 )(x x3 ) f2 + f3 + (x2 x0 )(x2 x1 )(x2 x3 ) (x3 x0 )(x3 x2 )(x3 x2 )

P1 (x1 ) = a0 + a1 x1 = f1 P1 (x2 ) = a0 + a1 x2 = f2 :

a0 = :

x2 f1 x1 f2 x2 x1

a1 =

f2 f1 x2 x1

P1 (x) =

x x2 x x1 f2 f1 x2 f1 x1 f2 + x = f1 + f2 x2 x1 x2 x1 x1 x2 x2 x1 = f1 L1 (x) + f2 L2 (x) L1 (x) = x x2 x1 x2

L2 (x) =

Li (x), Lagrange,

x x1 x2 x1

L1 (x1 ) = 1 , L1 (x2 ) = 0 ,

L2 (x1 ) = 0 L2 (x2 ) = 1

: 0 j = k Lj (xk ) = jk = 1 j = k jk Kronecker. n :n

Pn (x) = f0 L0 (x) + f1 L1 (x) + .... + fn Ln (x) =i=0

fi Li (x)

(3.1)

Lj (x) =

(x x1 )(x x2 )...(x xj1 )(x xj+1 )...(x xn ) (xj x1 )(xj x2 )...(xj xj1 )(xj xj+1 )...(xj xn )

(3.2)

52

3 &

3.1.2 Lagrange f (x) x :

E(x) = (x x0 )(x x1 )....(x xn )

f (n+1) () (n + 1)!

(3.3)

n x0 , ..., xn .

F (x) = f (x) P (x) C (x) C

(x) = (x x0 )(x x1 )(x x2 )...(x xn )

(3.4)

, f (xi ) = P (xi ) F (x) x = xi . C :

C=

f (t) P (t) (t)

t F (t) = 0 F (x) n + 2 . Rolle F (x) n + 1 F (x), F (x) n F (x) . F (n+1) (x) x = . P (n+1) (x) = 0

F (n+1) () 0 = f (n+1) () C (n + 1)!

C= :

f (n+1) () (n + 1)!

f (n+1) () (t) (n + 1)! t x0 , x1 , ..., xn x. f (t) P (t) = f (x) P (x) = y = cos(x) y x0 = 0, x1 = 0.5, x2 = 1. .

f (n+1) () (x) . (n + 1)!

(3.5)

3.1 LAGRANGE

53

1

f(x)=cos( x)

f(x), P(x)

P(x)=-2x+10

-1

0.00

0.25

0.50

0.75

1.00

x

3.1. y = cos(x) P (x) = 2x + 1.

:

xi fi

0 1

0.5 0.0

1 -1

Lagrange :

L1 =

(x x2 )(x x3 ) (x 0.5)(x 1) = = 2x2 3x + 1, (x1 x2 )(x1 x3 ) (0 0.5)(0 1) (x x1 ) (x x3 ) (x 0)(x 1) L2 = = = 4(x2 x) (x2 x1 )(x2 x3 ) (0.5 0)(0.5 1) (x 0)(x 0.5) (x x1 )(x x2 ) = = 2x2 x L3 = (x3 x1 )(x3 x2 ) (1 0)(1 0.5)

(3.6)

P (x) = 1 (2x2 3x + 1) 0.4 (x2 x) + (1) (2x2 x) = 2x + 1 , 3.1 . :

E(x) = x (x 0.5) (x 1) x = 0.25 : E(0.25) 0.24.

3 sin() 3!

54

3 &

3.2 xi xi , , . , fi = fi+1 fi . fi = fi fi1 1

f0 = f1 f0 , f1 = f2 f1 , ...

fi = fi+1 fi 2

(3.7)

2 f1 = (f1 ) = (f2 f1 ) = f2 f1 = (f3 f2 )(f2 f1 ) = f3 2f2 +f1

2 fi = fi+2 2fi+1 + fi 3

(3.8)

3 f1 = 2 (2 fi ) = f4 3f3 + 3f2 f1 3 fi = fi+3 3fi+2 + 3fi+1 fi n- (3.9)

n fi = fi+n nfi+n1 +

n(n 1) n(n 1)(n 2) fi+n2 fi+n3 +... (3.10) 2! 3!

, n- (xi , yi ). . , 3.2. n :

Pn (xs ) = f0 + s f0 + = f0 +n

s 1 s i

s(s 1)(s 2) s(s 1) 2 f0 + 3 f0 + ... 2! 3! s s f0 + 2 f0 + 3 f0 + ... 2 3(3.11)

=i=0

i f0

xs = x0 + s h. :

3.2 xi

55

x0.0 0.2 0.4 0.6 0.8 1.0

f (x)0.000

f (x)0.203

2 f (x) 3 f (x) 4 f (x) 5 f (x)

0.203 0.220 0.423 0.261 0.684 0.246 1.030 0.527 1.557

0.017 0.024 0.041 0.044 0.085 0.096 0.181 0.052 0.020 0.032

3.2.

s = 0 Pn (x0 ) = f0

s = 1 Pn (x1 ) = f0 + f0 = f1 s = 2 Pn (x2 ) = f0 + 2f0 + 2 f0 = f2

s = 0 s = 1 .

s = m, :n

P (xm ) =i=0

m i

i f0

s = m + 1,

P (xm+1 ) = P (xm ) + P (xm )n

=i=0

m in

n

i f0 + m i

i=0

m in

i+1 f0 m i1 i f0 + n+1 f0

= f0 +i=1 n

i f0 +

i=1

= f0 +i=1 n+1

m+1 i

i f0 + n+i f0(3.12)

=i=0

m+1 i

i f0

k+1 n

=

k n

+

k n1

.

56

3 &

: s , x :

s=3.2.1 xi

x x0 . h

(3.13)

Newton - Gregory ( x0 xn )

Pn (x) = f0 +

s 1

f0 +

s 2

2 f0 + ... +

s n

n f0

(3.14)

Newton - Gregory ( xn x0 )

Pn (x) = f0 +

s s+1 s+2 s+n1 f1 + 2 f2 + 3 f3 +...+ n fn 1 2 3 n(3.15)

3.3, Newton-Gregory.

x0.2 0.5 0.8 1.1 1.4 1.7

F (x)1.06894

f (x)0.11242

2 f (x)

3 f (x)

4 f (x)

5 f (x)

1.18136 0.12425 1.30561 0.13731 1.44292 0.15175 1.59467 0.16771 1.76238

0.01183 0.00123 0.01306 0.00138 0.01444 0.0152 0.01596 0.00014 0.00015 0.000001

3.3.

Newton Gregory x = 0.5 x = 1.4, x0 = 0.5

s s + 0.00138 2 3 = 1.18136 + 0.12425 s + 0.01306 s (s 1)/2 + 0.00138 s (s 1) (s 2)/6 = 0.9996 + 0.3354 x + 0.052 x2 + 0.085 x3 P3 (x) = 1.18136 + 0.12425 s + 0.01306

3.3 (HERMITE)

57

Newton Gregory x = 1.1 x = 0.2, x0 = 1.1

s+1 s+2 + 0.00123 2 3 = 1.44292 + 0.13731 s + 0.01306 s (s + 1)/2 + 0.00123 s (s + 1) (s + 2)/6 = 0.99996 + 0.33374 x + 0.05433 x2 + 0.007593 x3 P3 (x) = 1.44292 + 0.13731 s + 0.01306

3.3 (HERMITE) , . , n ,

P (x) = y(x) P (x) = y (x) x0, x1 , ..., xn 2n , 2n 1 . Hermite.n n

P2n1 (x) =i=1

Ai (x)yi +i=1

Bi (x)yi

(3.16)

Bi (x) = (x xi ) [Li (x)] Li (x) Lagrange.

Ai (x) = [1 2(x xi )L (xi )] [Li (x)]2 i2

(3.17) (3.18)

2n 1

y(x) p(x) =

y (2n+1) () 2 [(x x1 )(x x2 )....(x xn )] (2n + 1)!

(3.19)

58

3 &

k 0 1

xk 0 4

yk 0 2

yk 0 0

Lagrange :

L0 (x) =

x x1 x0 x1 x x0 L1 (x) = x1 x0 1 L0 (x) = x0 x1 1 L (x) = 1 x1 x0

x4 x4 = 04 4 x = 4 1 = 4 1 = 4 =

:

A0 (x) = [1 2 L (x x0 )] L2 = 1 2 0 0

1 4

(x 0) 2

x4 4 = 3 x 42

2

x 1 A1 (x) = [1 2 L (x x1 )] L2 = 1 2 (x 4) 0 1 4 4 B0 (x) = (x 0) x4 42

x x 2 4

2

=x

x4 4

2

B1 (x) = (x 4)

Hermite :

P (x) = (6 x)

x2 . 16

3.4 TAYLOR P (x) f (x) ( ) ( ). Taylor x0 , , , n, n , :

3.5 SPLINES

59

P (i) (x0 ) = f (i) (x0 ) i = 0, 1, ..., n Taylor :n

P (x) =i=0

f (i) (x) (x x0 )i i!

(3.20)

( ) :

EN (x) =

xN +1 f (N +1) () (N + 1)!

(3.21)

n x0 = 0 ex . ex ex ,

y0 = y0 = y0 = ... = y0 :n

(1)

(2)

(n)

=1

p(x) =i=1

1 n x2 1 x =1+x+ + ... + xn i! 2 n! e (n + 1)!

:

E = xn+1

Taylor 6 5 107 x [0, 1]. , x = 1 :

e n+1 n 10. , E = 0.00000005 10 .

3.5 SPLINES , .

60

3 &

6

4

2

0

Y-2 -4 -6 -1.0 -0.5 0.0 0.5 1.0

X

3.2. (3.22) 4 (3.23).

:

(-1,1) P (x) = 1 26x2 + 25x4 (3.23) (3.22). . . , . , 3 (xi , yi ) (xi+1 , yi+1 ), 4 , , , , (xi , yi ) . (xi , yi )

0 f (x) = 1 5|x| 0

1 x 0.2 0.2 < x < 0.2 0.2 x 1.0

(3.22)

(xi+1 , yi+1 )

3.5 SPLINES

61

y(x) = ai (x xi )3 + bi (x xi )2 + ci (x xi ) + di , :

yi = ai (xi xi )3 + bi (xi xi )2 + ci (xi xi ) + di = di

yi+1 = ai (xi+1 xi )3 + bi (xi+1 xi )2 + ci (xi+1 xi ) + di = ai h3 + bi h2 + ci hi + di i i hi = xi+1 xi . , , :

y (x) = 3ai (x xi )2 + 2bi (x xi ) + ci y (x) = 6ai (x xi ) + 2bi Si xi Si+1 xi+1 . :

Si = 6ai (xi xi ) + 2bi = 2bi Si+1 = 6ai (xi+1 xi ) + 2bi = 6ai hi + 2bi

bi =

Si , 2

ai =

Si+1 Si 6hi

(3.24)

yi+1 =

Si+1 Si 3 Si 2 h i + h i + ci h i + y i 6hi 2 yi+1 yi 2hi Si + hi Si+1 hi 6(3.25)

ci =

xi , yi = 3ai (xi xi )2 + 2bi (xi xi ) + ci = ci

yi = 3ai1 (xi xi1 )2 + 2bi1 (xi xi1 ) + ci1 = 3ai1 h2 + 2bi1 hi1 + ci1 i1

: yi =

2hi Si + hi Si+1 yi+1 yi hi 6 Si1 2hi1 Si1 + hi1 Si yi yi1 Si Si1 h2 + 2 hi1 + =3 i1 6hi1 2 hi1 6

62

3 &

hi1 Si1 + 2 (hi1 + hi ) Si + hi Si+1 = 6

yi+1 yi yi yi1 hi hi1

(3.26)

n + 1 , n 1 (x0 , xn ). , n 1 n + 1 Si . S0 Sn . . . . S0 = 0 Sn = 0 ( spline). . S0 = S1 Sn = Sn1 . . S0 S1 , S2 Sn Sn1 , Sn2 , Si x0 x1

S2 S1 (h0 + h1 )S1 h0 S2 S1 S0 = S0 = h0 h1 h1 xn1 xn

Sn Sn1 Sn1 Sn2 (hn2 + hn1 )Sn1 hn1 Sn2 = Sn = hn1 hn2 hn2IV. . f (x0 ) = A f (xn ) = B

y1 y0 A h0 yn yn1 hn1 Sn1 + 2hn Sn = 6 B hn1 2h0 S0 + h1 S1 = 6 n + 1 :

3.5 SPLINES

63

.......

, n 1 n + 1 Si . S0 Sn n1 n1 . S0 Sn : : S0 = 0 Sn = 0, :

y2 y1 y1 y0 h1 h0 y3 y2 y2 y1 h2 h1 =6 .... yn yn1 y y hn1 n1n2n2 h ....

h0 2(h0 + h1 ) h1 h1 2(h1 + h2 ) h2 h2 2(h2 + h3 ) h3 ....... ....... hn2 2(hn2 + hn1 ) hn1 .... =Y

S0 S1 S2 : Sn2 Sn1 Sn

S1 2(h0 + h1 ) h1 S2 h1 2(h1 + h2 ) h2 S3 = Y h2 2(h2 + h3 ) h3 .... Sn1 hn2 2(hn2 + hn1 ) : S0 = S1 Sn = Sn1 , :

: S0 Sn :

3h0 + 2h1 h1 S1 h1 S2 2(h1 + h2 ) h2 S3 = Y h2 2(h2 + h3 ) h3 ... ... hn2 2hn2 + 3hn1 Sn1

(h

0 +h1 )(h0 +2h1 )

h 1

h1

S1 2(h1 + h2 ) h2 S2 h2 2(h2 + h3 ) h3 S3 = Y .... .... h2 h2 n2 n1 (hn1 +hn2 )(hn1 +2hn2 ) Sn1hn2 hn2

h2 h2 1 0 h1

S0 Sn .

64

3 &

IV : f (x0 ) = A, f (xn ) = B

ai , bi , ci di :

2h0 h1 S1 h0 2(h0 + h1 ) h1 S2 S3 = Y h1 2(h1 + h2 ) h2 .... hn2 2hn1 Sn1

ai =

Si+1 Si 6hi Si bi = 2 2hi Si + hi Si+1 yi+1 yi ci = hi 6 di = yi

(3.27) (3.28) (3.29) (3.30)

I . III . IV , . spline (

x y

0 -8

1 -7

2 0

3 19

4 56

y = x3 8.) . :

410 36 S1 1 4 1 S2 = 72 014 S3 108

S0 S1 S2 S3 S4

=0 = 6.4285 = 10.2857 = 24.4285 =0

3.6

65

:

:

s1 1 4 01

0 S1 36 S1 = S0 = 4.8 1 S2 = 72 S2 = 1.2 s S3 108 S3 = 19.2 = S4

IV:

60 1 4 00

0 S1 36 S0 = 0 S1 = 6 1 S2 = 72 S2 = 12 S3 = 18 6 S3 108 S4 = 24 00 S0 6 0 0 S1 36 1 0 S2 = 72 4 1 S3 108 12 S4 66 S0 S1 S2 S3 S4 =0 =6 = 12 = 18 = 24

Si 2 y = x3 8 IV.

210 1 4 1 0 1 4 0 0 1 000

3.6 1. y(x) = sin x 2 (), x = 0, x1 = 0.5 x2 = 1. y(x) p(x). 2. y(x) = log(x + 1) (), x = 0, x1 = 4, x2 = 9. y(x) p(x). 3. Lagrange xk , yk .

xk yk

0 1

1 -1

4 1

6 -1

, x = 2, 3, 5. 4. Lagrange xk , yk . , x = 2 x = 3. 5. Newton , :

66

3 &

xk yk

0 0

1 16

4 48

5 88

6 0

xk yk

1 1

2 -1

3 1

4 -1

5 1

6. , y(x) = x4 x = 0, 1, 2. 7. , y(x) = x x = 0, 1, 4. 8. , :

xk yk

2 0

4 0

6 1

8 0

10 0

9. , :

xk yk

0 1

1 2

2 4

3 7

4 11

5 16

6 22

7 29

10. Hermite 3 ( p(x) = 2x2 x3 ):

xk0 1

yk0 1

yk

0 1

11. Hermite ( p(x) = x4 4x3 + 4x2 ):

xk0 1 2

yk0 1 0

yk

0 0 0

3.6

67

12. ( p(x) = x3 x2 + 1):

xk0 1

yk1 1

yk

-2 4

13. , p1 (0) = p (0) = 0 1 p2 (4) = 2 p (4) = 0, 2 (2, 1). p (2) = p (2). 1 2 14. Taylor n sin x cos x, x0 = 0. Taylor , x (0, /2) 15. 0.0 (sec) 0.302 0.2 0.185 0.3 0.106 0.4 0.093 0.5 0.24 0.6 0.579 0.7 0.561 0.8 0.468 1.0 0.302

t = 0.35 splines 0.05. 16. ( ) . 00 150 300 450 600 750 900 1 1855.4 m 1792.0 m 1608.2 m 1314.2 m 930.0 m 481.7 m 0.0 m g 9.7805 m/sec2 9.7839 m/sec2 9.7934 m/sec2 9.8063 m/sec2 9.8192 m/sec2 9.8287 m/sec2 9.8322 m/sec2

68

3 &

1 / (400 37) g .

4

, . , . , , . , . , y = y (x) .

4.1 y(x) P (x) . , Newton , x = x0 + sh

y(x) P (x) = y0 + sy0 +

s(s 1)(s 2) 3 s(s 1) 2 y0 + y0 + (4.1) 2! 3!

, :

2s 1 2 dP (x) 1 dP (s) 1 3s2 6s + 2 3 dy(x) y0 + = = = y0 + y0 + dx dx s ds h 2! 3!(4.2)

70

4

, x0 s = 0, x1 s = 1 ..., : y0 =

1 1 1 y0 2 y0 + 3 y0 + h 2 3

(4.3)

, : y0 =

y1 y0 + O(h) h 3y0 4y1 + y2 + O(h2 ) y0 = 2h 11y0 18y1 + 9y2 2y3 y0 = + O(h3 ) 6h ...

(4.4) (4.5) (4.6)

y(x) x0 . :

1 d2 P (s) 1 d2 P (x) = 2 = 2 2 y0 + (s 1)3 y0 + 2 dx h ds2 h y0 =

(4.7)

y0 2y1 + y2 + O(h) h2

(4.8)

y0 =

2y0 5y1 + 4y2 y3 + O(h2 ) h2

(4.9)

... . , (3.3) n ( n + 1 ) :

E(x) = (x x0 )(x x1 )(x x2 )...(x xn )

y n+1 () (n + 1)!

(4.10)

.. x = x0 , :

E (x0 ) = (x0 x1 )(x0 x2 )...(x0 xn )

y n+1 () (n + 1)!(4.11)

h = xi+1 xi ,

E (x0 ) = h(2h)...(nh)

y n+1 () y n+1 () = (1)n hn . (n + 1)! n+1

(4.12)

4.1

71

(4.4) O(h), (4.5) O(h2 ) (4.6) O(h3 ). , O(h(n1) ) (. 4.1). , (4.8) O(h) (4.9), O(h2 ). Newton , . x = x1 y0 , y1 , y2 y3 . Lagrange x0 , x1 , x2 ,... . Lagrange :

P (x) =

(x x1 ) (x x2 ) (x x3 ) (x x0 ) (x x2 ) (x x3 ) y0 + y1 (x0 x1 ) (x0 x2 ) (x0 x3 ) (x1 x0 ) (x1 x2 ) (x1 x3 ) (x x0 ) (x x1 ) (x x2 ) (x x0 ) (x x1 ) (x x3 ) y2 + (4.13) y3 + (x2 x0 ) (x2 x1 ) (x2 x3 ) (x3 x0 ) (x3 x1 ) (x3 x2 )

:

P (x) =

2y0 [(x x1 ) + (x x2 ) + (x x3 )] 6h3 2y1 + 3 [(x x0 ) + (x x2 ) + (x x3 )] 2h 2y2 + [(x x0 ) + (x x1 ) + (x x3 )] 2h3 2y0 + 3 [(x x0 ) + (x x1 ) + (x x2 )] 6h y0 2y1 + y2 h2

(4.14)

x = x1 :

P (x) =

(4.15)

y3 , , , O(h2 ), . , .

72

4

, . 4.1 y (x) P (x) .

4.1. f (x) p(x). x = 4 .

4.2 . , x0 x0 . : y(x0 ) = y0 =

y(x0 ) = y(x0 ) = y(x0 ) = y(x0 ) =

y1 y1 + O(h2 ) 2h y2 + 8y1 8y1 + y2 y0 = + O(h4 ) 12h y1 2y0 + y1 + O(h2 ) y0 = h2 y2 + 16y1 30y0 + 16y1 y2 y0 = + O(h4 ) 12h2 y2 2y1 + 2y1 y2 + O(h2 ) y0 = 2h3

(4.16) (4.17) (4.18) (4.19) (4.20)

4.2

73

y(x0 )(4) = y0 =

(4)

y2 4y1 + 6y0 4y1 + y2 + O(h2 ) h4

(4.21)

. Taylor x0 , x0 h x0 2h

h2 h3 h4 (4) y + y0 + y0 + ... 2 0 6 24 h2 h3 h4 (4) y(x0 h) y1 = y0 hy0 + y0 y0 + y0 ... 2 6 24 4 3 2 4 (4) 2 y(x0 + 2h) y2 = y0 + 2hy0 + 2h y0 + h y0 + h y0 + ... 3 3 4 3 2 4 (4) 2 y(x0 2h) y2 = y0 2hy0 + 2h y0 h y0 + h y0 ... 3 3 y(x0 + h) y1 = y0 + hy0 +

(4.22) (4.23) (4.24) (4.25)

. , (4.23) (4.22) (4.16), (4.18) Lagrange, (4.15). (4.22) (4.23) 8 (4.24) (4.25)

y2 y2 + 8 (y1 y1 )

(4.26)

(4.17) O(h4 ). h h3 , h5 . . f (x) 4.1 f (1) ( f (x) = ex ).

x f (x)

0.90 2.4596

1.00 2.7183 4.1.

1.11 3.0344

(h = x0 x1 = 0.1 h1 = x1 x0 = 0.11), .

74

4

. Lagrange , , . , . y(x0 + h) = y(1 + 0.11) y(x0 h1 ) = y(1 0.1). , : y1 = y (x0 + h) = y0 + hy0 +

y1

h2 h3 y + y0 + 2 0 6 h2 h3 = y (x0 h1 ) = y0 h1 y0 + 1 y0 1 y0 + 2 6

(4.27) (4.28)

h2 h2 1 O(h1 h) O(h2 ) y0 = 1 1 h2 y1 h2 y1 1 1 h1 + h

1 1 h h1

y0

(4.29)

h1 = 0.11 h = 0.1, , y0 = 2.7233 ( 2.7183).

1. . 2. (4.19), (4.20) (4.21) 3. f (x0 ) f (x0 ), f (x) = tan1 (x) x0 = 2. (4.16) (4.18) h = 0.1

5

. , . . , .

5.1 NEWTON-COTES . , f (x) (xi , yi ) . . :b a b

y(x)dx

Pn (x)dxa

(5.1)

Newton

Pn (xs ) = f0 + sf0 +

s(s 1)(s 2) 3 s(s 1) 2 f0 + f0 + ... 2! 3!

(5.2)

xs = x0 + sh.

76

5

. , b

E=a

En (xs )dx

(5.3)

En (xs ) =

s(s 1)(s 2)...(s n) n+1 (n+1) h f () (n + 1)!

(5.4)

[a, b]. . 1 :x1 x0 x1 x1 s=1

f (x)dx

P1 (xs )dx =x0 1 0

(f0 + sf0 ) dx = hx0 2 1 0 s=0

(f0 + sf0 ) dx

= hf0 s

+ hf0

s 2

1 = h f0 + f0 2(5.5)

h = (f0 + f1 ) 2

, x0 x1 f (x) (x0 , f (x0 )) (x1 , f (x1 )). . f (x) P (x) :

f (x) P (x) =

1 s(s 1)h2 f () x0 x1 2

(5.6)

, x1

E=x0

1 h3 s(s 1)h2 f ()dx = 2 2 s2 s3 6 41 0

s=1 s=0

s(s 1)f ()ds(5.7)

= h3 f (1 ) 1 [x0 , x1 ].

=

1 3 h f (1 ) 12

5.1 NEWTON-COTES

77

2 :x2 x0 x2 x2

f (x)dx

P2 (xs )dx =x0 s=2 x0

1 f0 + sf0 + s(s 1)2 f0 dx 2

= h

1 f0 + sf0 + s(s 1)2 f0 ds 2 s=0 1 = h 2f0 + 2f0 + 2 f0 3 h = (f0 + 4f1 + f2 ) 3

(5.8)

.

1 6

x2 x0

s(s 1)(s 2)3 f0 dx = 0 .

(5.9)

. :

E=

1 24

x2 x0

s(s1)(s2)(s3)h4 f (4) ()dx = ... =

1 5 (4) h f (1 ) (5.10) 90

x0 1 x2 . 3 :x3 x0 x3

f (x)dx

P3 (xs )dx =x0

3h (f0 + 3f1 + 3f2 + f3 ) . 8

(5.11)

:

E=

3 5 (4) h f (1 ) x0 1 x3 80

(5.12)

, , O(h5 ), 2 .

78

5

, 4 , 5 , O(h7 ). , , , Newton-Cotes.

x1

f (x)dx =x0 x2

1 3 (2) h (f0 + f1 ) h f (1 ) 2 12

f (x)dx =x0 x3

h 1 5 (4) (f0 + 4f1 + f2 ) h f (1 ) 3 90

f (x)dx =x0

3h 3 5 (4) (f0 + 3f1 + 3f2 + f3 ) h f (1 ) 8 80

5.1. Newton-Cotes 1, 2 3 Newton.

5.1.1 (a, b) , {a = x0 , ..., xn = b; n}, 5.1.

. , :b xn n1 xi+1

f (x)dx =a x0 x1

f (x)dx =i=0 xi x2

P1 (x)dxxn

=x0 n1

P1 (x)dx +x1

P1 (x)dx + ... +xn1

P1 (x)dx

=i=0

h (fi + fi+1 ) 2

:b

f (x)dx =a

h (f0 + 2f1 + ... + 2fn + fn+1 ) 2

(5.13)

.

5.1 NEWTON-COTES

79

f(x)

x

0

x

1

x

n

x

n+1

5.1. .

(xi , xi+1 ),

E=

h3 [f (1 ) + f (2 ) + ... + f (n )] 12

(5.14)

f (x) [a = x0 , b = xn ] , , nf (). :

E= h = (b a)/n.

h2 h3 nf () = (b a)f () 12 12

(5.15)

80

5

5.1.2 Simpson , (5.8) 2 . , (a, b). b=xn n2 xi+2 n2

f (x)dx =a=x0 i=0 xi

P2 (x)dx =i=0

h (fi + 4fi+1 + fi+2 ) 3

h . = (f1 + 4f2 + 2f3 + 4f4 + ... + 2fn2 + 4fn1 + fn ) (5.16) 3 , , (5.10). , :

E=

h5 n (4) b a 4 (4) f () = h f () x0 xn 90 2 180

(5.17)

5.1.3 Simpson (3/8) , b=xn

f (x) dx =a=x0

3h (f0 + 3f1 + 3f2 + 2f3 + ... + 2fn3 + 3fn2 + 3fn + fn+1 ) 8

(5.18) 3. :

E=

b a 4 (4) h f (1 ) x0 1 xn 80

(5.19)

5.1.4 Romberg , I1 , h = xi+1 xi I2 kh, . , h I1 , kh I2 , A :

5.1 NEWTON-COTES

81

xn

f (x)dx =x0

h (f0 + 2f1 + 2f2 + ... + 2fn1 + fn ) 2

b a 2 (2) h f (1 ) 12 xn h f (x)dx = (f0 + 4f1 + 2f2 + 4f3 + .... + 2fn2 + 4fn + fn+1 ) 3 x0 b a 5 (4) h f (1 ) 180 xn 3h f (x)dx = (f0 + 3f1 + 3f2 + 2f3 + 3f4 + 3f5 + .... + 3fn2 + 3fn + fn+1 ) 8 x0 b a 4 (4) h f (1 ) 80

5.2. .

h: kh:

A = I1 + ch2 A = I2 + ck 2 h2

c . , A c, :

A=

k 2 I1 I2 k2 1

c=

I2 I1 2 (1 k 2 ) h

(5.20)

I1 , I2 k . k = 1/2,

A = I2 +

1 (I2 I1 ) 3

(5.21)

, O(hn ), :

A = I2 +

I2 I1 . 2n 1

(5.22)

A I1 I2 . Simpson. . n+1 , (n + 1)/2

82

5

xi . , I1 I2 , (5.22). Romberg k = 1/2 Simpson. 3 . , I1 2h I2 h :

2h (f0 + f2 ) = h (f0 + f2 ) 2 h h h I2 = (f0 + f1 ) + (f1 + f2 ) = (f0 + 2f1 + f2 ) 2 2 2 I1 = , (5.21), :

A=

h 2h(f0 + 2f1 + f2 ) h(f0 + f2 ) = (f0 + 4f1 + f2 ) 3 3

Simpson, (5.8). , Romberg O(h3 ) O(h5 ). Simpson O(h5 ) O(h7 ). 5.1.5 Splines splines (xi , yi ) 3- . (xi , yi ) 3- splines . spline xi x xi+1 :

f (x) = ai (x xi )3 + bi (x xi )2 + ci (x xi ) + di ai , bi , ci di :

(5.23)

ai =

Si+1 Si Si , bi = , di = f (xi ) 6(xi+1 xi ) 2 f (xi+1 ) f (xi ) 2(xi+1 xi )Si + (xi+1 xi )Si+1 ci = xi+1 xi 6

(5.24)

5.1 NEWTON-COTES

83

, (x, y) 3- (5.23). ,

f (x) = 6ai (x xi ) f (xi ) = ci ,

f (x) = 3ai (x xi )2 + 2bi (x xi ) + ci

(5.25) (5.26)

n + 1 (xi , yi )

f (xi ) = 2bi

(5.27)

spline . splines. f (x) n- (xi , yi ) 3 [xi , xi+1 ], :xn n1

f (x)dx =x0 i=0 n1

ai bi ci (x xi )4 + (x xi )3 + (x xi )2 + di (x xi ) 4 3 2

xi+1 xi

=i=0

bi ci ai (xi+1 xi )4 + (xi+1 xi )3 + (xi+1 xi )2 + di (xi+1 xi ) . 4 3 2

h = xi+1 xi , spline. ,xn

f (x)dx =x0

h4 4

n1

ai +i=0

h3 3

n1

bi +i=0

h2 2

n1

n1

ci + hi=0 i=0

di

(5.28)

, . , , splines . ai , bi , ci di , . f (x) = sin(x) 0 x 1 splines x = 0, 0.25, 0.5, 0.75, 1.0.

84

5

i1 2 3 4

x0 0.25 0.5 0.75

Si0 -7.344 -10.3872 -7.344

ai-4.8960 -2.0288 2.0288 4.8960

bi0 -3.6720 -3.1936 -3.6720

ci3.1340 2.2164 0 -2.2164

di0 0.7071 1.0 0.7071

S1 = 0 S5 = 0 : :1

sin(x)dx =0

6.2514 (0.25) (0.25) (0) + (12.5376) + (3.1340) + 0.25 (2.4142) 4 3 2

3

2

= 0.6362

0.6366 0.0004 Simpson 0.0015, 4 .

5.2 f (x) (a, b) ( ):b a

f (x) dx = (b a) f ()

(5.29)

, f (x) (a, b). , f (x) . , , f (a) f (b). :b

I=a

f (x)dx c0 f (a) + c1 f (b)

(5.30)

c0 , c1 f (x) a b. , f (x) x. , f (x) = 1 f (x) = x, c0 c1 . :

5.2 b

85

f (x) = x f (x) = 1

a b a

x dx =

x2 2b

b

=a

1 2 b a2 c0 a + c1 b (5.31) 2(5.32)

1 dx = x|a = (b a) c0 1 + c1 1

, :

c0 =b a

ba 2

c1 =

ba 2

(5.33)

f (x)dx

ba [f (a) + f (b)] 2

(5.34)

, , . (5.30), ,b a

f (x)dx c0 f (a) + c1 f

a+b 2

+ c2 f (b)

(5.35)

f (x) 2 . , f (x) = 1, f (x) = x f (x) = x2 c0 , c1 c2 , :b

f (x)dx =a

ba f (a) + 4f 6

a+b 2

+ f (b)

(5.36)

Simpson . , , :b

f (x)dx = c0 f (a) + c1 f (b) + c2 f (a) + c3 f (b)a

(5.37)

:b

f (x) dx =a

(b a) (b a) (f (a) + f (b)) + (f (a) f (b)) 2 12

2

(5.38)

, [a, b]. Euler-Maclaurin ( ;)

86

5 xn

f (x) dx =x0

h [f (x0 ) + 2f (x1 ) + ... + 2f (xn1 ) + f (xn )] 2

h2 [f (xn ) f (x0 )] 12 h4 f (3) (xn ) f (3) (x0 ) + 720 h6 f (5) (xn ) f (5) (x0 ) 30240

(5.39)

6 1, 3 5 . Euler-Maclaurin /2

I=0

sin(x)dx

(5.39) ( ) :/2

sin(x)dx =0

+ + =

sin 0 + sin 4 2 2 cos 0 cos 22 12 2 4 sin 0 sin 4 720 2 2 6 cos 0 cos 26 30240 2 2 4 6 + + + 6 = 0.99996732 4 48 16 720 2 30240

5.2.1 Filon b b

f (x) sin(x)dx a a

f (x) cos(x)dx

. :

5.2 2 0

87

f (x) sin(x)dx A1 f (0) + A2 f () + A3 f (2)

f (x) = 1, f (x) = x f (x) = x2

f (x) = 1 f (x) = x f (x) = x2

0 = A1 + A2 + A3 2 = A2 + 2A3

4 2 = 2 A2 + 4 2 A3

A1 = 1, A2 = 0 A3 = 1. 2 0

f (x) sin xdx f (0) f (2) .

Filon :b a b a

y(x) sin(kx)dx h [Ay(a) cos(ka) Ay(b) cos(kb) + BSe + DSo ] (5.40) y(x) cos(kx)dx h [Ay(a) cos(ka) Ay(b) cos(kb) + BCe + DCo ] (5.41) 1 sin(2q) 2 sin2 (q) + q 2q 2 q3 1 cos2 (q) sin(2q) B= 2+ q q2 q3 4 sin(q) 4 cos(q) D= q3 q2 A=n

(5.42) (5.43) (5.44)

Se = y(a) sin(ka) y(b) sin(kb) + 2n

y(a + 2ih) sin(ka + 2iq) (5.45)i=0

So =i=1

y [a + (2i 1)h] sin [ka + (2i 1)q]n

(5.46)

Ce = y(a) sin(ka) y(b) sin(kb) + 2n

y(a + 2ih) sin(ka + 2iq) (5.47)i=0

Co =i=1

y [a + (2i 1)h] sin [ka + (2i 1)q]

(5.48)

q = kh.

88

5

2

I=0

ex/2 cos(100x)dx

(5.49)

4.783810813 105 , 5.3. Filon 4 Simpson 1000 .

n4 8 16 128 256 1024 2048

Simpson 1.91733833+0 -5.73192992-2 2.42801799-2 5.55127202-4 -1.30263888-4 4.77161559-5 4.78309107-5

Filon 4.77229440-5 4.72338540-5 4.72338540-5 4.78308678-5 4.78404787-5 4.78381120-5 4.78381084-5

5.3. Simpson Filon 5.49.

5.3 GAUSS [a, b] f (x) . a xi b Gauss. , [1, 1] :1 1

f (x)dx af (x1 ) + bf (x2 )

(5.50)

, a, b, x1 x2 . 4 , f (x) = 1, f (x) = x, f (x) = x2 f (x) = x3 . , (5.50), , :

5.3 GAUSS

89

, (5.50) :1 1

f (x) = 1 2 = a + b a=b=1 f (x) = x 0 = ax1 + bx2 f (x) = x2 2 = ax2 + bx2 x1 = x2 = 1 2 3 f (x) = x3 0 = ax3 + bx3 2 1

1 1/2 3

= 0.5773

f (x)dx f (0.5773) + f (0.5773)

f (x) x1 = 0.5773 x2 = 0.5773 . [1, 1],

t=

1 1 (b a)x + (b + a) 2 2

dt =

ba dx . 2

(5.51)

:/2

I=0

sin(x)dx

,

x=

1 2

t+ 2 2

=

(t + 1) dx = dt 4 4

, , :

I=

1 (t + 1) dt = [1.0 sin (0.10566 ) + 1.0 sin (0.39434 )] sin 4 1 4 4 = 0.99847

2 : 0.7854 Simpson 3 : 1.0023. 5.3.1 Gauss-Legendre Gauss . , n :

90

5 1 1 n

f (x)dx

Ai f (xi )i=1

(5.52)

2n 2n Ai xi . 2(n 1), 2n :

A1 xk + .... + An xk = 1 n

0 k = 1, 3, 5, ..., 2n 1 2 k+1

(5.53)

k = 2, 4, 6, ..., 2n 2

xi Legendre n (1, 1). Legendre :

(n + 1) Ln+1 (x) (2n + 1) xLn (x) + nLn1 (x) = 0 3 :

(5.54)

L0 (x) = 1,

L1 (x) = x L2 (x) = 2 1 x2

3 2 1 x 2 2

(5.55)

Ai :

Ai =

n2 [Ln1 (xi )]

2

(5.56)

n = 4 4- Legendre

P4 =

1 35x4 30x2 + 3 81/2

xi = (15 2 30)/35 (5.56) Ai . 5.4. , xi Ai . Gauss 4 :

I=

1 sin (t + 1) dt 4 1 4 0 = 0.34785 sin (1 + 861136) + 0.34785 sin (1 0.861136) 4 4 4 + 0.652145 sin (1 + 0.33998) + 0.652145 sin (1 0.33998) 4 4 = 1.000000... sin(x)dx =

/2

, Simpson 32 : 1.0000003 64 : 9.99999983.

5.3 GAUSS

91

n2 4 8

xi 0.57735027 0.86113631 0.33948104 0.96028986 0.79666648 0.52553241 0.18343464

Ai1.0000000 0.34785485 0.62214515 0.10122854 0.22381034 0.31370665 0.36268378

5.4. xi Ai Gauss-Legendre 2, 4 8 .

5.3.2 Gauss :b

I=a

w(x)y(x)dx

(5.57)

w(x) , Gauss Legendre . :b n

w(x)y(x)dx =a i=1

Ai y(xi )

(5.58)

xi Ai . 1 w(x) . Gauss-Legendre w(x) = 1 . Gauss-Laguerre n x

e0

y (x) dx

Ai y (xi )i=1

(5.59)

w(x) = ex xi Laguerre :

Ln (x) = ex

dn ex xn dxn

(5.60)

Ai :1

Abramowitz-Stegun.

92

5

Ai =

(n!)2 xi [L (xi )]2 n

(5.61)

, ( 5.5).

n2 4

xi0.58578644 3.41421356 0.32254769 1.74576110 4.53662030 9.39507091 0.22284660 1.18893210 2.99273633 5.77514357 9.83746742 15.98287398

Ai0.85355339 0.14644661 0.60315410 0.35741869 0.03888791 0.00053929 0.10122854 0.41700083 0.11337338 0.01039920 0.00026102 0.00000090

6

5.5. xi Ai Gauss-Laguerre 2, 4 6 .

Gauss-Hermite

ex y(x)dx

2

n

Ai y(xi )i=12

(5.62)

, w(x) = ex xi Hermite. Hermite

Hn (x) = (1)n ex

2

2 dn ex dxn

(5.63)

Ai :

Ai =

2n+1 n! [Hn (xi )] 2

(5.64)

xi , Ai , ( 5.6). Gauss-Chebyshev 1 1

y(x) dx 2 n 1x

n

y(xi )i=1

(5.65)

1 w (x) = 1 x2

(5.66)

5.4

93

n2 4 6

xi 0.70710678 0.52464762 1.65068012 0.43607741 1.33584907 2.35060497

Ai0.88622693 0.80491409 0.08131284 0.72462960 0.15706732 0.00453001

5.6. xi Ai Gauss-Hermite 2, 4 6 .

xi Chebyshev

Tn (x) = cos [n arccos(x)]

(5.67)

xi = cos

(2i 1) . 2n n

(5.68)

Ai

Ai =

(5.69)

n .

5.4 1. Simpson 3/8 ( 5.12). 2. x, log(x), 1/x h = 0.1 h = 0.05 [1, 1.3]. . 3. Romberg . 4. , 1 Simpson . 5. 0 sin xdx 6 , Simpson. . 6. Romberg . 2.4 7. 1.8 dx , 5 , h ; ./2

94

5

8. Simpson.

1 k sin2 (x)dx k = 0.5 9. 0 k = 0.25 4 . 10. x2 /16 + y 2 /25 = 1 6 . 11. h

/2

y(x)dx = h [a1 y(h) + a0 y(0) + a1 y(h)]h

+ h2 [b1 y (h) + b0 y (0) + b1 y (h)]

(5.70)

. 12. h

y(x)dx =0

h h3 (y0 + y1 ) (y + y1 ) 2 24 0

(5.71)

4 . 2 13. Filon 0 ex sin(10x)dx. Simpson [ (1 e2 )/101 ]. 14. Gauss-Legendre SimpsonRomberg 0.856589940). ;/2 0

log(1+x)dx ( -

6

. () . , , . , , . - . x (t) y (t) . , ( ) ,

dx = a (y + b) x dt dy = c (x + d) y dt a, b, c, d , . . . Mathematica .

6.1 xi + h ( h ) -

96

6

xi 6.1.1 Taylor Taylor. , ,

y = f (x, y)

y (x0 ) = y0

(6.1)

, y (x) x = x0 + h, Taylor, :

y (x0 + h) = y (x0 ) + hy (x0 ) +

h2 y (x0 ) + 2

(6.2)

y = f (x, y), y (x) :

y = f (x, y) ,

y = f (x, y) , .

(6.3)

y (x0 + h) . :

y = x + y

(6.4)

y(0) = 1. y(1) ( : y = 2ex x 1). 4 :

y (x0 ) = y (0) = 0 + 1 = 1 y = 1 + y = 1 + x + y = 1 + y y (x0 ) = 1 + y (0) = 2 y = 1 + y y (x0 ) = 1 + y (0) = 2 y (4) = 1 + y y (4) (x0 ) = 1 + y (0) = 2, (6.2) :

6.1

97

y (0 + h) = 1 + h + h2 +

h4 h3 + + 3 12

(6.5)

Taylor, :

E=

y (5) () 5 h 0 < < h 5!

(6.6)

x 0 0.1 0.2 0.3 0.4 : 1.0

y 1 1.110342 1.24280 1.39968 1.383467 3.416667

y 1 1.110342 1.24281 1.39972 1.383649 3.436564

*

1.7 107 5.5 106 4.3 105 1.8 104 2 102

3.7 107 6.2 107 9.1 107 4.2 106

6.1. , , h = 0.1 , h = 0.2 , . , h (4 ). , , (5 ).

6.1.2 Euler & Euler - Heun Euler Taylor. 1 h. :

y (x0 + h) = y (x0 ) + hy (x0 ) :

(6.7)

E=

y () 2 h 0 < < h 2

(6.8)

, . : y (x0 + h) = y (x0 ) + h (x0 + y (x0 )) (6.9) , :

yn+1 = yn + h (xn + yn )

(6.10)

Euler-Heun - ( ).

98

6

3

y=f(x)

2

=y-y}

1

y'0

y

(x ,y )0 0

1y1

h

0 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

x

6.1. Euler y = x + y y0 = y(0) = 1 h = 0.4. y = f (x), y(x0 + h) = y1 , x1 = x0 + h = 0 + 0.4 = 0.4. = y y1 . , y2 .

, Euler (6.7), y (x) y (x0 + h), y1 , , , y1 y (x0 ) x0 x0 + h. , : yn+1 = yn + hyn + O(h2 ) h yn+1 = yn + y + yn+1 + O(h3 ) 2 n

(6.11) (6.12)

Euler-Heun ( h3 ) Euler h2 ( ). 6.1.3 . , .

6.1

99

1. , .. , . , , . 2. , 3. h. & . yn Yn . , :

n = yn Yn

(6.13)

, , Euler. :

n+1 = yn+1 Yn+1 = yn + hf (xn , yn ) Yn hf (xn , Yn ) (f (xn , yn ) f (xn , Yn )) n = n (1 + hfy (xn , yn )) = n + h yn Yn f = (1 + hk) n k = (6.14) y . |1 + hk| 1 Euler . , |1 + hk| < 1 . : 2 < hk < 0 (6.15) f /y 0 a . Dung. ) q(t) q (t). .. q(0) = 0 q (0) = 0.001. (b = 0.05, a = 1) (b = 0.05, a = 4). . ) . ) . ) ( ) q a 7. -

q (t) = aq(t) bq 3 (t) q (t) + Q0 cos(t) q(0) = 0 q (0) = 0.001. a = 0.4, b = 0.5, = 0.2 = 1/8 Q0 = 0 Q0 = 0.1. ) q(t) q (t). ) , Fourier .

7

. , n- (xi , yi ). . .

7.1 , . P (x) f (x) yi f (xi ) . , n

S=i=0

[yi P (xi )]2

(7.1)

P (x) ( ). 7.1.1 , : P (x) = ax + b. , :n

S=i=0

[yi axi b]

2

(7.2)

112

7

a b S . :

S = 0 a :

S =0 b

(7.3)

S xi (yi axi b) = 0 = 2 a i=0(7.4)

n

S (yi axi b) = 0 = 2 b i=0 :

n

(n + 1) b + xi b +,

xi a = x2 i a=

yi(7.5)

xi yi

s0 = n + 1 u0 = yi

s1 = u1 =

xi xi yi

s2 =

x2 i(7.6)

(7.5) :

s0 b + s1 a = u0 s1 b + s2 a = u1

a=

s0 u 1 s1 u 0 s0 s2 s2 1

b=

s2 u 0 s1 u 1 . s0 s2 s2 12

(7.7)

, :

s0 s2 s2 = (n + 1) 1

x2 i

xi

=i 0

2

(7.8)

S a b, :

=

2S 2S a2 b2

2S ab

2

>0

(7.9)

7.1 2 2

113

, , S , S a2 b2 a b, (7.7). , :

2S = 2s2 > 0, a2

2S = 2s0 > 0 b22

2S = 2s1 ab

(7.10)

= 2s2 2s0 (2s1 ) = 4 s0 s2 s2 > 0 1

(7.11)

, (7.7) . P (x) = AeMx ( A M ) :

x P (x)

1 7

2 11

3 17

4 27

z = log(P ) B = log(A) z = B + M x, (xi , zi )

x z

1 1.9459

2 2.3979

3 2.8332

4 3.2958

s0 = 4, s1 = 10, s2 = 30, v0 = 10.48 v1 = 28.44 M 0.45, B = 1.5 A = 4.48. : P (x) = 4.48e0.45x. :

x y

0.05 0.956

0.11 0.890

0.15 0.832

0.31 0.717

0.46 0.571

0.52 0.539

0.70 0.378

0.74 0.370

0.82 0.306

. n = 9, s0 = 10, s1 =

xi = 3.86, s2 =

x2 = 2.3252 i

114

7

u0 =

yi = 5.559,

u1 =

xi yi = 1.82506

a = 0.83496,

b = 0.97577

, P (x) = 0.97577 0.83496x.

7.1. .

7.1.2 . , :n

S=i=0

yi ax2 bxi c i

2

(7.12)

S S S = = =0 a b c

(7.13)

s0 s1 s2 c u0 s1 s2 s3 b = u 1 s2 s3 s4 a u2

(7.14)

7.1

115

s3 =

x3 , i

s4 =

x4 u2 = i

x2 yi i

s0 , s1 , s2 , v0 , v1 (7.6). n 1 :

sk uk :

s0 s1 : : sn

s1 ... ... sn a0 u0 s2 ... ... sn+1 a1 u1 : = : : : : : : : sn+1 ... ... s2n an un sk = xk uk = i xk yi i

(7.15)

:

P (x) = a0 + a1 x + ... + an xn . 2 ,

P (x) = 0.996 0.995x + 0.186x2 :

P (x) = 1.0 1.06x + 0.37x2 0.14x3 .7.1.3 y (x) [1, 1] :

P (x) = am Pm (x) + am1 Pm1 (x) + ... + a0 P0 (x) Pk (x) Legendre k ak .1 :1

Legendre Rodrigues

Pk (x) =

1 2k k!

dk 2 (x 1)k dxk1 (3x2 2

: P0 (x) = 1, P1 (x) = x, P2 =

1) ...

116

7 1

S=1

[y (x) am Pm (x) am1 Pm1 (x) ... a0 P0 (x)] dx

2

(7.16)

:

S =0 ak :

S = ak

1

[y (x) am Pm (x) am1 Pm1 (x) a0 P0 (x)] Pk (x) dx = 01

Legendre , :1

1

Pm (x) Pk (x)dx = 1

0 k = m(7.17)2 k+1

k=m

S = ak

[y (x) ak Pk (x)] Pk (x) dx = 01

ak :

2k + 1 ak = 2

1

y (x) Pk (x) dx1

(7.18)

, [1, 1] [a, b], :

x=

1 1 (b a) t + (b + a) 2 2

(7.19)

y(t) = sin(t) [0, ] . : t = [0, ] [1, 1] : 2 (x+1)

y = sin :

(x + 1) . 2

7.2

117

1 a0 = 2 3 a1 = 2 5 a2 = 2

1

sin1 1

2 (x + 1) dx = 2 (x + 1) x dx = 0 2 1 10 (x + 1) (3x2 1) dx = 2 2 1 12 2

sin1 1

sin1

:

y(x) =

2 10 +

1

12 2

1 3x2 1 2

:

y=

10 2 +

1

12 2

6 t 2 2

2

1 2

7.2 y (x) (xi , yi ) y(x). . y(x) y(x) x :

= |y (x) y(x)|

(7.20)

y(x) = ax + b , Chebyshev (x0 , y0 ), (x1 , y1 ) (x2 , y2 ). xi :

|i | = |yi (axi + b)| (x2 , y2 ) : y (1) (x) =

(7.21)

(x0 , y0 ),

y0 x2 y2 x0 y2 y0 x+ x2 x0 x2 x0

(7.22)

118

7

C'

y y

2 1

{1

B

C

B'

A'

y

(2)

(x)

y

0

A y(1)

(x)

x

0

x

1

x

2

7.2. .

:

0 = 0,

(x0 , y0 )(7.23)

1 = |y1 y (1) (x1 )| = |y1 ax1 b| (x1 , y1 ) 2 = 0 (x2 , y2 )

, 0 = 1 = 2 ,

y (2) =

y2 y0 x+c x2 x0

(7.24)

c AA = BB , :

y0

y2 y0 x0 + c x2 x0

=

y2 y0 x1 + c y1 x2 x0

(7.25)

:

y (2) (x) =

y0 (x1 + x2 ) + y1 (x2 x0 ) y2 (x0 + x1 ) y2 y0 x+ x2 x0 2 (x2 x0 )(2)

(7.26)(2)

0 = y0 y0 :

, 1 = y1 y1

(2)

2 = y2 y2

|0 | = |1 | = |2 | :

(7.27)

7.3

119

= y0 y0 =

(2)

y0 (x2 x1 ) y1 (x2 x0 ) + y2 (x1 x0 ) 2 (x2 x0 )

(7.28)

, . , ...

7.3 f (x) , n m . :

f (x) RN (x)

a0 + a1 x + a2 x2 + ... + an xn 1 + b1 x + b2 x2 + ... + bm xm

(7.29)

N = n + m RN (x) Maclaurin f (x) ( RN (x) f (x) N ).

f (x) RN (x) c0 + c1 x + ... + cN xN = c0 + c1 x + ... + cN xN

a0 + a1 x + ... + an xn 1 + b1 x + ... + bm xm (1 + b1 x + ... + bm xm ) (a0 + a1 x + ... + an xn ) 1 + b1 x + ... + bm xm

N f (x) RN (x) x = 0, N . N + 1 . x = 0 f (0) = RN (0) c0 a0 = 0 N N

b 3 c0 + b 2 c1 + b 1 c2 + c3 a 3 = 0

b 1 c0 + c1 a 1 = 0 b 2 c0 + b 1 c1 + c2 a 2 = 0

. . .

(7.30)

bm cnm + bm1 cnm+1 + ... + cn an = 0 bm cnm+1 + bm1 cnm+2 + ... + cn+1 = 0 bm cnm+2 + bm1 cnm+3 + ... + cn+2 = 0. . . (7.31)

bm cN m + bm1 cN m+1 + ... + cN = 0 N N .

120

7

N . Pad . e Pade, R9 (x), tan1 (x) 2 . Maclaurin tan1 (x) :

1 1 1 1 tan1 (x) = x x3 + x5 x7 + x9 3 5 7 9

f (x) R9 (x) = 1 1 1 (x 1 x3 + 5 x5 7 x7 + 9 x9 )(1+b1 x+b2 x2 +b3 x3 +b4 x4 )(a0 +a1 x+a2 x2 +...+a5 x5 ) 31+b1 x+b2 x2 +b3 x3 +b4 x4

:

a0 = 0,

1 a3 = + b 2 3 1 1 1 a4 = b 1 + b 3 , a5 = b 2 + b 4 3 5 3 1 1 1 1 1 b1 b3 = 0, + b2 b4 = 0 5 2 7 5 3 1 1 1 1 1 b2 + b4 = 0 b1 + b3 = 0, 7 5 9 7 5 a1 = 1, a2 = b 1 , a0 = 0, a1 = 1, a2 = 0 64 7 a3 = 9 , a4 = 0, a5 = 945 10 b1 = 0, b2 = 9 , b3 = 0, b4 =

(7.32)

:

5 21 .

:

tan1 x R9 (x) =

7 64 x + 9 x3 + 945 x5 5 1 + 10 x2 + 21 x4 9

, x = 1 R9 (1) = 0.7856, 0.7854 Maclaurin 0.8349! Maclaurin .2

R5,4 (x) , 3 6 .

7.3

121

7.3.1 f (x) (xi , f (xi )), k , Lagrange 3. k :

a0 + a1 xi + a2 x2 + ... + an xn i i = f (xi ) 1 + b1 xi + b2 x2 + ... + bm xm i i

(7.33)

k m + n + 1. :

a0 + a1 x1 + ... + an xn (f1 x1 ) b1 ... (f1 xm ) bm = f1 1 1 :: a0 + a1 xi + ... + an xn (fi xi ) b1 ... (fi xm ) bm = fi i i :: a0 + a1 xk + ... + an xn (fk xk ) b1 ... (fk xm ) bm = fk k k k k a0, a1 , ..., an b1 , b2 , ..., bm . 2 . :

x y

-1 1

0 2

1 -1

, , 3 (n + m + 1 = 3). :

R1,1 (x) =

a0 + a1 x 1 + b1 x

: :

a0 + (1) a1 (1) b1 = 1 a0 + 0 a1 0 b 1 = 2 a0 + 1 a1 (1) b1 = 1 R1,1 (x) = 2x . 1 2x

a0 = 2 a1 = 1 b1 = 2

122

7

( )

R0,2 (x) =

:

2 a0 R (x) = 2 1 + b1 x + b2 x 1 2x x2

R2,0 (x) =

a0 + a1 x + a2 x2 R2,0 (x) = 2 x 2x2 . 1

7.4 1. :

xi yi

1 1.2

2 1.5

3 1.4

4 1.6

5 2.1

6 2.0

7 2.2

8 2.5

9 3.0

2. - ( ) . 3. - ( ) (0, 0), (1, 0) (2, 1). 4. 2 y = ex /2 . 2 5. y(t) = keat +bt k, a, b :

t y(t)

0.1 2.7

0.2 2.9

0.3 3.1

0.4 3.4

0.5 3.6

0.6 3.9

0.7 4.3

0.8 4.6

0.9 5.0

1.0 5.6

( : ). 6. y = ex R3,3 (x) ( 3 ). Maclaurin 6 x = 1. ; 7. y = cos(x) y = sin(x) R3,5 (x)) y = 1/x sin(x) R4,6 (x). 8. :

7.4 0 0.83 1 1.06 2 1.25 4 4.15

123

8

(). . . x y

2 , x2

2 , xy

2 y 2

(8.1)

, , .

A

2 2 2 +B + C 2 + D x, y, , , 2 x xy y x x

=0

(8.2)

B 2 4AC : , B 2 4AC < 0 , B 2 4AC = 0 , B 2 4AC > 0

A, B C x, y . Laplace 2 = 0 (8.3) Poisson

2 = f (x, y)

(8.4)

126

8

B = 0, A = C = 1, .

2

1 2 =0 c2 t2

(8.5)

2

=0 t

(8.6)

. / . , - . . - .

8.1 8.1.1 Laplace Laplace

2 u uxx + uyy = 0

(8.7)

. . Poisson 2 u uxx + uyy = g(x, y) (8.8) Helmoltz

uxx + uyy + f (x, y)u = g(x, y)

(8.9)

g(x, y) f (x, y) . Laplace. , u(x, y). (4.18) uxx (xi , yj )

[uxx ]i,j = uyy

ui1,j 2ui,j + ui+1,j h2

(8.10)

8.1

127

ui,j+1

ui-1,j -4ui,j

ui+1,j

Laplace

ui,j-1

8.1. Laplace.

[uyy ]i,j =

ui,j1 2ui,j + ui,j+1 h2

(8.11)

Laplace

2 u

ui1,j + ui+1,j + ui,j1 + ui,j+1 4ui,j =0 h2

(8.12)

i = 2, ..., n 1 j = 2, ..., m 1. Laplace, h2 . (8.12) u(x, y) (xi , yj ) (xi+1 , yj ), (xi1 , yj ), (xi , yj1 ) (xi , yj+1 ), 8.1. , (8.12)

ui,j =

1 (ui1,j + ui+1,j + ui,j1 + ui,j+1 ) 4

(8.13)

u(x, y) (xi , yj ) (xi+1 , yj ), (xi1 , yj ), (xi , yj1 ) (xi , yj+1 ).

u(x1 , yj ) = u1,j 2 j m 1 u(xi , y1 ) = ui,1 2 i n 1

128

8

u(xn , yj ) = un,j 2 j m 1 u(xi , ym ) = ui,m 2 i n 1

(8.14)

u(x, y), (n 2) (n 2) (n 2)2 .

u2,5

u3,5

u4,5

u1,4

u2,4

u3,4

u4,4

u5,4

u1,3

u2,3

u3,3

u5,3 u4,3 u5,2

u1,2

u2,2

u3,2

u4,2

u2,1

u3,1

u4,1

8.2. 5 5 Laplace.

8.2 9 u(x, y), u2,2 , u2,3 , u2,4 , u3,2 , u3,3 , u3,4 , u4,2 , u4,3 u4,4 . (8.13) 9 9 .

4u2,2 +u3,2 +u2,3 u2,2 4u3,2 +u4,2 u3,2 4u4,2 u2,2 4u2,3 u3,2 +u2,3 u4,2 u2,3

+u3,3 +u4,3 +u3,3 +u2,4 4u3,3 +u4,3 +u3,3 4u4,3 4u2,4 u3,3 +u2,4 u4,3

+u3,4 +u4,4 +u3,4 4u3,4 +u4,4 +u3,4 4u4,4

(8.15) .

= u2,1 u1,2 = u3,1 = u4,1 u5,2 = u1,3 =0 = u5,3 = u2,5 u1,4 = u3,5 = u4,5 u5,4

8.1

129

Laplace 2 u = 0 {(x, y) : 0 x 4, 0 y 4} u(x, y) (x, y). :

u(x, 0) = 20 u(x, 4) = 180 0 < x < 4 u(0, y) = 80 u(4, x) = 0 0 < y < 4 3 9 9

u2,5=180 u3,5=180

u4,5=180

u1,4=80

u2,4

u3,4

u4,4

u5,4=0

u1,3=80

u2,3

u3,3

u5,3=0

u4,3u5,2=0

u1,2=80

u2,2

u3,2

u4,2

u2,1=20

u3,1=20

u4,1=20

8.3. 5 5 .

4u2,2 +u3,2 +u2,3 u2,2 4u3,2 +u4,2 u3,2 4u4,2 u2,2 4u2,3 u3,2 +u2,3 u4,2 u2,3

+u3,3 +u4,3 +u3,3 +u2,4 4u3,3 +u4,3 +u3,3 4u4,3 4u2,4 u3,3 +u2,4 u4,3

+u3,4 +u4,4 +u3,4 4u3,4 +u4,4 +u3,4 4u4,4

= 100 = 20 = 20 = 80 =0 =0 = 260 = 180 = 180

(8.16)

130

8

:

u2,2 = 55.7143, u3,2 = 43.2143, u4,2 = 27.1429, u2,3 = 79.6429, u3,3 = 70.000, u4,3 = 45.3571, u2,4 = 112.857, u3,4 = 111.786 u4,4 = 84.2857.

150 uHi,jL 100 50 0 2.5 5 7.5 i 10 12.5

12.5 10 7.5 j 5 2.5

8.4. Laplace . 15 15.

Laplace (4.18). Laplace

(8.17) i = 2, ..., n 1 j = 2, ..., m 1. Laplace, h4 . 1. Laplace u(0, y) = 0 0 y 1

2 u

ui1,j+1 + 4ui,j+1 + ui+1,j+1 + 4ui1,j 20ui,j + 4ui+1,j + ui1,j1 + 4ui,j1 + ui+1,j1 =0 6h2

u(1, y) = 0 0 y 1 u(x, 1) = 0 0 x 1

8.2

131

u(x, 0) = x(1 x) 0 x 1( : 4 4 u2,2 = 1/12, u3,2 = 1/12, u2,3 = 1/36, u3,3 = 1/36). 8 8 .

2. (4.18) 9 , (8.17), Laplace. . 3. 9 20 20 . 4. 6cm 8cm u(x, y) (0)

2 u + 2 = 0. u(x, y), 1cm . ( Poisson). : 2cm u2,2 = u3,2 = u2,4 = u3,4 = 4.56 u2,3 = u3,3 = 5.72

8.2 8.2.1 . a . :

2 u(x, t) 2 u(x, t) = c2 t2 x2

0 < x < a

0