数值迭代法.doc

雅可比迭代：function x1=Jacobi(A,b,x0)D=diag(diag(A);L=-tril(A,-1);U=-triu(A,1);BJ=D(L+U);gJ=Db;x1=BJ*x0+gJ; A=-5 -1 22 6 -32 1 7; b=1 2 32; x0=0 0 0; k=0; x1=Jacobi(A,b,x0); while norm(x1-x0,inf)=1e-6,x0=x1;x1=Jacobi(A,b,x0);k=k+1;end x1 x1 = 1.0000 2.0000 4.0000 kk =19GS迭代：function x1=GS(A,b,x0)D=diag(diag(A);L=-tril(A,-1);U=-triu(A,1);BG=(D-L)U;gG=(D-L)b;x1=BG*x0+gG; A=-5 -1 22 6 -32 1 7;b=1 2 32; x0=0 0 0; k=0; x1=GS(A,b,x0); while norm(x1-x0,inf)=1e-6,x0=x1;x1=GS(A,b,x0);k=k+1;end x1x1 = 1.0000 2.0000 4.0000 kk = 11SOR迭代：function x1=SOR(A,b,w,x0)D=diag(diag(A);L=-tril(A,-1);U=-triu(A,1);Bs=(D-w*L)(1-w)*D+w*U);gs=(D-w*L)(w*b);x1=Bs*x0+gs; A=-5 -1 22 6 -32 1 7; b=1 2 32; for i=1:19,w=0.1*i;k=0;x0=0 0 0;x1=SOR(A,b,w,x0);while norm(x1-x0,inf)=1e-6,x0=x1;x1=SOR(A,b,w,x0);k=k+1;endx1kwend结果如下：收敛最快的是w=0.9 或1. 当w=1.5 、1.6、 1.7 、1.8 、1.9时发散。X1wkX1wk 1.0000 2.0000 4.00000.1000134 1.0000 2.0000 4.00001.100018 1.0000 2.0000 4.00000.200070 1.0000 2.0000 4.00001.200030 1.0000 2.0000 4.00000.300046 1.0000 2.0000 4.00001.300055 1.0000 2.0000 4.00000.400034 1.0000 2.0000 4.00001.4000183 1.0000 2.0000 4.00000.500026 NaN NaN NaN1.50008890 1.0000 2.0000 4.00000.600021 NaN NaN NaN1.60003218 1.0000 2.0000 4.00000.700017 NaN NaN NaN1.70002072 1.0000 2.0000 4.00000.800014 NaN NaN NaN1.80001578 1.0000 2.00