实验四实验报告线性方程组的迭代解法.docx
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实验四实验报告线性方程组的迭代解法.docx
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实验四实验报告线性方程组的迭代解法
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)
function[x,k,index]=Jacobi(A,b,ep,it_max)
%求解线性方程组的Jacobi迭代法,其中,
%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.
ifnargin<4it_max=100;end
ifnargin<3ep=1e-5;end
n=length(A);k=0;
x=zeros(n,1);y=x;index=1;
whilel
fori=1:
n
y(i)=b(i)
forj=1:
n
ifj=i
y(i)=y(i)-A(i,j)*x(j);
end
end
ifabs(A(i,i))<1e-10|k==it_max
index=0;return;
end
y(i)=y(i)/A(i,i);
end
ifnorm(y-x,inf) break; end e=y;k=k+1; end 2.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: function[x,k,index]=Jacobi(A,b,ep,it_max) %£¥Çó½âÏßÐÔ·½³Ì×éµÄJacobiµü´ú·¨,ÆäÖÐ, %£¥AΪ·½³Ì×éµÄϵÊý¾ØÕó; %£¥bΪ·½³Ì×éµÄÓÒ¶ËÏî; %£¥epΪ¾«¶ÈÒªÇó£¬È±Ê¡Öµ1e-5; %£¥it_maxΪ×î´óÑ¡´ú´ÎÊý£¬È±Ê¡Öµ100; %£¥xΪ·½³Ì×éµÄ½â; %£¥kΪµü´ú´ÎÊý; %£¥indexΪָ±ê±äÁ¿index=1±íʾµü´úÊÕÁ²µ½Ö¸¶¨ÒªÇó, %£¥index=0±íʾµü´úʧ°Ü. ifnargin<4it_max=100; end ifnargin<3ep=1e-5; end n=length(A); k=0; x=zeros(n,1); y=x; index=1; while1 fori=1: n y(i)=b(i) forj=1: n ifj~=i y(i)=y(i)-A(i,j)*x(j); end end ifabs(A(i,i))<1e-10|k==it_max index=0; return; end y(i)=y(i)/A(i,i); end ifnorm(y-x,inf) break; end x=y;k=k+1; end 主函数: A=[430;33-1;0-14] b=[24;30;-24] [x,k,index]=Jacobi(A,b) 输出结果: A= 430 33-1 0-14 b= 24 30 -24 y= 24 0 0 y= 6 30 0 y= 6 10 -24 y= 24 10 -6 y= -1.5000 30.0000 -6.0000 y= -1.5000 2.0000 -24.0000 y= 24.0000 2.0000 -3.5000 y= 4.5000 30.0000 -3.5000 y= 4.5000 10.3333 -24.0000 y= 24.0000 10.3333 -5.5000 y= -1.7500 30.0000 -5.5000 y= -1.7500 3.6667 -24.0000 y= 24.0000 3.6667 -3.4167 y= 3.2500 30.0000 -3.4167 y= 3.2500 10.6111 -24.0000 y= 24.0000 10.6111 -5.0833 y= -1.9583 30.0000 -5.0833 y= -1.9583 5.0556 -24.0000 y= 24.0000 5.0556 -3.3472 y= 2.2083 30.0000 -3.3472 y= 2.2083 10.8426 -24.0000 y= 24.0000 10.8426 -4.7361 y= -2.1319 30.0000 -4.7361 y= -2.1319 6.2130 -24.0000 y= 24.0000 6.2130 -3.2894 y= 1.3403 30.0000 -3.2894 y= 1.3403 11.0355 -24.0000 y= 24.0000 11.0355 -4.4468 y= -2.2766 30.0000 -4.4468 y= -2.2766 7.1775 -24.0000 y= 24.0000 7.1775 -3.2411 y= 0.6169 30.0000 -3.2411 y= 0.6169 11.1962 -24.0000 y= 24.0000 11.1962 -4.2056 y= -2.3972 30.0000 -4.2056 y= -2.3972 7.9812 -24.0000 y= 24.0000 7.9812 -3.2009 y= 0.0141 30.0000 -3.2009 y= 0.0141 11.3302 -24.0000 y= 24.0000 11.3302 -4.0047 y= -2.4977 30.0000 -4.0047 y= -2.4977 8.6510 -24.0000 y= 24.0000 8.6510 -3.1674 y= -0.4883 30.0000 -3.1674 y= -0.4883 11.4418 -24.0000 y= 24.0000 11.4418 -3.8372 y= -2.5814 30.0000 -3.8372 y= -2.5814 9.2092 -24.0000 y= 24.0000 9.2092 -3.1395 y= -0.9069 30.0000 -3.1395 y= -0.9069 11.5349 -24.0000 y= 24.0000 11.5349 -3.6977 y= -2.6511 30.0000 -3.6977 y= -2.6511 9.6743 -24.0000 y= 24.0000 9.6743 -3.1163 y= -1.2557 30.0000 -3.1163 y= -1.2557 11.6124 -24.0000 y= 24.0000 11.6124 -3.5814 y= -2.7093 30.0000 -3.5814 y= -2.7093 10.0619 -24.0000 y= 24.0000 10.0619 -3.0969 y= -1.5464 30.0000 -3.0969 y= -1.5464 11.6770 -24.0000 y= 24.0000 11.6770 -3.4845 y= -2.7577 30.0000 -3.4845 y= -2.7577 10.3849 -24.0000 y= 24.0000 10.3849 -3.0808 y= -1.7887 30.0000 -3.0808 y= -1.7887 11.7308 -24.0000 y= 24.0000 11.7308 -3.4038 y= -2.7981 30.0000 -3.4038 y= -2.7981 10.6541 -24.0000 y= 24.0000 10.6541 -3.0673 y= -1.9906 30.0000 -3.0673 y= -1.9906 11.7757 -24.0000 y= 24.0000 11.7757 -3.3365 y= -2.8318 30.0000 -3.3365 y= -2.8318 10.8784 -24.0000 y= 24.0000 10.8784 -3.0561 y= -2.1588 30.0000 -3.0561 y= -2.1588 11.8131 -24.0000 y= 24.0000 11.8131 -3.2804 y= -2.8598 30.0000 -3.2804 y= -2.8598 11.0654 -24.0000 y= 24.0000 11.0654 -3.0467 y= -2.2990 30.0000 -3.0467 y= -2.2990 11.8442 -24.0000 y= 24.0000 11.8442 -3.2337 y= -2.8832 30.0000 -3.2337 y= -2.8832 11.2211 -24.0000 y= 24.0000 11.2211 -3.0389 y= -2.4159 30.0000 -3.0389 y= -2.4159 11.8702 -24.0000 y= 24.0000 11.8702 -3.1947 y= -2.9026 30.0000 -3.1947 y= -2.9026 11.3509 -24.0000 y= 24.0000 11.3509 -3.0325 y= -2.5132 30.0000 -3.0325 y= -2.5132 11.8918 -24.0000 y= 24.0000 11.8918 -3.1623 y= -2.9189 30.0000 -3.1623 y= -2.9189 11.4591 -24.0000 y= 24.0000 11.4591 -3.0270 y= -2.5943 30.0000 -3.0270 y= -2.5943 11.9099 -24.0000 y= 24.0000 11.9099 -3.1352 y= -2.9324 30.0000 -3.1352 y= -2.9324 11.5493 -24.0000 y= 24.0000 11.5493 -3.0225 y= -2.6620 30.0000 -3.0225 y= -2.6620 11.9249 -24.0000 y= 24.0000 11.9249 -3.1127 y= -2.9437 30.0000 -3.1127 y= -2.9437 11.6244 -24.0000 y= 24.0000 11.6244 -3.0188 y= -2.7183 30.0000 -3.0188 y= -2.7183 11.9374 -24.0000 y= 24.0000 11.9374 -3.0939 y= -2.9530 30.0000 -3.0939 y= -2.9530 11.6870 -24.0000 y= 24.0000 11.6870 -3.0157 y= -2.7652 30.0000 -3.0157 y= -2.7652 11.9478 -24.0000 y= 24.0000 11.9478 -3.0783 y= -2.9609 30.0000 -3.0783 y= -2.9609 11.7392 -24.0000 y= 24.0000 11.7392 -3.0130 y= -2.8044 30.0000 -3.0130 y= -2.8044 11.9565 -24.0000 y= 24.0000 11.9565 -3.0652 y= -2.9674 30.0000 -3.0652 y= -2.9674 11.7826 -24.0000 y= 24.0000 11.7826 -3.0109 y= -2.8370 30.0000 -3.0109 y= -2.8370 11.9638 -24.0000 y= 24.0000 11.9638 -3.0543 y= -2.9728 30.0000 -3.0543 y= -2.9728 11.8189 -24.0000 y= 24.0000 11.8189 -3.0091 y= -2.8641 30.0000 -3.0091 y= -2.8641 11.9698 -24.0000 y= 24.0000 11.9698 -3.0453 y= -2.9774 30.0000 -3.0453 y= -2.9774 11.8491 -24.0000 y= 24.0000 11.8491 -3.0075 y= -2.8868 30.0000 -3.0075 y= -2.8868 11.9748 -24.0000 y= 24.0000 11.9748 -3.0377 y= -2.9811 30.0000 -3.0377 y= -2.9811 11.8742 -24.0000 y= 24.0000 11.8742 -3.0063 y= -2.9057 30.0000 -3.0063 y= -2.9057 11.9790 -24.0000 y= 24.0000 11.9790 -3.0314 y= -2.9843 30.0000 -3.0314 y= -2.9843 11.8952 -24.0000 y= 24.0000 11.8952 -3.0052 y= -2.9214 30.0000 -3.0052 y= -2.9214 11.9825 -24.0000 y= 24.0000 11.9825 -3.0262 y= -2.9869 30.0000 -3.0262 y= -2.9869 11.9126 -24.0000 y= 24.0000 11.9126 -3.0044 y= -2.9345 30.0000 -3.0044 y= -2.9345 11.9854 -24.0000 y= 24.0000 11.9854 -3.0218 y= -2.9891 30.0000 -3.0218 y= -2.9891 11.9272 -24.0000 y= 24.0000 11.9272 -3.0036 y= -2.9454 30.0000 -3.0036 y= -2.9454 11.9879 -24.0000 y= 24.0000 11.9879 -3.0182 y= -2.9909 30.0000 -3.0182 y= -2.9909 11.9393 -24.0000 y= 24.0000 11.9393 -3.0030 y= -2.9545 30.0000 -3.0030 y= -2.9545 11.9899 -24.0000 y= 24.0000 11.9899 -3.0152 y= -2.9924 30.0000 -3.0152 y= -2.9924 11.9494 -24.0000 y= 24.0000 11.9494 -3.0025 y= -2.9621 30.0000 -3.0025 y= -2.9621 11.9916 -24.0000 y= 24.0000 11.9916 -3.0126 y= -2.9937 30.0000 -3.0126 y= -2.9937 11.9579 -24.0000 y= 24.0000 11.9579 -3.0021 y= -2.9684 30.0000 -3.0021 y= -2.9684 11.9930 -24.0000 y= 24.0000 11.9930 -3.0105 y= -2.9947 30.0000 -3.0105 y= -2.9947 11.9649 -24.0000 y= 24.0000 11.9649 -3.0018 y= -2.9737 30.0000 -3.0018 y= -2.9737 11.9941 -24.0000 y= 24.0000 11.9941 -3.0088 y= -2.9956 30.0000 -3.0088 y= -2.9956 11.9707 -24.0000 y= 24.0000 11.9707 -3.0015 y= -2.9781 30.0000 -3.0015 y= -2.9781 11.9951 -24.0000 y= 24.0000 11.9951 -3.0073 y= -2.9963 30.0000 -3.0073 y= -2.9963 11.9756 -24.0000 y= 24.0000 11.9756 -3.0012 y= -2.9817 30.0000 -3.0012 y= -2.9817 11.9959 -24.0000 y= 24.0000 11.9959 -3.0061 y= -2.9970 30.0000 -3.0061 y= -2.9970 11.9797 -24.0000 y= 24.0000 11.9797 -3.0010 y= -2.9848 30.0000 -3.0010 y= -2.9848 11.9966 -24.0000 y= 24.0000 11.9966 -3.0051 y= -2.9975 30.0000 -3.0051 y= -2.9975 11.9831 -24.0000 y= 24.0000 11.9831 -3.0008 y= -2.9873 30.0000 -3.0008 y= -2.9873 11.9972 -24.0
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