实验5线性代数方程组的数值解法.docx

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,-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=Jacob(x0,A,b,m);x2=Gauss(x0,A,b,m);结果输出数据：k01234567x各分量10.5833330.5277780.4994210.4895830.4851390.4832390.48237410.750.6388890.6024310.5860340.5791620.5760460.57463910.8333330.7152780.6701390.6495950.6405450.6363960.63448710.8333330.7430560.6932870.6715860.6611850.65640.65413710.8333330.750.7042820.6817130.6709150.6657180.66323810.8333330.750.7077550.6855710.6747160.6693670.66676810.8333330.750.7083330.6870660.6762720.6709240.66827410.8333330.750.7083330.6874520.676850.6715350.66888310.8333330.750.7083330.68750.6770390.6717670.66912710.8333330.750.7083330.68750.6770790.6718440.6692110.8333330.750.7083330.68750.6770790.6718440.6692110.8333330.750.7083330.68750.6770390.6717670.66912710.8333330.750.7083330.6874520.676850.6715350.66888310.8333330.750.7083330.6870660.6762720.6709240.66827410.8333330.750.7077550.6855710.6747160.6693670.66676810.8333330.750.7042820.6817130.6709150.6657180.66323810.8333330.7430560.6932870.6715860.6611850.65640.65413710.8333330.7152780.6701390.6495950.6405450.6363960.63448710.750.6388890.6024310.5860340.5791620.5760460.57463910.5833330.5277780.4994210.4895830.4851390.4832390.482374k89101112131415x各分量值0.481980.4817980.4817130.4816720.4816530.4816440.481640.4816380.5739880.5736860.5735430.5734760.5734440.5734290.5734210.5734180.6335970.6331790.6329810.6328870.6328430.6328210.6328110.6328060.6530710.6525650.6523240.6522080.6521530.6521260.6521130.6521070.6620480.6614770.6612020.6610690.6610050.6609740.6609590.6609520.6655040.664890.6645910.6644460.6643760.6643410.6643250.6643160.6669720.6663330.6660190.6658650.665790.6657530.6657350.6657270.6675650.6669130.666590.6664310.6663530.6663140.6662950.6662860.6678060.6671480.6668210.6666590.6665780.6665390.6665190.666510.667890.667230.6669010.6667380.6666570.6666170.6665970.6665870.667890.667230.6669010.6667380.6666570.6666170.6665970.6665870.6678060.6671480.6668210.6666590.6665780.6665390.6665190.666510.6675650.6669130.666590.6664310.6663530.6663140.6662950.6662860.6669720.6663330.6660190.6658650.665790.6657530.6657350.6657270.6655040.664890.6645910.6644460.6643760.6643410.6643250.6643160.6620480.6614770.6612020.6610690.6610050.6609740.6609590.6609520.6530710.6525650.6523240.6522080.6521530.6521260.6521130.6521070.6335970.6331790.6329810.6328870.6328430.6328210.6328110.6328060.5739880.5736860.5735430.5734760.5734440.5734290.5734210.5734180.481980.4817980.4817130.4816720.4816530.4816440.481640.4816383.3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00.3333330.4166670.4537040.4689430.4758230.47892800.3333330.4722220.5277780.5526620.5637860.56891100.3333330.50.5717590.6045520.6195670.62655900.3333330.50.5810190.6184410.6361560.64449900.3333330.50.5833330.623650.6430040.65230800.3333330.50.5833330.6248070.6450620.65494700.3333330.50.5833330.6250.6456890.65588300.3333330.50.5833330.6250.6458170.65616200.3333330.50.5833330.6250.6458330.65623500.3333330.50.5833330.6250.6458330.65624900.3333330.50.5833330.6250.6458330.65624900.3333330.50.5833330.6250.6458330.65623500.3333330.50.5833330.6250.6458170.65616200.3333330.50.5833330.6250.6456890.65588300.3333330.50.5833330.6248070.6450620.65494700.3333330.50.5833330.623650.6430040.65230800.3333330.50.5810190.6184410.6361560.64449900.3333330.50.5717590.6045520.6195670.62655900.3333330.4722220.5277780.5526620.5637860.56891100.3333330.4166670.4537040.4689430.4758230.478928k78910111213x各分量值0.4803650.4810350.481350.4814990.481570.4816040.4816210.571290.5724060.5729330.5731840.5733040.5733610.5733890.6298380.6313880.6321240.6324760.6326440.6327250.6327640.6484660.6503560.6512610.6516960.6519050.6520060.6520550.6567780.6589310.659970.6604720.6607150.6608330.6608910.6597530.6620920.6632290.6637830.6640520.6641840.6642480.6608970.6633560.6645610.6651520.6654410.6655820.6656520.6612860.6638180.6650670.6656820.6659850.6661340.6662070.6614130.6639840.6652580.6658880.6661990.6663520.6664280.6614470.6640350.665320.6659570.6662720.6664270.6665040.6614470.6640350.665320.6659570.6662720.6664270.6665040.6614130.6639840.6652580.6658880.6661990.6663520.6664280.6612860.6638180.6650670.6656820.6659850.6661340.6662070.6608970.6633560.6645610.6651520.6654410.6655820.6656520.6597530.6620920.6632290.6637830.6640520.6641840.6642480.6567780.6589310.659970.6604720.6607150.6608330.6608910.6484660.6503560.6512610.6516960.6519050.6520060.6520550.6298380.6313880.6321240.6324760.6326440.6327250.6327640.571290.5724060.5729330.5731840.5733040.5733610.5733890.4803650.4810350.481350.4814990.481570.4816040.481621k141516x各分量值0.4816280.4816320.4816340.5734020.5734090.5734120.6327830.6327920.6327970.6520790.652090.6520960.6609180.6609320.6609390.6642790.6642940.6643020.6656860.6657020.665710.6662420.666260.6662690.6664650.6664830.6664920.6665410.666560.6665690.6665410.666560.6665690.6664650.6664830.6664920.6662420.666260.6662690.6656860.6657020.665710.6642790.6642940.6643020.6609180.6609320.6609390.6520790.652090.6520960.6327830.6327920.6327970.5734020.5734090.5734120.4816280.4816320.481634【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102.8333331.5277780.910880.6753470.5689780.522036104.52.1388891.2743060.8863810.7175440.640258105.3333332.6527781.5555561.0549770.8293470.724922105.3333332.9305561.7037041.1498840.8864450.763505105.33333331.7928241.2042820.9221080.786414105.33333331.8275461.2324460.9416070.799154105.33333331.8333331.245660.9515170.806292105.33333331.8333331.2495180.9561470.809893105.33333331.8333331.250.9578910.811558105.33333331.8333331.250.9582930.812201105.33333331.8333331.250.9582930.812201105.33333331.8333331.250.9578910.811558105.33333331.8333331.2495180.9561470.809893105.33333331.8333331.245660.9515170.806292105.33333331.8275461.2324460.9416070.799154105.33333331.7928241.2042820.9221080.786414105.3333332.9305561.7037041.1498840.8864450.763505105.3333332.6527781.5555561.0549770.8293470.724922104.52.1388891.2743060.8863810.7175440.640258102.8333331.5277780.910880.6753470.5689780.522036k78910111213x各分量值0.5004530.4904920.4858280.4836320.482590.4820940.4818560.6047850.5882290.5804540.5767750.5750250.5741890.5737880.6763310.6534790.6426720.6375310.6350750.6338970.6333310.7051740.6775090.6642990.6579760.6549370.6534730.6527660.7213780.6900960.6750370.6677680.6642540.6625520.6617260.7299010.6962110.6798360.671870.6679940.6661050.6651850.7346720.699520.6822740.6738250.6696870.6676610.6666690.7372550.7012870.683520.674760.6704460.6683240.6672820.7385570.7022020.6841550.6752150.6707960.6686150.667540.7390810.702590.6844230.6754030.6709340.6687250.6676350.7390810.702590.6844230.6754030.6709340.6687250.6676350.7385570.7022020.6841550.6752150.6707960.6686150.667540.7372550.7012870.683520.674760.6704460.6683240.6672820.7346720.699520.6822740.6738250.6696870.6676610.6666690.7299010.6962110.6798360.671870.6679940.6661050.6651850.7213780.6900960.6750370.6677680.6642540.6625520.6617260.7051740.6775090.6642990.6579760.6549370.6534730.6527660.6763310.6534790.6426720.6375310.6350750.6338970.6333310.6047850.5882290.5804540.5767750.5750250.5741890.5737880.5004530.4904920.4858280.4836320.482590.4820940.481856k14151617181920x各分量值0.4817420.4816870.4816610.4816480.4816420.4816390.4816370.5735950.5735020.5734570.5734350.5734250.5734190.5734170.6330580.6329250.6328610.632830.6328150.6328080.6328040.6524240.6522580.6521780.6521380.6521190.652110.6521060.6613250.661130.6610350.6609890.6609660.6609550.660950.6647370.6645180.6644110.6643590.6643330.6643210.6643150.6661830.6659460.665830.6657730.6657450.6657310.6657240.666770.6665190.6663950.6663350.6663050.6662910.6662840.6670110.6667510.6666240.6665610.666530.6665150.6665080.6670980.6668330.6667030.6666390.6666080.6665930.6665850.6670980.6668330.6667030.6666390.6666080.6665930.6665850.6670110.6667510.6666240.6665610.666530.6665150.6665080.666770.6665190.6663950.6663350.6663050.6662910.6662840.6661830.6659460.665830.6657730.6657450.6657310.6657240.6647370.6645180.6644110.6643590.6643330.6643210.6643150.6613250.661130.6610350.6609890.6609660.6609550.660950.6524240.6522580.6521780.6521380.6521190.652110.6521060.6330580.6329250.6328610.632830.6328150.6328080.6328040.5735950.5735020.5734570.5734350.5734250.5734190.5734170.4817420.4816870.4816610.4816480.4816420.4816390.481637【分析】可见，当初值置为10时，并不改变其收敛性，但是达到结果所需要的迭代次数更长了，要20次。3.4用高斯方法重新计算输出结果如下:k012345x各分量值10.5833330.5088730.4900550.4843390.48251910.6805560.6060640.5838240.5768050.5745310.745370.6685310.6444190.6366190.63406310.7642750.6887170.6641480.6560850.65342210.7728270.6978880.673180.6650060.66229510.7758270.7013390.6766140.6684020.66567110.777040.7027680.6780520.6698260.66708710.7774920.7033290.6786230.6703930.66765110.7776690.7035550.6788560.6706250.66788210.7777360.7036450.6789490.6707180.66797510.7777620.703680.6789870.6707560.66800910.7777720.7036940.6790020.6707710.6679810.7777750.70370.6790080.6707290.66776610.7777770.7037020.6790110.6703140.66707910.7777770.7037030.6784330.6685950.66538410.7777780.7037030.6746720.6642460.66168210.7777780.6967590.6624390.6541940.65254910.7777780.6666670.6385650.6339530.63303910.6944440.5881560.5761160.5739360.57352210.5138890.4869150.4825660.4818190.481674k67891011x各分量值0.4819270.4817320.4816680.4816460.4816390.4816370.5737840.5735370.5734550.5734280.5734190.5734160.6332190.632940.6328470.6328160.6328060.6328030.652540.6522470.652150.6521180.6521070.6521030.6613940.6610950.6609950.6609620.6609510.6609470.6647630.664460.664360.6643260.6643140.6643110.6661750.6658720.665770.6657360.6657240.665720.6667380.6664330.666330.6662940.6662820.6662790.6669670.666660.6665540.6665170.6665050.6665020.6670560.6667390.6666290.6665930.6665820.6665790.6670670.6667350.6666240.6665910.6665810.6665790.6669830.6666430.6665390.6665110.6665030.6665010.6667150.6663940.6663070.6662840.6662790.6662770.6660720.6658060.665740.6657230.6657190.6657180.6645640.6643690.6643230.6643120.6643090.6643090.6611120.6609830.6609540.6609470.6609450.6609450.6521990.6521230.6521060.6521020.6521020.6521010.6328520.6328120.6328040.6328020.6328010.6328010.5734380.573420.5734160.5734150.5734150.5734150.4816440.4816380.4816360.4816360.4816360.4816363.4.2改变各分量初值为0k012345x各分量值00.3333330.4336420.4658780.4764270.47990800.3888890.5129890.5534660.5668030.57121800.4259260.5645580.6101930.6252930.63030400.4367280.5807220.6283940.6442170.64947700.4416150.5880570.6366990.6528740.65825700.443330.5907640.6398210.6561530.66159200.4440230.5918770.641120.6575230.66298800.4442810.5923090.6416310.6580660.66354300.4443820.5924810.6418380.6582870.6637700.444420.5925480.641920.6583760.66386100.4444350.5925750.6419530.6584120.66389800.4444410.5925860.6419670.6584260.66391300.4444430.592590.6419720.6584320.66389800.4444440.5925920.6419740.6584350.66371400.4444440.5925920.6419750.6581780.66280800.4444440.5925920.6419750.6565070.65994600.4444440.5925930.6388890.6493550.65150600.4444440.5925930.6255140.6313140.63248700.4444440.5555560.5700450.5727430.57327300.4444440.4753090.4804670.48140.481586k67891011x各分量值0.4810620.4814450.4815720.4816150.4816290.4816340.5726840.5731710.5733340.5733880.5734060.5734120.631970.6325240.6327090.632770.6327910.6327980.6512280.651810.6520040.6520690.6520910.6520980.660050.6606470.6608460.6609120.6609340.6609410.6634040.6640080.6642090.6642760.6642980.6643050.664810.6654160.6656190.6656860.6657080.6657150.6653690.6659770.666180.6662460.6662680.6662740.6655970.6662060.6664080.6664730.6664920.6664980.6656890.6662970.6664930.6665540.6665710.6665760.6657250.6663210.6665050.6665580.6665730.6665760.6657210.6662820.6664420.6664850.6664970.6664990.6656170.6661050.6662330.6662660.6662740.6662760.665210.6655930.6656880.6657110.6657160.6657180.6639530.6642250.6642890.6643040.6643080.6643080.6607180.6608930.6609330.6609420.6609440.6609450.651970.6520720.6520940.65210.6521010.6521010.6327330.6327860.6327980.63280.6328010.6328010.5733840.5734080.5734130.5734140.5734150.5734150.4816250.4816330.4816350.4816360.4816360.4816363.4.3 改变各分量初值为10k01234567x各分量值103.3055561.4437370.8570470.6668280.6043350.5836780.576826103.620371.6042950.9524550.7385490.6678880.6444630.636681103.7121911.6606740.9859350.7628920.6889280.6643540.656181103.7537291.6863711.0015120.774190.6986310.6734930.665125103.7683041.6965161.0077540.7786420.7023850.6769930.668535103.7741951.7007861.0104390.7805490.7039770.6784680.669968103.7763911.7025131.0115490.7813310.7046210.6790590.670535103.7772481.7032221.0120150.781660.7048890.6792990.670741103.7775741.7035091.0122080.7817960.7049980.6793540.670722103.77771.7036251.0122890.7818530.7050030.6791470.670456103.7777481.7036721.0123220.7818760.7045870.6783480.669888103.7777661.7036911.0123360.7814040.7025820.6765940.668997103.7777731.7036991.0123420.7772280.6973610.6738320.667723103.7777761.7037021.0065570.7623440.6885690.6700670.665668103.7777771.7037030.9689420.7338930.6773070.6646540.661793103.7777781.6342590.8743890.6977440.6619380.6542630.652586103.7777781.3333330.7560170.6576980.6380090.633930.633052102.9444440.8815590.6307640.5846720.5757610.573920.573527101.1388890.5913710.5014620.4855870.4824610.4818140.4