实验5-线性代数方程组的数值解法

试验5线性代数方程组的数值解法

化工系毕啸天2010011811

【试验目的】

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。

【试验内容】

题目3

已知方程组ArR,其中ACR20X20,定义为

-3-1/2-1/4

-1/23-1/2-1/4

-1/4-1/23-1/2

A

•••-1/4

-1/4-1/23-1/2

-1/4-1/23

试通过迭代法求解此方程组，相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。试验要求：

（1）选取不同的初始向量x（o）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）取定右端向量力和初始向量V。），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，要求迭代误差满意||x（k+i）-x（k）|L<l（rS,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)=(l,l,…,1)T,b=(l,l,…,W，迭代要求为误差满意||x(k+l)—X(k)118V

10-5,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

functionx=Jacobi(xO,A,b,m)

D=diag(diag(A));

U=-triu(A,l);

L=-tril(A/-l)；

B1=D\(L+U);

fl=D\b;

x(：,l)=xO;

x(:,2)=Bl*x(:,l)+fl;

k=l;

whilenorm((x(:,k+l)-x(:,k)),inf)>m

x(:,k+2)=Bl*x(:,k+l)+fl;

k=k+l;

end

end

functionx=Gauss(xO,A,b,n)

D=diag(diag(A));

U=-triu(A,l)；

L=-tril(A,-l);

B2=(D-L)\U;

f2=(D-L)\b;

x(：,l)=xO;

x(:,2)=B2*x(:,l)+f2;

k=l;

whilenorm((x(:,k+l)-x(:zk))/inf)>m

x(:,k+2)=B2*x(:/k+l)+f2;

k=k+l；

end

end

Al=3.*eye(20,20);

A2=sparse(l:19,2:20,-1/2,20,20);

A3=sparse(l:18,3:20,-1/4,20,20);

AA=A1+A2+A3+A2'+A3';

A=full(AA);

b=ones(20,l);%输入自选右端项向量b

x0=ones(20zl);%输入自选初始向量x0

m=le-5;

xl=Jacob(xO,A,b,m);

x2=Gauss(xO,A/b,m);

结果输出数据:

k01234567

10.5833330.5277780.4994210.4895830.4851390.4832390.482374

10.750.6388890.6024310.5860340.5791620.5760460.574639

10.8333330.7152780.6701390.6495950.6405450.6363960.634487

10.8333330.7430560.6932870.6715860.6611850.65640.654137

x各重量10.8333330.750.7042820.6817130.6709150.6657180.663238

10.8333330.750.7077550.6855710.6747160.6693670.666768

10.8333330.750.7083330.6870660.6762720.6709240.668274

10.8333330.750.7083330.6874520.676850.6715350.668883

10.8333330.750.7083330.68750.6776790.6717670.669127

10.8333330.750.7083330.68750.6770790.6718440.66921

10.8333330.750.7083330.68750.6770790.6718440.66921

10.8333330.750.7083330.68750.6770390.6717670.669127

10.8333330.750.7083330.6874520.676850.6715350.668883

10.8333330.750.7083330.6870660.6762720.6709240.668274

10.8333330.750.7077550.6855710.6747160.6693670.666768

10.8333330.750.7042820.6817130.6709150.6657180.663238

10.8333330.7430560.6932870.6715860.6611850.65640.654137

10.8333330.7152780.6701390.6495950.6405450.6363960.634487

10.750.6388890.6024310.5860340.5791620.5760460.574639

10.5833330.5277780.4994210.4895830.4851390.4832390.482374

k89101112131415

0.481980.4817980.4817130.4816720.4816530.4816440.481640.481638

0.5739880.5736860.5735430.5734760.5734440.5734290.5734210.573418

0.6335970.6331790.6329810.6328870.6328430.6328210.6328110.632806

0.6530710.6525650.6523240.6522080.6521530.6521260.6521130.652107

0.6620480.6614770.6612020.6610690.6610050.6609740.6609590.660952

0.6655040.664890.6645910.6644460.6643760.6643410.6643250.664316

0.6669720.6663330.6660190.6658650.665790.6657530.6657350.665727

0.6675650.6669130.666590.6664310.6663530.6663140.6662950.666286

0.6678060.6671480.6668210.6666590.6665780.6665390.6665190.66651

x各重量0.667890.667230.6669010.6667380.6666570.6666170.6665970.666587

值0.667890.667230.6669010.6667380.6666570.6666170.6665970.666587

0.6678060.6671480.6668210.6666590.6665780.6665390.6665190.66651

0.6675650.6669130.666590.6664310.6663530.6663140.6662950.666286

0.6669720.6663330.6660190.6658650.665790.6657530.6657350.665727

0.6655040.664890.6645910.6644460.6643760.6643410.6643250.664316

0.6620480.6614770.6612020.6610690.6610050.6609740.6609590.660952

0.6530710.6525650.6523240.6522080.6521530.6521260.6521130.652107

0.6335970.6331790.6329810.6328870.6328430.6328210.6328110.632806

0.5739880.5736860.5735430.5734760.5734440.5734290.5734210.573418

0.481980.4817980.4817130.4816720.4816530.4816440.481640.481638

3.3变更迭代初始值

3.3.1将xO各重量初值置为0

增加一句代码为:x0=zeros(20,l);

k0123456

0C.3333330.4166670.4537040.4689430.4758230.478928

X各重量0C.3333330.4722220.5277780.5526620.5637860.568911

值0C.3333330.50.5717590.6045520.6195670.626559

0C.3333330.50.5810190.6184410.6361560.644499

0C.3333330.50.5833330.623650.6430040.652308

0C.3333330.50.5833330.6248070.6450620.654947

0C.3333330.50.5833330.6250.6456890.655883

0C.3333330.50.5833330.6250.6458170.656162

0C.3333330.50.5833330.6250.6458330.656235

0C.3333330.50.5833330.6250.6458330.656249

0C.3333330.50.5833330.6250.6458330.656249

0C.3333330.50.5833330.6250.6458330.656235

0C.3333330.50.5833330.6250.6458170.656162

0C.3333330.50.5833330.6250.6456890.655883

0C.3333330.50.5833330.6248070.6450620.654947

0C.3333330.50.5833330.623650.6430040.652308

0C.3333330.50.5810190.6184410.6361560.644499

0C.3333330.50.5717590.6045520.6195670.626559

0C.3333330.4722220.5277780.5526620.5637860.568911

0C.3333330.4166670.4537040.4689430.4758230.478928

k78910111213

0.4803650.4810350.481350.4814990.481570.4816040.481621

0.571290.5724060.5729330.5731840.5733040.5733610.573389

0.6298380.6313880.6321240.6324760.6326440.6327250.632764

0.6484660.6503560.6512610.6516960.6519050.6520060.652055

0.6567780.6589310.659970.6604720.6607150.6608330.660891

0.6597530.6620920.6632290.6637830.6640520.6641840.664248

0.6608970.6633560.6645610.6651520.6654410.6655820.665652

0.6612860.6638180.6650670.6656820.6659850.6661340.666207

0.6614130.6639840.6652580.6658880.6661990.6663520.666428

x各重量0.6614470.6640350.665320.6659570.6662720.6664270.666504

值0.6614470.6640350.665320.6659570.6662720.6664270.666504

0.6614130.6639840.6652580.6658880.6661990.6663520.666428

0.6612860.6638180.6650670.6656820.6659850.6661340.666207

0.6608970.6633560.6645610.6651520.6654410.6655820.665652

0.6597530.6620920.6632290.6637830.6640520.6641840.664248

0.6567780.6589310.659970.6604720.6607150.6608330.660891

0.6484660.6503560.6512610.6516960.6519050.6520060.652055

0.6298380.6313880.6321240.6324760.6326440.6327250.632764

0.571290.5724060.5729330.5731840.5733040.5733610.573389

0.4803650.4810350.481350.4814990.481570.4816040.481621

k141516

0.4816280.4816320.481634

x各重量0.5734020.5734090.573412

值0.6327830.6327920.632797

0.6520790.652090.652096

0.6609180.6609320.660939

0.6642790.6642940.664302

0.6656860.6657020.66571

0.6662420.666260.666269

0.6664650.6664830.666492

0.6665410.666560.666569

0.6665410.666560.666569

0.6664650.6664830.666492

0.6662420.666260.666269

0.6656860.6657020.66571

0.6642790.6642940.664302

0.6609180.6609320.660939

0.6520790.652090.652096

0.6327830.6327920.632797

0.5734020.5734090.573412

0.4816280.4816320.481634

【分析】

从数据中可以看出，当迭代的初值变更了，达到相同精度所须要的迭代次数也变更了。

3.3.2将各重量初始值置为10

干脆在下面给出数据结果

k0123456

102.8333331.5277780.91C880.6753470.5689780.522036

104.52.1388891.2743060.8863810.7175440.640258

105.3333332.6527781.5555561.0549770.8293470.724922

105.3333332.9305561.7037041.1498840.8864450.763505

105.33333331.7928241.2042820.9221080.786414

1()5.33333331.8275461.2324460.9416070.799154

105.33333331.8333331.245660.9515170.806292

105.33333331.8333331.2495180.9561470.809893

105.33333331.8333331.250.9578910.811558

X各重量105.33333331.8333331.250.9582930.812201

值105.33333331.8333331.250.9582930.812201

105.33333331.8333331.250.9578910.811558

105.33333331.8333331.2495180.9561470.809893

105.33333331.8333331.245660.9515170.806292

105.33333331.8275461.2324460.9416070.799154

105.33333331.7928241.2042820.9221080.786414

105.3333332.9305561.7037041.1498840.8864450.763505

105.3333332.6527781.5555561.0549770.8293470.724922

104.52.1388891.2743060.8863810.7175440.640258

102.8333331.5277780.91C880.6753470.5689780.522036

k78910111213

0.5004530.4904920.4858280.4836320.482590.4820940.481856

0.6047850.5882290.5804540.5767750.5750250.5741890.573788

0.6763310.6534790.6426720.6375310.6350750.6338970.633331

0.7051740.6775090.6642990.6579760.6549370.6534730.652766

0.7213780.6900960.6750370.6677680.6642540.6625520.661726

0.7299010.6962110.6798360.671870.6679940.6661050.665185

0.7346720.699520.6822740.6738250.6696870.6676610.666669

0.7372550.7012870.683520.674760.6704460.6683240.667282

0.7385570.7022020.6841550.6752150.6707960.6686150.66754

X各重量0.7390810.702590.6844230.6754030.6709340.6687250.667635

值0.7390810.702590.6844230.6754030.6709340.6687250.667635

0.7385570.7022020.6841550.6752150.6707960.6686150.66754

0.7372550.7012870.683520.674760.6704460.6683240.667282

0.7346720.699520.6822740.6738250.6696870.6676610.666669

0.7299010.6962110.6798360.671870.6679940.6661050.665185

0.7213780.6900960.6750370.6677680.6642540.6625520.661726

0.7051740.6775090.6642990.6579760.6549370.6534730.652766

0.6763310.6534790.6426720.6375310.6350750.6338970.633331

0.6047850.5882290.5804540.5767750.5750250.5741890.573788

0.5004530.4904920.4858280.4836320.482590.4820940.481856

k141516171S1920

0.4817420.4816870.4816610.4816480.4816420.4816390.481637

0.5735950.5735020.5734570.5734350.5734250.5734190.573417

0.6330580.6329250.6328610.632830.6328150.6328080.632804

0.6524240.6522580.6521780.6521380.6521190.652110.652106

0.6613250.661130.6610350.6609890.6609660.6609550.66095

0.6647370.6645180.6644110.6643590.6643330.6643210.664315

0.6661830.6659460.665830.6657730.6657450.6657310.665724

0.666770.6665190.6663950.6663350.6663050.6662910.666284

0.6670110.6667510.6666240.6665610.666530.6665150.666508

x各重量0.6670980.6668330.6667030.6666390.6666080.6665930.666585

值0.6670980.6668330.6667030.6666390.6666080.6665930.666585

0.6670110.6667510.6666240.6665610.666530.6665150.666508

0.666770.6665190.6663950.6663350.6663050.6662910.666284

0.6661830.6659460.665830.6657730.6657450.6657310.665724

0.6647370.6645180.6644110.6643590.6643330.6643210.664315

0.6613250.661130.6610350.6609890.6609660.6609550.66095

0.6524240.6522580.6521780.6521380.6521190.652110.652106

0.6330580.6329250.6328610.632830.6328150.6328080.632804

0.5735950.5735020.5734570.5734350.5734250.5734190.573417

0.4817420.4816870.4816610.4816480.4816420.4816390.481637

【分析】可见，当时值置为10时，并不变更其收敛性，但是达到结果所须要的迭代次数更

长了,要20次c

3.4用高斯方法重新计算

输出结果如下：

k012345

10.5833330.5088730.490C550.4843390.482519

10.6805560.6060640.5838240.5768050.57453

10.745370.6685310.6444190.6366190.634063

10.7642750.6887170.6641480.6560850.653422

10.7728270.6978880.673180.6650060.662295

10.7758270.7013390.6766140.6684020.665671

10.777040.7027680.678C520.6698260.667087

10.7774920.7033290.6786230.6703930.667651

10.7776690.7035550.6788560.6706250.667882

X各重量10.7777360.7036450.6789490.6707180.667975

值10.7777620.703680.6789870.6707560.668009

10.7777720.7036940.679C020.6707710.66798

10.7777750.70370.679C080.6707290.667766

10.7777770.7037020.679C110.6703140.667079

10.7777770.7037030.6784330.6685950.665384

10.7777780.7037030.6746720.6642460.661682

10.7777780.6967590.6624390.6541940.652549

10.7777780.6666670.6385650.6339530.633039

10.6944440.5881560.5761160.5739360.573522

10.5138890.4869150.4825660.4818190.481674

k67出91011

X各重量

0.4819270.4817320.4816680.4816460.4816390.481637

值

0.5737840.5735370.5734550.5734280.5734190.573416

0.6332190.632940.6328470.6328160.6328060.632803

0.652540.6522470.652150.6521180.6521070.652103

0.6613940.6610950.6609950.6609620.6609510.660947

0.6647630.664460.664360.6643260.6643140.664311

0.6661750.6658720.665770.6657360.6657240.66572

0.6667380.6664330.666330.6662940.6662820.666279

0.6669670.666660.6665540.6665170.6665050.666502

0.6670560.6667390.6666290.6665930.6665820.666579

0.6670670.6667350.6666240.6665910.6665810.666579

0.6669830.6666430.6665390.6665110.6665030.666501

0.6667150.6663940.6663070.6662840.6662790.666277

0.6660720.6658060.665740.6657230.6657190.665718

0.6645640.6643690.6643230.6643120.6643090.664309

0.6611120.6609830.6609540.6609470.6609450.660945

0.6521990.6521230.6521060.6521020.6521020.652101

0.6328520.6328120.6328040.6328020.6328010.632801

0.5734380.573420.5734160.5734150.5734150.573415

0.4816440.4816380.4816360.4816360.4816360.481636

3.4.2变更各重量初1直为0

k012345

00.3333330.4336420.4658780.4764270.479908

00.3888890.5129890.5534660.5668030.571218

00.4259260.5645580.6101930.6252930.630304

00.4367280.5807220.6283940.6442170.649477

00.4416150.5880570.6366990.6528740.658257

00.443330.5907640.6398210.6561530.661592

00.4440230.5918770.641120.6575230.662988

00.4442810.5923090.6416310.6580660.663543

00.4443820.5924810.6418380.6582870.66377

X各重量00.444420.5925480.641920.6583760.663861

值00.4444350.5925750.6419530.6584120.663898

00.4444410.5925860.6419670.6584260.663913

00.4444430.592590.6419720.6584320.663898

00.4444440.5925920.6419740.6584350.663714

00.4444440.5925920.641S750.6581780.662808

00.4444440.5925920.641S750.6565070.659946

00.4444440.5925930.6388890.6493550.651506

00.4444440.5925930.6255140.6313140.632487

00.4444440.5555560.570C450.5727430.573273

00.4444440.4753090.4804670.48140.481586

k67891011

0.4810620.4814450.4815720.4816150.4816290.481634

0.5726840.5731710.5733340.5733880.5734060.573412

0.631970.6325240.6327090.632770.6327910.632798

0.6512280.651810.6520040.652C690.6520910.652098

0.660050.6606470.6608460.6609120.6609340.660941

0.6634040.6640080.6642090.6642760.6642980.664305

0.664810.6654160.6656190.6656860.6657080.665715

X各重里

0.6653690.6659770.666180.6662460.6662680.666274

值

0.6655970.6662060.6664080.6664730.6664920.666498

0.6656890.6662970.6664930.6665540.6665710.666576

0.6657250.6663210.6665050.6665580.6665730.666576

0.6657210.6662820.6664420.6664850.6664970.666499

0.6656170.6661050.6662330.6662660.6662740.666276

0.665210.6655930.6656880.6657110.6657160.665718

0.6639530.6642250.6642690.6643040.6643080.664308

0.6607180.6608930.6609330.6609420.6609440.660945

0.651970.6520720.6520940.65210.6521010.652101

0.6327330.6327860.6327980.63280.6328010.632801

0.5733840.5734080.5734130.5734140.5734150.573415

0.4816250.4816330.4816350.4816360.4816360.481636

3.4.3变更各重量初值为10

k01234567

103.3055561.4437370.857C470.6668280.6043350.5836780.576826

103.620371.6042950.9524550.7385490.6678880.6444630.636681

103.7121911.6606740.9859350.7628920.6889280.6643540.656181

103.7537291.6863711.0015120.774190.6986310.6734930.665125

103.7683041.6965161.0077540.7786420.7023850.6769930.668535

103.7741951.7007861.0104390.7805490.7039770.6784680.669968

103.7763911.7025131.0115490.7813310.7046210.6790590.670535

103.7772481.7032221.012C150.781660.7048890.6792990.670741

103.7775741.7035091.0122080.7817960.7049980.6793540.670722

X各重量

103.77771.7036251.0122890.7818530.7050030.6791470.670456

值

103.7777481.7036721.0123220.7818760.7045870.6783480.669888

103.7777661.7036911.0123360.7814040.7025820.6765940.668997

103.7777731.7036991.0123420.7772280.6973610.6738320.667723

103.7777761.7037021.0065570.7623440.6885690.6700670.665668

103.7777771.7037030.9689420.7338930.6773070.6646540.661793

103.7777781.6342590.8743890.6977440.6619380.6542630.652586

103.7777781.3333330.756C170.6576980.6380090.633930.633052

102.9444440.8815590.6307640.5846720.5757610.573920.573527

101.1388890.5913710.5014620.4855870.4824610.4818140.481675

k891011121314

0.574550.5737930.573540.5734570.5734290.5734190.573416

0.6340930.6332310.6329440.6328490.6328170.6328060.632803

0.653460.6525540.6522520.6521510.6521170.6521060.652103

0.6623380.6614090.6610990.6609950.6609610.660950.660946

0.6657170.6647760.6644620.6643570.6643240.6643130.66431

0.6671320.6661830.6658660.6657640.6657320.6657220.665719

0.6676820.6667270.6664150.6663180.6662890.666280.666278

x各重量

0.6678660.666920.6666230.6665350.666510.6665030.666501

值

0.6678510.6669470.6666810.6666060.6665850.666580.666578

0.6676950.6668840.6666590.6665990.6665830.6665790.666578

0.6674130.6667380.6665610.6665160.6665040.6665010.666501

0.666970.666450.666320.6662880.666280.6662780.666277

0.6662070.6658370.6657470.6657250.665720.6657180.665718

0.664630.6643850.6643270.6643130.664310.6643090.664309

0.6611410.6609910.6609560.6609470.6609460.6609450.660945

0.6522120.6521270.6521070.6521030.6521020.6521010.652101

0.6328580.6328140.6328040.6328020.6328010.6328010.632801

0.573440.573420.5734160.5734150.5734150.5734150.573415

0.4816450.4816380.4816360.4816360.4816360.4816360.481636

3.5两种迭代法运算次数的比较

X重量初值0110

雅可比方法151620

高斯方法111114

【小结】视察上表所比较的两种算法所须要的次数，可以得到结论：高斯-赛德尔迭代算法

比雅可比算法有更快的收敛速度。雅可比迭代公式形式相对较为简洁；高斯-赛德尔迭代计

算出的上一个结果可以存储在下一次迭代中，存储便利，因此会比雅可比方法收敛更快。进

一步地，我们接下来会变更右端向量、变更A的主对角线元素再视察结果的变更。

3.6变更b的值

3.6.1将b各重量的值置为1.5

b=1.5.*ones(20,l);

我们先给出雅口J比方法的结果，共需9次迭代

k01234

10.750.7361110.7262730.724055

10.9166670.8750.8663190.862365

110.9652780.956C190.951871

110.9930560.9837960.980806

1110.9959490.993538

1110.9994210.997975

11110.999566

11110.999952

11111

X各重量11111

值11111

11111

11110.999952

11110.999566

1110.9994210.997975

1110.9959490.993538

110.9930560.9837960.980806

110.9652780.956C190.951871

10.9166670.8750.8663190.862365

10.750.736111