常微分方程的解線性方程組的迭代法

實(shí)驗(yàn)五解線性方程組的迭代法【實(shí)驗(yàn)容】對(duì)1、設(shè)線性方程組42-3-1210000、(/586—5-3650100x21242-2-132-1031x330-215-13-1194x42-426-167-3323%5=386-8571726-35x64602-13-425301x7131610-11—917342-122x838462-713920124x91900-18-3-24-863-D丫kx"k-21k7一1,0,0,3,1,-1,1,2,x*=G,2、設(shè)對(duì)稱正定系數(shù)陣線性方程組42一40240022—142一40240022—1-21320-4-1141-8-356=G,0216-1-4-3341—8-1224-10-3-1,0,23-3-44111-42,1,025-3-101142-1,0063-3-4219x1x2x3X4X5x6x7x80-620239-22-15453、三對(duì)角形線性方程組4-100000000-14-14-100000000-14-100000000-14-1000000=(2,00-14-1000001,000-14-100000000-14-10000,00000-14-100000000-1

4-1

01,—2,0000000-14-100000000-14X1X2X3X4X5X6X7X8X9X103,0,1, -1方75-1326-1214-45-5試分別選用Jacobi迭代法,Gauss-Seidol迭代法和SOR^^it算其解?！緦?shí)驗(yàn)方法或步驟】1、體會(huì)迭代法求解線性方程組，并能與消去法加以比較;2、分別對(duì)不同精度要求，如￡=10-3,10-4,10-5由迭代次數(shù)體會(huì)該迭^法的收斂快慢;3、對(duì)方程組2,3使用50口方法時(shí)，選取松弛因子3=0.8,0.9,1,1.1,1.2等，試看對(duì)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(jì)程序和計(jì)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?yàn)锳=[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=Columns1through4TOC\o"1-5"\h\z4 2 -3 -18 6 -5 -34 2 -2 -10 -2 1 5-4 2 6 -16 -8 50 2 -1 316 10 -11 -96 2 -70 0 -1 8Columns5through8TOC\o"1-5"\h\z5 0 12-10-13-117-3317 2 6-4 2 5 317 34 2 -113 9 2 0-3 -24 -8 6Columns9through100000319423TOC\o"1-5"\h\z-3 5012212 43 -151232346133819-21>>x0=ones(10,1);>>[x,n]=Jacobi(A,b,x0)match得到的結(jié)果為Warning:FunctioncallJacobiinvokesinexactmatchd:\MATLAB7\work\jacobi.m.不收斂！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 -4 0420022 -1 21320

i(0x,q,v)!qo3Bf=[u^]<<f(I,8)sauo=ox<<SVST-ZZ-60?9-61ZZPTP-IE-01-61ZZPTP-IE-01-￡ ￡-9 Sp- E-T 0T-TT PP ZZP- l-E- 8-￡ 9E- SV- E-T- 8-9 TT VI0 0Z 0￡ VT 33- 0T- P-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]'Columns1through54-1000-14-100TOC\o"1-5"\h\z0 -1 4 -1 00 0 -1 4 -10 0 0 -1 40 0 0 0 -100000000000000000000Columns6through1000000000000000000000-1 0 0 0 04-1000-1 4 -1 0 0=X(0x,q,v)!qo3Bf=[u^]<<;(l,OT)S9uo=Ox<<s-Sp-PlZl-9Z￡T-SLTOC\o"1-5"\h\zV T- 0 0 0T- V T- 0 0T-V T- 0T-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?tīng)栐闯绦騠unction[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:^tt^！');return;endend解上面3個(gè)方程組的程序分別為

第一個(gè)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]'4286420 -24286420 -2-4 2860216 1046003 -15 -32 -1156 -1-8 5-1 3-11 -92 -7-1 82165321 3677 174 217 3413 93 -2400011 01 13 326532 -120-8 60000319423-3 5012212 43 -11232346133819-21x0=zeros(10,1);>>[x,n]=gauseidel(A,b,x0)Warning:7^^!x=1.0e+247*0.0165-0.02710.2202-0.4576-0.5951

0.3138-0.43812.24500.04137.4716n=200即迭代200次后此方程可能不收斂方程2的程序?yàn)锳=[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]'4222-4 -10 4222-4 -10 -221430200-4 0-1 214 116-8 -1-3 -45 -3634213-8 -3-1 -422 44 11-10 1-3 -4002056-3 3-10 -31 -414 22 190-620239-22-1545>>x0=zeros(8,1);>>[x,n]=gauseidel(A,b,x0)x=3.5441-4.88771.94820.45251.4283-1.1606-0.10811.674344即在迭代44次后可求的方程的近似解第3個(gè)的程序?yàn)?gt;>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]'TOC\o"1-5"\h\z4 -1-1 40 -1000000-1 04 -1-1 40 -1000000-1 04 -100000000000000000000(0x,q,v)|dppsne6=[u^]<<;(l,OT)sojaz=ox<<S-Sp-PlZl-9Z￡T-SLTOC\o"1-5"\h\zV T-T- V0 T-0 00 00 0T- 0V T-T- 什0 T-0 00 00 0T- 0什 T-0 00 00 00 00 00 00 00 00 00 02.00001.0000-3.0000-0.00001.0000-2.00003.0000-0.00001.0000-1.0000n=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?yàn)閒unction[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:7tttt！');return;

endend解第一個(gè)方程組選取抄松弛因子為1程序?yàn)椋篈=[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]'4286420 -24286420 -2-4 2860216 1046003 -15 -32 -1156 -1-8 5-1 3-11 -92 -7-1 82165321 3677 174 217 3413 93 -2400011 01 13 326532 -120-8 60000319423-3 5012212 43 -151232346133819-21x0=[0000000000],;[x,n]=SOR(A,b,x0,1)Warning:不收斂！x=1.0e+247*0.0165-0.02710.2202-0.4576-0.59510.3138-0.43812.24500.04137.4716200迭代次數(shù)太多可能不收斂解第2個(gè)的程序?yàn)锳=[42-404200;22-121320;-4-1141-8-356;0-216-4-33;21-8-1224-10-3;43-3-44111-4;025-3-101142;0063-3-4219;],b=[0-620239-22-1545]'4222TOC\o"1-5"\h\z-4 -1-4 0-1 214 14213-8 -3002056TOC\o"1-5"\h\z0 -2 1 6 -1 -4 -3 32 1 -8 -1 22 4 -10 -34 3 -3 -4 4 11 1 -40 2 5 -3 -10 1 14 20 0 6 3 -3 -4 2 19b=0-620239-22-1545>>x0=[00000000]';[x,n]=SOR(A,b,x0,0.8)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=3.5441-4.88771.94820.45251.4283-1.1606-0.10811.674344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(gè)的程序?yàn)?gt;>A=[4-100000000;-14-10000000;0-14-1000000；00-14-100000;000-14-10000;0000-14-10