实验4 Jacobi迭代法和GS迭代2.doc

实验题目1 Jacobi迭代法用Jacobi迭代法求解线性方程组，保留四位有效数字(err1e-4)，其中A=8 -1 1;2 10 -1;1 1 -5; b=1 ;4; 3。2Gauss-Seidel迭代法用Gauss-Seidel迭代法求解线性方程组，保留四位有效数字(err1e-4)，其中A=8 -1 1;2 10 -1;1 1 -5; b=1 ;4; 3。3分别用Jacobi迭代法和Gauss-Seidel迭代法求解方程组要求精度，初始，最大迭代次数N=25，试比较这几种迭代法的迭代次数和收敛速度。1.程序：#include#includeint main()int n=3,i,k,j,mm=1000;/最大迭代次数;float t,xmmn,dxn,dx_max=1,err=1e-4;/精度for(i=0;in;i+) x0i=0;float a33=8,-1,1,2,10,-1,1,1,-5;float b3=1,4,3;for(i=0;in;i+) bi=bi/aii;for(i=0;in;i+) for(j=0;j1e-8;k+) dx_max=0; for(i=0;in;i+) t=0; for(j=0;jn;j+) t+=aij*xkj; xk+1i=bi-t; printf(%0.4f ,xk+1i); dxi=xk+1i-xki; dx_max+=pow(dxi,2); printf( %d,k+1); printf(n); printf(n);程序结果：精度1e-4时，解x3=0.1250 0.4000 -0.6000 10.2500 0.3150 -0.4950 20.2262 0.3005 -0.4870 30.2234 0.3061 -0.4947 40.2251 0.3058 -0.4941 50.2250 0.3056 -0.4938 60.2249 0.3056 -0.4939 70.2249 0.3056 -0.4939 82.程序：#include#includeint main()int n=3,i,k,j,mm=1000;/最大迭代次数;float t,xmmn,dxn,dx_max=1,err=1e-4;/精度for(i=0;in;i+) x0i=0;float a33=8,-1,1,2,10,-1,1,1,-5;float b3=1,4,3;for(i=0;in;i+) bi=bi/aii;for(i=0;in;i+) for(j=0;j1e-8;k+) dx_max=0; for(i=0;in;i+) if(i=0) t=0; for(j=i+1;jn;j+) t+=aij*xkj; xk+1i=bi-t; else t=0; for(j=0;j=i-1;j+)t+=aij*xk+1j; for(j=i+1;jn;j+)t+=aij*xkj; xk+1i=bi-t; printf(%0.4f ,xk+1i); dxi=xk+1i-xki; dx_max+=pow(dxi,2); printf(n); printf(n);程序结果：精度1e-4时，解x3=0.1250 0.3750 -0.50000.2344 0.3031 -0.49250.2245 0.3059 -0.49390.2250 0.3056 -0.49390.2249 0.3056 -0.49393. Jacobi迭代法程序：#include#includeint main()int n=6,i,k,j,mm=1000;/最大迭代次数;float t,xmmn,dxn,dx_max=1,err=1e-5;/精度for(i=0;in;i+) x0i=0;float a66=4,-1,0,-1,0,0,-1,4,-1,0,-1,0,0 ,-1 ,4 ,0 ,0 ,-1,-1, 0, 0 ,4 ,-1 ,0, 0 ,-1 ,0 ,-1, 4 ,-1,0 ,0 ,-1, 0, -1 ,4;float b6=0 ,5 ,0 ,6 ,-2, 6;for(i=0;in;i+) bi=bi/aii;for(i=0;in;i+) for(j=0;jpow(err,2);k+) dx_max=0; for(i=0;in;i+) t=0; for(j=0;jn;j+) t+=aij*xkj; xk+1i=bi-t; printf(%0.4f ,xk+1i); dxi=xk+1i-xki; dx_max+=pow(dxi,2); printf( %d,k+1); printf(n); printf(n);程序结果：a的对角变0处理后的(a66,b6)=0.0000 -0.2500 0.0000 -0.2500 0.0000 0.0000 0.0000-0.2500 0.0000 -0.2500 0.0000 -0.2500 0.0000 1.25000.0000 -0.2500 0.0000 0.0000 0.0000 -0.2500 0.0000-0.2500 0.0000 0.0000 0.0000 -0.2500 0.0000 1.50000.0000 -0.2500 0.0000 -0.2500 0.0000 -0.2500 -0.50000.0000 0.0000 -0.2500 0.0000 -0.2500 0.0000 1.5000精度1e-005时，解x6=0.0000 1.2500 0.0000 1.5000 -0.5000 1.5000 10.6875 1.1250 0.6875 1.3750 0.5625 1.3750 20.6250 1.7344 0.6250 1.8125 0.4688 1.8125 30.8867 1.6797 0.8867 1.7734 0.8398 1.7734 40.8633 1.9033 0.8633 1.9316 0.8066 1.9316 50.9587 1.8833 0.9587 1.9175 0.9417 1.9175 60.9502 1.9648 0.9502 1.9751 0.9296 1.9751 70.9850 1.9575 0.9850 1.9699 0.9787 1.9699 80.9819 1.9872 0.9819 1.9909 0.9743 1.9909 90.9945 1.9845 0.9945 1.9890 0.9923 1.9890 100.9934 1.9953 0.9934 1.9967 0.9907 1.9967 110.9980 1.9944 0.9980 1.9960 0.9972 1.9960 120.9976 1.9983 0.9976 1.9988 0.9966 1.9988 130.9993 1.9979 0.9993 1.9985 0.9990 1.9985 140.9991 1.9994 0.9991 1.9996 0.9988 1.9996 150.9997 1.9993 0.9997 1.9995 0.9996 1.9995 160.9997 1.9998 0.9997 1.9998 0.9995 1.9998 170.9999 1.9997 0.9999 1.9998 0.9999 1.9998 180.9999 1.9999 0.9999 1.9999 0.9998 1.9999 191.0000 1.9999 1.0000 1.9999 1.0000 1.9999 201.0000 2.0000 1.0000 2.0000 0.9999 2.0000 211.0000 2.0000 1.0000 2.0000 1.0000 2.0000 221.0000 2.0000 1.0000 2.0000 1.0000 2.0000 231.0000 2.0000 1.0000 2.0000 1.0000 2.0000 241.0000 2.0000 1.0000 2.0000 1.0000 2.0000 25Gauss-Seidel迭代法程序：#include#includeint main()int n=6,i,k,j,mm=25;/最大迭代次数;float t,xmmn,dxn,dx_max=1,err=1e-5;/精度for(i=0;in;i+) x0i=0;float a66=4,-1,0,-1,0,0,-1,4,-1,0,-1,0,0 ,-1 ,4 ,0 ,0 ,-1,-1, 0, 0 ,4 ,-1 ,0, 0 ,-1 ,0 ,-1, 4 ,-1,0 ,0 ,-1, 0, -1 ,4;float b6=0 ,5 ,0 ,6 ,-2, 6;for(i=0;in;i+) bi=bi/aii;for(i=0;in;i+) for(j=0;jpow(err,2)&kmm;k+) dx_max=0; for(i=0;in;i+) if(i=0) t=0; for(j=i+1;jn;j+) t+=aij*xkj; xk+1i=bi-t; else t=0; for(j=0;j=i-1;j+)t+=aij*xk+1j; for(j=i+1;jn;j+)t+=aij*xkj; xk+1i=bi-t; printf(%0.4f ,xk+1i); dxi=xk+1i-xki; dx_max+=pow(dxi,2); printf( %dn,k+1); printf(n);程序结果：a对角0化处理后的(a66,b61)=0.0000 -0.2500 0.0000 -0.2500 0.0000 0.0000 0.0000-0.2500 0.0000 -0.2500 0.0000 -0.2500 0.0000 1.25000.0000 -0.2500 0.0000 0.0000 0.0000 -0.2500 0.0000-0.2500 0.0000 0.0000 0.0000 -0.2500 0.0000 1.50000.0000 -0.2500 0.0000 -0.2500 0.0000 -0.2500 -0.50000.0000 0.0000 -0.2500 0.0000 -0.2500 0.0000 1.5000精度1e-005时，解x6=0.0000 1.2500 0.3125 1.5000 0.1875 1.6250 10.6875 1.5469 0.7930 1.7188 0.7227 1.8789 20.8164 1.8330 0.9280 1.8848 0.8992 1.9568 30.9294 1.9391 0.9740 1.9572 0.9633 1.9843 40.9741 1.9778 0.9905 1.9843 0.9866 1.9943 50.9905 1.9919 0.9966 1.9943 0.9951 1.9979 60.9966 1.9971 0.9987 1.9979 0.9982 1.9992 70.9987 1.9989 0.9995 1.9992 0.9994 1.9997 80.9995 1.9996 0.99