数值分析李庆杨版习题及答案(7)

{ xc=(x[i]+x[i+1])/2; I(xc,x[i],x[i+1]);

printf("I[%d]=%f\n",i+1,I(xc,x[i],x[i+1])); }

for(i=0;i<10;i++) { xx=(x[i]+x[i+1])/2; f(xx);

printf("f[%d]=%f\n",i+1,f(xx)); } }

21. Ih(x)在每个小区间[xk,xk 1]上为

Ih(x)

x xk 12x xk2

xk xk 1 (xk 1 xk)x xkxk 1.

xk xk 1xk 1 xk

2

(xk 1 xk)2h2

|R(x)| |Ih(x) f(x)| |x (xk 1 xk)x xkxk 1| .

44

3

22. f (x) 4x, 则Ih(x)在每个小区间[xk,xk 1]上表示为

x xk 1

Ih(x) x x

k 1 k

2

2

x xk

1 2 xk 1 xk 4 x xk

xk x x

k k 1

2

2

x xk 1 4 1 2 xk 1

x xkk 1

x xk 1 x xk 33

4 (x x)x 4 (x x)xkkk 1k 1. x x x x k 1 k k k 1

|R(x)| |f(4)( )(x xk)2(x xk 1)2/4!|

xk 1 xkh42xk 1 xk2

4! ( xk)( xk 1)! .

2216

23. h1 x2 x1 0.05, h2 x3 x2 0.09,

h3 x4 x3 0.06, h4 x5 x4 0.08.

hi 1

i

hi hi 1

yi 1 yiyi yi 1 6 i h hhhii 1i 1i

则三次样条插值函数表达式为

mmymym

Si(x) i 1(xi x)3 i(x xi 1)3 (i 1 i 1hi)(xi x) (i ihi)(x xi 1)

6hi6hihi6hi6

i) 由S (0.25) 1.0000,S (0.53) 0.6868，得

1 0.6429, 2 0.4, 3 0.5714

0 6(f[x0,x1] 1) 0.276， 1 4.3157, 2 3.264, 3 2.43

4 6(0.6868 f[x3,x4]) 0.1692 关于m0,m1,m2,m3,m4的方程组为

2

f(x) C[a,b],所以 24. i) 因为