常微分方程的解線性方程組的迭代法

./實驗五解線性方程組的迭代法[實驗容]對1、設(shè)線性方程組2、設(shè)對稱正定系數(shù)陣線性方程組3、三對角形線性方程組試分別選用Jacobi迭代法,Gauss-Seidol迭代法和SOR方法計算其解。[實驗方法或步驟]1、體會迭代法求解線性方程組,并能與消去法加以比較；2、分別對不同精度要求,如由迭代次數(shù)體會該迭代法的收斂快慢；3、對方程組2,3使用SOR方法時,選取松弛因子ω=0.8,0.9,1,1.1,1.2等,試看對算法收斂性的影響,并能找出你所選用的松弛因子的最佳者；4、給出各種算法的設(shè)計程序和計算結(jié)果。程序：用雅可比方法求的程序：function[x,n]=jacobi<A,b,x0,eps,varargin>ifnargin==3eps=1.0e-6;M=200;elseifnargin==5M=varargin{1};endD=diag<diag<A>>;L=-tril<A,-1>;U=-triu<A,1>;B=D\<L+U>;f=D\b;x=B*x0+f;n=1;whilenorm<x-x0>>=epsx0=x;x=B*x0+f;n=n+1;if<n>=M>diso<'不收斂！'>return;endend解1的程序為A=[42-3-1210000;86-5-3650100;42-2-132-1031;0-215-13-1194;-426-167-3323;86-8571726-35;02-13-425301;1610-11-917342-122;462-713920124;00-18-3-24-863-1;],b=[51232346133819-21]'A=Columns1through442-3-186-5-342-2-10-215-426-186-8502-131610-11-9462-700-18Columns5through82100650132-10-13-1167-3371726-425317342-113920-3-24-86Columns9through100000319423-3501221243-1b=51232346133819-21>>x0=ones<10,1>;>>[x,n]=Jacobi<A,b,x0>得到的結(jié)果為Warning:FunctioncallJacobiinvokesinexactmatchd:\MATLAB7\work\jacobi.m.不收斂！x=1.0e+124*-0.1794-0.3275-0.70941.59901.03110.32910.24644.39050.4927-2.6574n=200即迭代了200次而且可能不收斂A=42-40420022-121320-4-1141-8-3560-216-1-4-3321-8-1224-10-343-3-44111-4025-3-1011420063-3-4219b=0-620239-22-1545>>x0=ones<8,1>;>>[x,n]=Jacobi<A,b,x0>不收斂！x=1.0e+047*0.96271.0084-0.4954-0.59790.30970.6872-0.0666-0.2629n=200此方程可能不收斂A=[4-100000000;-14-10000000;0-14-1000000;00-14-100000;000-14-10000;0000-14-1000;00000-14-100;000000-14-10;0000000-14-1;00000000-14;],b=[75-1326-1214-45-5]'A=Columns1through54-1000-14-1000-14-1000-14-1000-140000-100000000000000000000Columns6through1000000000000000000000-100004-1000-14-1000-14-1000-14-1000-14b=75-1326-1214-45-5>>x0=ones<10,1>;>>[x,n]=Jacobi<A,b,x0>x=2.00001.0000-3.00000.00001.0000-2.00003.00000.00001.0000-1.0000n=22得到結(jié)果為迭代了22次得到近似解為x=2.00001.0000-3.00000.00001.0000-2.00003.00000.00001.0000-1.0000用高斯賽德爾源程序function[x,n]=gauseidel<A,b,x0,eps,M>ifnargin==3eps=1.0e-6;M=200;elseifnargin==4M=200;elseifnargin<3errorreturn;endD=diag<diag<A>>;L=-tril<A,-1>;U=-triu<A,1>;G=<D-L>\U;f=<D-L>\b;x=G*x0+f;n=1;whilenorm<x-x0>>=epsx0=x;x=G*x0+f;n=n+1;if<n>=M>disp<'Warning:不收斂！'>;return;endend解上面3個方程組的程序分別為第一個方程A=[42-3-1210000;86-5-3650100;42-2-132-1031;0-215-13-1194;-426-167-3323;86-8571726-35;02-13-425301;1610-11-917342-122;462-713920124;00-18-3-24-863-1;],b=[51232346133819-21]'A=42-3-121000086-5-365010042-2-132-10310-215-13-1194-426-167-332386-8571726-3502-13-4253011610-11-917342-122462-71392012400-18-3-24-863-1b=51232346133819-21x0=zeros<10,1>;>>[x,n]=gauseidel<A,b,x0>Warning:不收斂！x=1.0e+247*0.0165-0.02710.2202-0.4576-0.59510.3138-0.43812.24500.04137.4716n=200即迭代200次后此方程可能不收斂方程2的程序為A=[42-404200;22-121320;-4-1141-8-356;0-216-1-4-33;21-8-1224-10-3;43-3-44111-4;025-3-101142;0063-3-4219;],b=[0-620239-22-1545]'A=42-40420022-121320-4-1141-8-3560-216-1-4-3321-8-1224-10-343-3-44111-4025-3-1011420063-3-4219b=0-620239-22-1545>>x0=zeros<8,1>;>>[x,n]=gauseidel<A,b,x0>x=3.5441-4.88771.94820.45251.4283-1.1606-0.10811.6743n=44即在迭代44次后可求的方程的近似解第3個的程序為>>A=[4-100000000;-14-10000000;0-14-1000000;00-14-100000;000-14-10000;0000-14-1000;00000-14-100;000000-14-10;0000000-14-1;00000000-14;],b=[75-1326-1214-45-5]'A=4-100000000-14-100000000-14-100000000-14-100000000-14-100000000-14-100000000-14-100000000-14-100000000-14-100000000-14b=75-1326-1214-45-5>>x0=zeros<10,1>;>>[x,n]=gauseidel<A,b,x0>x=2.00001.0000-3.0000-0.00001.0000-2.00003.0000-0.00001.0000-1.0000n=12在迭代12次后得到方程的近似解用超松弛迭代的源程序為function[x,n]=SOR<A,b,x0,w,eps,M>ifnargin==4eps=1.0e-6;M=200;elseifnargin<4errorreturnelseifnargin==5M=200;endif<w<=0||w>=2>error;return;endD=diag<diag<A>>;L=-tril<A,-1>;U=-triu<A,1>;B=inv<D-L*w>*<<1-w>*D+w*U>;f=w*inv<<D-L*w>>*b;x=B*x0+f;n=1;whilenorm<x-x0>>=epsx0=x;x=B*x0+f;n=n+1;if<n>=M>disp<'Warning:不收斂！'>;return;endend解第一個方程組選取抄松弛因子為1程序為：A=[42-3-1210000;86-5-3650100;42-2-132-1031;0-215-13-1194;-426-167-3323;86-8571726-35;02-13-425301;1610-11-917342-122;462-713920124;00-18-3-24-863-1;],b=[51232346133819-21]'A=42-3-121000086-5-365010042-2-132-10310-215-13-1194-426-167-332386-8571726-3502-13-4253011610-11-917342-122462-71392012400-18-3-24-863-1b=51232346133819-21x0=[0000000000]';[x,n]=SOR<A,b,x0,1>Warning:不收斂！x=1.0e+247*0.0165-0.02710.2202-0.4576-0.59510.3138-0.43812.24500.04137.4716n=200迭代次數(shù)太多可能不收斂解第2個的程序為A=[42-404200;22-121320;-4-1141-8-356;0-216-1-4-33;21-8-1224-10-3;43-3-44111-4;025-3-101142;0063-3-4219;],b=[0-620239-22-1545]'A=42-40420022-121320-4-1141-8-3560-216-1-4-3321-8-1224-10-343-3-44111-4025-3-1011420063-3-4219b=0-620239-22-1545>>x0=[00000000]';[x,n]=SOR<A,b,x0,0.8>x=3.5441-4.88771.94820.45251.4283-1.1606-0.10811.6743n=52得到結(jié)果為迭代了52次得到近似解松弛因子為0.8[x,n]=SOR<A,b,x0,0.9>x=3.5441-4.88771.94820.45251.4283-1.1606-0.10811.6743n=47得到結(jié)果為迭代了47次得到近似解松弛因子為0.9>[x,n]=SOR<A,b,x0,1>x=3.5441-4.88771.94820.45251.4283-1.1606-0.10811.6743n=44得到結(jié)果為迭代了44次得到近似解松弛因子為1.0解第3個的程序為>>A=[4-100000000;-14-10000000;0-14-1000000;00-14-100000;000-14-10000;0000-14-1000;00000-14-100;000000-14-10;0000000-14-1;00000000-14;],b=[75-1326-1214-45-5]'A=4-1000000