数值分析简明教程课后习题答案(18)

详细的《数值分析简明教程》第二版课后习题答案

【解】欧先写出高斯-赛德尔迭代：

3(k) ~(k 1)

x x2 4 1

4

3~(k)1(k)9(k)1(k) ~(k 1)

x x x 5 x2 x3 2 213

44164

9(k)1(k)5 ~(k 1)1~(k)

x x 3 x2 x3 2 3

464162

引入松弛因子，得

(1)

1(k)5~(k 1) (k 1)(k)(k 1)~x (1 )x x x1 x1

11 1

44

1(k)5~(k 1) (k 1)(k)

(1 )x2 ~x2(k 1) x2 x2 x2

44

1(k)5~(k 1) (k 1)(k)(k 1)~x (1 )x x x3 x3

3 33

44

将方程组（1）代入（2），并化简

(2)

1(k)15(k) (k 1)

x x1 x2 5 1

416

(k 1)29(k)5(k)5

x2 x3 x2

64162

45(k)11(k)25 (k 1)

x x2 x3 3

256648

(3)