5线性代数方程组数值解法.docx

1、5线性代数方程组数值解法实验5 线性代数方程组的数值解法化工系毕啸天 2018011811【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A，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：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；，将A的主对角线元素

2、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，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=(1,1,1T，b=(1,1,1T，迭代要求为误差满足，编写雅各比、高斯-赛德尔迭代法的函数，迭代求解。3.2 程序代码function x = Jacobi( x0,A,b,m D=diag(diag(A。 U=-triu(A,1。 L=-tril(A,-1。 B1=D(L+U。f1=Db。x(:,1=x0。 x(:,2=B1*x(:,1+f1。k=1。while norm(x(:,k+1-x(:,k,infm x(:,k+2=B1*x(:,

3、k+1+f1。 k=k+1。endendfunction x = Gauss( x0,A,b,m D=diag(diag(A。U=-triu(A,1。L=-tril(A,-1。B2=(D-LU。f2=(D-Lb。x(:,1=x0。x(:,2=B2*x(:,1+f2。k=1。while norm(x(:,k+1-x(:,k,infm 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

4、=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=Jacob(x0,A,b,m。x2=Gauss(x0,A,b,m。结果输出数据：k01234567x各分量10.5833330.5277780.4994210.4895830.4851390.4832390.48237410.750.6388890.6024310.5860340.5791620.5760460.57463910.8333330.7152780.6701390.6495950.6405450.6363960.63448710.8333330.7

5、430560.6932870.6715860.6611850.65640.65413710.8333330.750.7042820.6817130.6709150.6657180.66323810.8333330.750.7077550.6855710.6747160.6693670.66676810.8333330.750.7083330.6870660.6762720.6709240.66827410.8333330.750.7083330.6874520.676850.6715350.66888310.8333330.750.7083330.68750.6770390.6717670.6

6、6912710.8333330.750.7083330.68750.6770790.6718440.6692110.8333330.750.7083330.68750.6770790.6718440.6692110.8333330.750.7083330.68750.6770390.6717670.66912710.8333330.750.7083330.6874520.676850.6715350.66888310.8333330.750.7083330.6870660.6762720.6709240.66827410.8333330.750.7077550.6855710.6747160.

7、6693670.66676810.8333330.750.7042820.6817130.6709150.6657180.66323810.8333330.7430560.6932870.6715860.6611850.65640.65413710.8333330.7152780.6701390.6495950.6405450.6363960.63448710.750.6388890.6024310.5860340.5791620.5760460.57463910.5833330.5277780.4994210.4895830.4851390.4832390.482374k8910111213

8、1415x各分量值0.481980.4817980.4817130.4816720.4816530.4816440.481640.4816380.5739880.5736860.5735430.5734760.5734440.5734290.5734210.5734180.6335970.6331790.6329810.6328870.6328430.6328210.6328110.6328060.6530710.6525650.6523240.6522080.6521530.6521260.6521130.6521070.6620480.6614770.6612020.6610690.661

9、0050.6609740.6609590.6609520.6655040.664890.6645910.6644460.6643760.6643410.6643250.6643160.6669720.6663330.6660190.6658650.665790.6657530.6657350.6657270.6675650.6669130.666590.6664310.6663530.6663140.6662950.6662860.6678060.6671480.6668210.6666590.6665780.6665390.6665190.666510.667890.667230.66690

10、10.6667380.6666570.6666170.6665970.6665870.667890.667230.6669010.6667380.6666570.6666170.6665970.6665870.6678060.6671480.6668210.6666590.6665780.6665390.6665190.666510.6675650.6669130.666590.6664310.6663530.6663140.6662950.6662860.6669720.6663330.6660190.6658650.665790.6657530.6657350.6657270.665504

11、0.664890.6645910.6644460.6643760.6643410.6643250.6643160.6620480.6614770.6612020.6610690.6610050.6609740.6609590.6609520.6530710.6525650.6523240.6522080.6521530.6521260.6521130.6521070.6335970.6331790.6329810.6328870.6328430.6328210.6328110.6328060.5739880.5736860.5735430.5734760.5734440.5734290.573

12、4210.5734180.481980.4817980.4817130.4816720.4816530.4816440.481640.4816383.3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00.3333330.4166670.4537040.4689430.4758230.47892800.3333330.4722220.5277780.5526620.5637860.56891100.3333330.50.5717590.6045520.6195670.62655900.3333330.50.5810190.61

13、84410.6361560.64449900.3333330.50.5833330.623650.6430040.65230800.3333330.50.5833330.6248070.6450620.65494700.3333330.50.5833330.6250.6456890.65588300.3333330.50.5833330.6250.6458170.65616200.3333330.50.5833330.6250.6458330.65623500.3333330.50.5833330.6250.6458330.65624900.3333330.50.5833330.6250.64

14、58330.65624900.3333330.50.5833330.6250.6458330.65623500.3333330.50.5833330.6250.6458170.65616200.3333330.50.5833330.6250.6456890.65588300.3333330.50.5833330.6248070.6450620.65494700.3333330.50.5833330.623650.6430040.65230800.3333330.50.5810190.6184410.6361560.64449900.3333330.50.5717590.6045520.6195

15、670.62655900.3333330.4722220.5277780.5526620.5637860.56891100.3333330.4166670.4537040.4689430.4758230.478928k78910111213x各分量值0.4803650.4810350.481350.4814990.481570.4816040.4816210.571290.5724060.5729330.5731840.5733040.5733610.5733890.6298380.6313880.6321240.6324760.6326440.6327250.6327640.6484660.

16、6503560.6512610.6516960.6519050.6520060.6520550.6567780.6589310.659970.6604720.6607150.6608330.6608910.6597530.6620920.6632290.6637830.6640520.6641840.6642480.6608970.6633560.6645610.6651520.6654410.6655820.6656520.6612860.6638180.6650670.6656820.6659850.6661340.6662070.6614130.6639840.6652580.66588

17、80.6661990.6663520.6664280.6614470.6640350.665320.6659570.6662720.6664270.6665040.6614470.6640350.665320.6659570.6662720.6664270.6665040.6614130.6639840.6652580.6658880.6661990.6663520.6664280.6612860.6638180.6650670.6656820.6659850.6661340.6662070.6608970.6633560.6645610.6651520.6654410.6655820.665

18、6520.6597530.6620920.6632290.6637830.6640520.6641840.6642480.6567780.6589310.659970.6604720.6607150.6608330.6608910.6484660.6503560.6512610.6516960.6519050.6520060.6520550.6298380.6313880.6321240.6324760.6326440.6327250.6327640.571290.5724060.5729330.5731840.5733040.5733610.5733890.4803650.4810350.4

19、81350.4814990.481570.4816040.481621k141516x各分量值0.4816280.4816320.4816340.5734020.5734090.5734120.6327830.6327920.6327970.6520790.652090.6520960.6609180.6609320.6609390.6642790.6642940.6643020.6656860.6657020.665710.6662420.666260.6662690.6664650.6664830.6664920.6665410.666560.6665690.6665410.666560.

20、6665690.6664650.6664830.6664920.6662420.666260.6662690.6656860.6657020.665710.6642790.6642940.6643020.6609180.6609320.6609390.6520790.652090.6520960.6327830.6327920.6327970.5734020.5734090.5734120.4816280.4816320.481634【分析】从数据中可以看出，当迭代的初值变化了，达到相同精度所需要的迭代次数也变化了。3.3.2将各分量初始值置为10直接在下面给出数据结果k0123456x各分量

21、值102.8333331.5277780.910880.6753470.5689780.522036104.52.1388891.2743060.8863810.7175440.640258105.3333332.6527781.5555561.0549770.8293470.724922105.3333332.9305561.7037041.1498840.8864450.763505105.33333331.7928241.2042820.9221080.786414105.33333331.8275461.2324460.9416070.799154105.33333331.833333

22、1.245660.9515170.806292105.33333331.8333331.2495180.9561470.809893105.33333331.8333331.250.9578910.811558105.33333331.8333331.250.9582930.812201105.33333331.8333331.250.9582930.812201105.33333331.8333331.250.9578910.811558105.33333331.8333331.2495180.9561470.809893105.33333331.8333331.245660.9515170

23、.806292105.33333331.8275461.2324460.9416070.799154105.33333331.7928241.2042820.9221080.786414105.3333332.9305561.7037041.1498840.8864450.763505105.3333332.6527781.5555561.0549770.8293470.724922104.52.1388891.2743060.8863810.7175440.640258102.8333331.5277780.910880.6753470.5689780.522036k78910111213x

24、各分量值0.5004530.4904920.4858280.4836320.482590.4820940.4818560.6047850.5882290.5804540.5767750.5750250.5741890.5737880.6763310.6534790.6426720.6375310.6350750.6338970.6333310.7051740.6775090.6642990.6579760.6549370.6534730.6527660.7213780.6900960.6750370.6677680.6642540.6625520.6617260.7299010.6962110

25、.6798360.671870.6679940.6661050.6651850.7346720.699520.6822740.6738250.6696870.6676610.6666690.7372550.7012870.683520.674760.6704460.6683240.6672820.7385570.7022020.6841550.6752150.6707960.6686150.667540.7390810.702590.6844230.6754030.6709340.6687250.6676350.7390810.702590.6844230.6754030.6709340.6687250.6676350.7385570.7022020.6841550.6752150.6707960.6686150.667540.