实验四实验报告线性方程组的迭代解法

1、实验四 线性方程组的迭代解法一、 实验目的(1) 学会用Jacobi迭代法、Gauss- Seidel迭代法和超松弛迭代法求线性方程组解(2) 学会对各种迭代法作收敛性分析，研究求方程组解的最优迭代方法(3) 按照题目要求完成实验内容，写出相应的Matlab程序，给出实验结果(4) 对实验结果进行分析讨论(5) 写出相应的实验报告二、实验内容1.熟悉Jacobi迭代法，并编写Matlab程序matlab程序按照算法(Jacobi迭代法)编写Matlab程序(Jacobi.m)functionx,k,index=Jacobi(A,b,ep,it_max) 求解线性方程组的Jacobi迭代法,其中

2、, A为方程组的系数矩阵; b为方程组的右端项; ep为精度要求，缺省值1e-5; it_max为最大选代次数，缺省值100; x为方程组的解; k为迭代次数; index为指标变量 index=1表示迭代收敛到指定要求, , index=0表示迭代失败.if nargin<4 it_max=100;endif nargin<3 ep=1e-5;endn=length(A);k=0;x=zeros(n,1);y=x;index=1;while l for i=1:ny(i)=b(i) for j=1:n if j = i y(i)=y(i)-A(i,j)*x(j); end end

3、 if abs(A(i,i)<1e-10|k= it_max index=0; return; end y(i)=y(i)/A(i,i); end if norm(y-x,inf)<ep break; end e=y;k=k+1;end2. 熟悉Gauss-Seidel迭代法，并编写Matlab程序练习题1. 用Jacobi迭代法求方程组的解。修改上述程序，得到正确的源程序：functionx,k,index=Jacobi(A,b,ep,it_max)%£¥ Çó½âÏßÐÔ

4、3;½³Ì×éµÄJacobiµü´ú·¨,ÆäÖÐ,%£¥ AΪ·½³Ì×éµÄϵÊý¾ØÕó%£¥ bΪ·½³Ì×é&#

5、181;ÄÓÒ¶ËÏî%£¥ epΪ¾«¶ÈÒªÇó£¬È±Ê¡Öµ1e-5;%£¥ it_maxΪ×î´óÑ¡´ú´ÎÊý£¬È

6、77;Ê¡Öµ100;%£¥ xΪ·½³Ì×éµÄ½â%£¥ kΪµü´ú´ÎÊý%£¥ indexΪָ±ê±äÁ¿ index=1±íÊ&#

7、190;µü´úÊÕÁ²µ½Ö¸¶¨ÒªÇó,%£¥ index=0±íʾµü´úʧ°Ü.if nargin<4 it_max=100;endif nargin<3 ep=1e-5;endn=length(A);k=0;x=zeros(n,1);y=x;index

8、=1;while 1 for i=1:ny(i)=b(i) for j=1:n if j=i y(i)=y(i)-A(i,j)*x(j); end end if abs(A(i,i)<1e-10|k=it_max index=0; return; end y(i)=y(i)/A(i,i); end if norm(y-x,inf)<ep break; end x=y;k=k+1;end主函数：A=4 3 0;3 3 -1;0 -1 4b=24;30;-24x,k,index=Jacobi(A,b)输出结果：A = 4 3 0 3 3 -1 0 -1 4b = 24 30 -24y

9、= 24 0 0y = 6 30 0y = 6 10 -24y = 24 10 -6y = -1.5000 30.0000 -6.0000y = -1.5000 2.0000 -24.0000y = 24.0000 2.0000 -3.5000y = 4.5000 30.0000 -3.5000y = 4.5000 10.3333 -24.0000y = 24.0000 10.3333 -5.5000y = -1.7500 30.0000 -5.5000y = -1.7500 3.6667 -24.0000y = 24.0000 3.6667 -3.4167y = 3.2500 30.0000

10、 -3.4167y = 3.2500 10.6111 -24.0000y = 24.0000 10.6111 -5.0833y = -1.9583 30.0000 -5.0833y = -1.9583 5.0556 -24.0000y = 24.0000 5.0556 -3.3472y = 2.2083 30.0000 -3.3472y = 2.2083 10.8426 -24.0000y = 24.0000 10.8426 -4.7361y = -2.1319 30.0000 -4.7361y = -2.1319 6.2130 -24.0000y = 24.0000 6.2130 -3.28

11、94y = 1.3403 30.0000 -3.2894y = 1.3403 11.0355 -24.0000y = 24.0000 11.0355 -4.4468y = -2.2766 30.0000 -4.4468y = -2.2766 7.1775 -24.0000y = 24.0000 7.1775 -3.2411y = 0.6169 30.0000 -3.2411y = 0.6169 11.1962 -24.0000y = 24.0000 11.1962 -4.2056y = -2.3972 30.0000 -4.2056y = -2.3972 7.9812 -24.0000y =

12、24.0000 7.9812 -3.2009y = 0.0141 30.0000 -3.2009y = 0.0141 11.3302 -24.0000y = 24.0000 11.3302 -4.0047y = -2.4977 30.0000 -4.0047y = -2.4977 8.6510 -24.0000y = 24.0000 8.6510 -3.1674y = -0.4883 30.0000 -3.1674y = -0.4883 11.4418 -24.0000y = 24.0000 11.4418 -3.8372y = -2.5814 30.0000 -3.8372y = -2.58

13、14 9.2092 -24.0000y = 24.0000 9.2092 -3.1395y = -0.9069 30.0000 -3.1395y = -0.9069 11.5349 -24.0000y = 24.0000 11.5349 -3.6977y = -2.6511 30.0000 -3.6977y = -2.6511 9.6743 -24.0000y = 24.0000 9.6743 -3.1163y = -1.2557 30.0000 -3.1163y = -1.2557 11.6124 -24.0000y = 24.0000 11.6124 -3.5814y = -2.7093

14、30.0000 -3.5814y = -2.7093 10.0619 -24.0000y = 24.0000 10.0619 -3.0969y = -1.5464 30.0000 -3.0969y = -1.5464 11.6770 -24.0000y = 24.0000 11.6770 -3.4845y = -2.7577 30.0000 -3.4845y = -2.7577 10.3849 -24.0000y = 24.0000 10.3849 -3.0808y = -1.7887 30.0000 -3.0808y = -1.7887 11.7308 -24.0000y = 24.0000

15、 11.7308 -3.4038y = -2.7981 30.0000 -3.4038y = -2.7981 10.6541 -24.0000y = 24.0000 10.6541 -3.0673y = -1.9906 30.0000 -3.0673y = -1.9906 11.7757 -24.0000y = 24.0000 11.7757 -3.3365y = -2.8318 30.0000 -3.3365y = -2.8318 10.8784 -24.0000y = 24.0000 10.8784 -3.0561y = -2.1588 30.0000 -3.0561y = -2.1588

16、 11.8131 -24.0000y = 24.0000 11.8131 -3.2804y = -2.8598 30.0000 -3.2804y = -2.8598 11.0654 -24.0000y = 24.0000 11.0654 -3.0467y = -2.2990 30.0000 -3.0467y = -2.2990 11.8442 -24.0000y = 24.0000 11.8442 -3.2337y = -2.8832 30.0000 -3.2337y = -2.8832 11.2211 -24.0000y = 24.0000 11.2211 -3.0389y = -2.415

17、9 30.0000 -3.0389y = -2.4159 11.8702 -24.0000y = 24.0000 11.8702 -3.1947y = -2.9026 30.0000 -3.1947y = -2.9026 11.3509 -24.0000y = 24.0000 11.3509 -3.0325y = -2.5132 30.0000 -3.0325y = -2.5132 11.8918 -24.0000y = 24.0000 11.8918 -3.1623y = -2.9189 30.0000 -3.1623y = -2.9189 11.4591 -24.0000y = 24.00

18、00 11.4591 -3.0270y = -2.5943 30.0000 -3.0270y = -2.5943 11.9099 -24.0000y = 24.0000 11.9099 -3.1352y = -2.9324 30.0000 -3.1352y = -2.9324 11.5493 -24.0000y = 24.0000 11.5493 -3.0225y = -2.6620 30.0000 -3.0225y = -2.6620 11.9249 -24.0000y = 24.0000 11.9249 -3.1127y = -2.9437 30.0000 -3.1127y = -2.94

19、37 11.6244 -24.0000y = 24.0000 11.6244 -3.0188y = -2.7183 30.0000 -3.0188y = -2.7183 11.9374 -24.0000y = 24.0000 11.9374 -3.0939y = -2.9530 30.0000 -3.0939y = -2.9530 11.6870 -24.0000y = 24.0000 11.6870 -3.0157y = -2.7652 30.0000 -3.0157y = -2.7652 11.9478 -24.0000y = 24.0000 11.9478 -3.0783y = -2.9

20、609 30.0000 -3.0783y = -2.9609 11.7392 -24.0000y = 24.0000 11.7392 -3.0130y = -2.8044 30.0000 -3.0130y = -2.8044 11.9565 -24.0000y = 24.0000 11.9565 -3.0652y = -2.9674 30.0000 -3.0652y = -2.9674 11.7826 -24.0000y = 24.0000 11.7826 -3.0109y = -2.8370 30.0000 -3.0109y = -2.8370 11.9638 -24.0000y = 24.

21、0000 11.9638 -3.0543y = -2.9728 30.0000 -3.0543y = -2.9728 11.8189 -24.0000y = 24.0000 11.8189 -3.0091y = -2.8641 30.0000 -3.0091y = -2.8641 11.9698 -24.0000y = 24.0000 11.9698 -3.0453y = -2.9774 30.0000 -3.0453y = -2.9774 11.8491 -24.0000y = 24.0000 11.8491 -3.0075y = -2.8868 30.0000 -3.0075y = -2.

22、8868 11.9748 -24.0000y = 24.0000 11.9748 -3.0377y = -2.9811 30.0000 -3.0377y = -2.9811 11.8742 -24.0000y = 24.0000 11.8742 -3.0063y = -2.9057 30.0000 -3.0063y = -2.9057 11.9790 -24.0000y = 24.0000 11.9790 -3.0314y = -2.9843 30.0000 -3.0314y = -2.9843 11.8952 -24.0000y = 24.0000 11.8952 -3.0052y = -2

23、.9214 30.0000 -3.0052y = -2.9214 11.9825 -24.0000y = 24.0000 11.9825 -3.0262y = -2.9869 30.0000 -3.0262y = -2.9869 11.9126 -24.0000y = 24.0000 11.9126 -3.0044y = -2.9345 30.0000 -3.0044y = -2.9345 11.9854 -24.0000y = 24.0000 11.9854 -3.0218y = -2.9891 30.0000 -3.0218y = -2.9891 11.9272 -24.0000y = 2

24、4.0000 11.9272 -3.0036y = -2.9454 30.0000 -3.0036y = -2.9454 11.9879 -24.0000y = 24.0000 11.9879 -3.0182y = -2.9909 30.0000 -3.0182y = -2.9909 11.9393 -24.0000y = 24.0000 11.9393 -3.0030y = -2.9545 30.0000 -3.0030y = -2.9545 11.9899 -24.0000y = 24.0000 11.9899 -3.0152y = -2.9924 30.0000 -3.0152y = -

25、2.9924 11.9494 -24.0000y = 24.0000 11.9494 -3.0025y = -2.9621 30.0000 -3.0025y = -2.9621 11.9916 -24.0000y = 24.0000 11.9916 -3.0126y = -2.9937 30.0000 -3.0126y = -2.9937 11.9579 -24.0000y = 24.0000 11.9579 -3.0021y = -2.9684 30.0000 -3.0021y = -2.9684 11.9930 -24.0000y = 24.0000 11.9930 -3.0105y =

26、-2.9947 30.0000 -3.0105y = -2.9947 11.9649 -24.0000y = 24.0000 11.9649 -3.0018y = -2.9737 30.0000 -3.0018y = -2.9737 11.9941 -24.0000y = 24.0000 11.9941 -3.0088y = -2.9956 30.0000 -3.0088y = -2.9956 11.9707 -24.0000y = 24.0000 11.9707 -3.0015y = -2.9781 30.0000 -3.0015y = -2.9781 11.9951 -24.0000y =

27、 24.0000 11.9951 -3.0073y = -2.9963 30.0000 -3.0073y = -2.9963 11.9756 -24.0000y = 24.0000 11.9756 -3.0012y = -2.9817 30.0000 -3.0012y = -2.9817 11.9959 -24.0000y = 24.0000 11.9959 -3.0061y = -2.9970 30.0000 -3.0061y = -2.9970 11.9797 -24.0000y = 24.0000 11.9797 -3.0010y = -2.9848 30.0000 -3.0010y =

28、 -2.9848 11.9966 -24.0000y = 24.0000 11.9966 -3.0051y = -2.9975 30.0000 -3.0051y = -2.9975 11.9831 -24.0000y = 24.0000 11.9831 -3.0008y = -2.9873 30.0000 -3.0008y = -2.9873 11.9972 -24.0000y = 24.0000 11.9972 -3.0042y = -2.9979 30.0000 -3.0042y = -2.9979 11.9859 -24.0000y = 24.0000 11.9859 -3.0007y

29、= -2.9894 30.0000 -3.0007y = -2.9894 11.9976 -24.0000y = 24.0000 11.9976 -3.0035y = -2.9982 30.0000 -3.0035y = -2.9982 11.9882 -24.0000y = 24.0000 11.9882 -3.0006y = -2.9912 30.0000 -3.0006y = -2.9912 11.9980 -24.0000y = 24.0000 11.9980 -3.0029y = -2.9985 30.0000 -3.0029y = -2.9985 11.9902 -24.0000y

30、 = 24.0000 11.9902 -3.0005y = -2.9927 30.0000 -3.0005y = -2.9927 11.9984 -24.0000y = 24.0000 11.9984 -3.0024y = -2.9988 30.0000 -3.0024y = -2.9988 11.9918 -24.0000y = 24.0000 11.9918 -3.0004y = -2.9939 30.0000 -3.0004y = -2.9939 11.9986 -24.0000y = 24.0000 11.9986 -3.0020y = -2.9990 30.0000 -3.0020y

31、 = -2.9990 11.9932 -24.0000y = 24.0000 11.9932 -3.0003y = -2.9949 30.0000 -3.0003y = -2.9949 11.9989 -24.0000y = 24.0000 11.9989 -3.0017y = -2.9991 30.0000 -3.0017y = -2.9991 11.9943 -24.0000y = 24.0000 11.9943 -3.0003y = -2.9957 30.0000 -3.0003y = -2.9957 11.9991 -24.0000y = 24.0000 11.9991 -3.0014

32、y = -2.9993 30.0000 -3.0014y = -2.9993 11.9953 -24.0000y = 24.0000 11.9953 -3.0002y = -2.9965 30.0000 -3.0002y = -2.9965 11.9992 -24.0000y = 24.0000 11.9992 -3.0012y = -2.9994 30.0000 -3.0012y = -2.9994 11.9961 -24.0000y = 24.0000 11.9961 -3.0002y = -2.9970 30.0000 -3.0002y = -2.9970 11.9993 -24.000

33、0y = 24.0000 11.9993 -3.0010y = -2.9995 30.0000 -3.0010y = -2.9995 11.9967 -24.0000y = 24.0000 11.9967 -3.0002y = -2.9975 30.0000 -3.0002y = -2.9975 11.9995 -24.0000y = 24.0000 11.9995 -3.0008y = -2.9996 30.0000 -3.0008y = -2.9996 11.9973 -24.0000y = 24.0000 11.9973 -3.0001y = -2.9979 30.0000 -3.000

34、1y = -2.9979 11.9995 -24.0000y = 24.0000 11.9995 -3.0007y = -2.9997 30.0000 -3.0007y = -2.9997 11.9977 -24.0000y = 24.0000 11.9977 -3.0001y = -2.9983 30.0000 -3.0001y = -2.9983 11.9996 -24.0000y = 24.0000 11.9996 -3.0006y = -2.9997 30.0000 -3.0006y = -2.9997 11.9981 -24.0000y = 24.0000 11.9981 -3.00

35、01y = -2.9986 30.0000 -3.0001y = -2.9986 11.9997 -24.0000y = 24.0000 11.9997 -3.0005y = -2.9998 30.0000 -3.0005y = -2.9998 11.9984 -24.0000y = 24.0000 11.9984 -3.0001y = -2.9988 30.0000 -3.0001y = -2.9988 11.9997 -24.0000y = 24.0000 11.9997 -3.0004y = -2.9998 30.0000 -3.0004y = -2.9998 11.9987 -24.0

36、000y = 24.0000 11.9987 -3.0001x = -2.9998 11.9987 -3.0001k = 100index = 0练习题2. 用Gauss-Seidel迭代法求解上题的线性方程组，取。源程序：function x,k,index=GaussSeidel(A,b,ep,it_max)if nargin<4 it_max=100;endif nargin<3 ep=1e-5;endn=length(A);k=0;x=zeros(n,1);y=x;e=x;index=1;while 1 for i=1:n y(i)=b(i) for j=1:n if j

37、= i y(i)=y(i)-A(i,j)*x(j); end end if abs(A(i,i)<1e-10|k= it_max index=0; return; end y(i)=y(i)/A(i,i); e(i)=y(i)-x(i); x(i)=y(i); end if norm(e,inf)<ep break; end x=y;k=k+1;end x;return;主函数：A=4 3 0;3 3 -1;0 -1 4b=24;30;-24x,k,index=GaussSeidel(A,b)输出结果：A = 4 3 0 3 3 -1 0 -1 4b = 24 30 -24y =

38、24 0 0y = 6 30 0y = 6 4 -24y = 24 4 -5y = 3 30 -5y = 3.0000 5.3333 -24.0000y = 24.0000 5.3333 -4.6667y = 2.0000 30.0000 -4.6667y = 2.0000 6.4444 -24.0000y = 24.0000 6.4444 -4.3889y = 1.1667 30.0000 -4.3889y = 1.1667 7.3704 -24.0000y = 24.0000 7.3704 -4.1574y = 0.4722 30.0000 -4.1574y = 0.4722 8.1420 -24.0000y = 24.0000 8.1420 -3.9645y = -0.1065 30.0000 -3.9645y = -0.1065 8.7850 -24.0000y = 24.0000 8.7850 -3.8038y = -0.5887 30.0000 -3.8038y = -0.5887 9.3208 -24.0000y = 24.0000 9.