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1、数学实验线性代数方程组的数值解法2010011811 毕啸天实验5 线性代数方程组的数值解法化工系 毕啸天 2010011811【实验目的】1. 学会用MATLAB 软件数值求解线性代数方程组，对迭代法的收敛性和解的稳定性作初步分析；2. 通过实例学习用线性代数方程组解决简化的实际问题。【实验内容】题目3已知方程组Ax=b，其中AR2020，定义为试通过迭代法求解此方程组，认识迭代法收敛的含义以及迭代初值和方程组系数矩阵性质对收敛速度的影响。实验要求：（1）选取不同的初始向量x(0)和不同的方程组右端项向量b ，给定迭代误差要求，用雅可比迭代法和高斯-赛德尔迭代法计算，观测得到的迭代向量序列是
2、否均收敛？若收敛，记录迭代次数，分析计算结果并得出你的结论；（2）取定右端向量b和初始向量x(0)，将A的主对角线元素成倍增长若干次，非主对角线元素不变，每次用雅可比迭代法计算，要求迭代误差满足x(k+1)-x(k)10-5，比较收敛速度，分析现象并得出你的结论。3.1 模型分析选取初始向量x(0) =(1,1,1)T ，b=(1,1,1)T，迭代要求为误差满足x(k+1)-x(k)m x(:,k+2)=B1*x(:,k+1)+f1; k=k+1;endendfunction x = Gauss( x0,A,b,m )D=diag(diag(A);U=-triu(A,1);L=-tril(A,
3、-1);B2=(D-L)U;f2=(D-L)b;x(:,1)=x0;x(:,2)=B2*x(:,1)+f2;k=1;while norm(x(:,k+1)-x(:,k),inf)m x(:,k+2)=B2*x(:,k+1)+f2; k=k+1;endendA1=3.*eye(20,20); A2=sparse(1:19,2:20,-1/2,20,20);A3=sparse(1:18,3:20,-1/4,20,20);AA=A1+A2+A3+A2+A3;A=full(AA);b=ones(20,1); %输入自选右端项向量bx0=ones(20,1); %输入自选初始向量x0m=1e-5;x1=
4、Jacob(x0,A,b,m);x2=Gauss(x0,A,b,m);结果输出数据：k01234567x各分量10.5833330.5277780.4994210.4895830.4851390.4832390.48237410.750.6388890.6024310.5860340.5791620.5760460.57463910.8333330.7152780.6701390.6495950.6405450.6363960.63448710.8333330.7430560.6932870.6715860.6611850.65640.65413710.8333330.750.7042820.
5、6817130.6709150.6657180.66323810.8333330.750.7077550.6855710.6747160.6693670.66676810.8333330.750.7083330.6870660.6762720.6709240.66827410.8333330.750.7083330.6874520.676850.6715350.66888310.8333330.750.7083330.68750.6770390.6717670.66912710.8333330.750.7083330.68750.6770790.6718440.6692110.8333330.
6、750.7083330.68750.6770790.6718440.6692110.8333330.750.7083330.68750.6770390.6717670.66912710.8333330.750.7083330.6874520.676850.6715350.66888310.8333330.750.7083330.6870660.6762720.6709240.66827410.8333330.750.7077550.6855710.6747160.6693670.66676810.8333330.750.7042820.6817130.6709150.6657180.66323
7、810.8333330.7430560.6932870.6715860.6611850.65640.65413710.8333330.7152780.6701390.6495950.6405450.6363960.63448710.750.6388890.6024310.5860340.5791620.5760460.57463910.5833330.5277780.4994210.4895830.4851390.4832390.482374k89101112131415x各分量值0.481980.4817980.4817130.4816720.4816530.4816440.481640.4
8、816380.5739880.5736860.5735430.5734760.5734440.5734290.5734210.5734180.6335970.6331790.6329810.6328870.6328430.6328210.6328110.6328060.6530710.6525650.6523240.6522080.6521530.6521260.6521130.6521070.6620480.6614770.6612020.6610690.6610050.6609740.6609590.6609520.6655040.664890.6645910.6644460.664376
9、0.6643410.6643250.6643160.6669720.6663330.6660190.6658650.665790.6657530.6657350.6657270.6675650.6669130.666590.6664310.6663530.6663140.6662950.6662860.6678060.6671480.6668210.6666590.6665780.6665390.6665190.666510.667890.667230.6669010.6667380.6666570.6666170.6665970.6665870.667890.667230.6669010.6
10、667380.6666570.6666170.6665970.6665870.6678060.6671480.6668210.6666590.6665780.6665390.6665190.666510.6675650.6669130.666590.6664310.6663530.6663140.6662950.6662860.6669720.6663330.6660190.6658650.665790.6657530.6657350.6657270.6655040.664890.6645910.6644460.6643760.6643410.6643250.6643160.6620480.6
11、614770.6612020.6610690.6610050.6609740.6609590.6609520.6530710.6525650.6523240.6522080.6521530.6521260.6521130.6521070.6335970.6331790.6329810.6328870.6328430.6328210.6328110.6328060.5739880.5736860.5735430.5734760.5734440.5734290.5734210.5734180.481980.4817980.4817130.4816720.4816530.4816440.481640
12、.4816383.3改变迭代初始值3.3.1将x0各分量初值置为0增加一句代码为:x0=zeros(20,1);k0123456x各分量值00.3333330.4166670.4537040.4689430.4758230.47892800.3333330.4722220.5277780.5526620.5637860.56891100.3333330.50.5717590.6045520.6195670.62655900.3333330.50.5810190.6184410.6361560.64449900.3333330.50.5833330.623650.6430040.65230800.3333330.50.5833330.6248070.6450620.65494700.3333330.50.5833330.6250.6456890.65588300.3333330.50.5833330.6250.6458170.65616200.3333330.50.5833330.6250.6458330.65623500.3333330.50.5833330.6250.645833
