北京理工数值分析作业

第一早

1.题

wc=lE-8;

i=l；

a(l,l)=5A(l/2);

a(l,2)=5A(l/4);

whileabs(a(l,i+l)-a(l,i))>wc

i=i+l;

a(1,i)=sqrt(a(1,i-1));

a(1,i+1)=sqrt(a(1,i));

fprintf('a(1,%d)=%12.10f\n',i,a(1,i));

end

fprintf。一共要计算％d次计算

结果

a(1,2)=1.4953487812

a(1,3)=1.2228445450

a(1,4)=1.1058230170

a(1,5)=1.0515811985

a(1,6)=1.0254663322

a(1,7)=1.0126531154

a(1,8)=1.0063066707

a(1,9)=1.0031483792

a(1,10)=1.0015729525

a(1,11)=1.0007861672

a(1,12)=1.0003930064

a(1,13)=1.0001964839

a(1,14)=1.0000982371

a(1,15)=1.0000491174

a(1,16)=1.0000245584

a(1,17)=1.0000122791

a(1,18)=1.0000061395

a(1,19)=1.0000030698

a(1,20)=1.0000015349

a(1,21)=1.0000007674

a(1,22)=1.0000003837

a(1,23)=1.0000001919

a(1,24)=1.0000000959

a(1,25)=1.0000000480

a(1,26)=1.0000000240

a(1,27)=1.0000000120

一共要进行27次计算

4题

Maltab代码：

当把区间[-10,10]分为100等份时

»x=-10:20/100:10;

»y=x;

»[x,y]=meshgrid(x,y);

»z=exp(-abs(x))+cos(x+y)+l./(xA2+yA2+l);

»mesh(z)

>>title('区间分成100等分的三维图像,);

»xlabel('x');

»ylabel('y');

»zlabel('z');

当把区间[-10,10]分为200等份时,

»x=-10:20/200:10;

»y=x;

»[x,y]=meshgrid(x,y);

»z=exp(-abs(x))+cos(x+y)+l./(xA2+yA2+l);

»mesh(z)

>>title('区间分成200等分的三维图像,);

»xlabel('x');

»ylabel('y');

»zlabel('z');

当把区间[-10,10]分为400等份时，编程如下,

»x=-10:20/400:10;

»y=x;

»[x,y]=meshgrid(x,y);

»z=exp(-abs(x))+cos(x+y)+l./(xA2+yA2+l);

»mesh(z)

>>title('区间分成400等分的三维图像，)；

»xlabel('x');

»ylabel('y');

»zlabel('z')

第一早

1.题

a=-2*ones(1,8);

a(l,D=O;

b=5*ones(1,8);

b(l,1)=2;

c=-2*ones(1,7);

disp。请输入电压值V，)

V=input('V=')

disp。请输入电阻值R')

R=input('R=')

d=zeros(1,8);

d(l,1)=(V/R);

fori=2:8

a(l,i)=a(l,i)/b(l,i-l);

b(l,i)=b(l,i)-c(l,i-l)*a(l,i);

d(l,i)=d(l,i)-a(l,i)*d(l,i-l);

end

d(l,8)=d(l,8)/b(l,8);

fori=7:-1:1

d(l,i)=(d(l,i)-c(l,i)*d(l,i+l))/b(l,i);

end

disp(1电流值如下:1);

fori=l:8

disp([111int2str(i)1=1num2str(d(i))]);

㊀nd

fprintf(1\n1);

结果：

请输入电压值V

V=10

V=

10

请输入电阻值R

R=10

R

10

电流值如下：

II=0.99995

12=0.49995

13=0.24993

工4=0.12487

15=0.062254

16=0.030761

17=0.014648

18=0.0058592

3.题

改进的平方根法

formatrational

%A=[335;359;5917]

A=[4-2-4;-21710;-410

n=length(A);

L=zeros(n,n);

U=zeros(n,n);

sum=0;

forj=l:n

U(l,j)=A(l,j);

end

fori=2:n

end

forr=2:n

forj=r:n

fork=l:r-1

sum=sum+L(r,k)*U(k,j);

end

U(r,j)=A(r,j)-sum;

sum=0;

㊀nd

sum=0;

fori=r+l:n

fork=l:r-1

sum=sum+L(irk)*U(k,r);

end

L(i,r)=(A(i,r)-sum)/U(r,r);

sum=O;

end

sum=O;

end

L=L+eye(n);

U；

D=eye(n,n);

D=D.*U;

L*D*L'

结果：

A=

4-2-4

-21710

-4109

ans=

4-2-4

-21710

-4109

第二早

1题

%Jacobi迭代算法

function[x]=jacobi_zhao(A,b,x_0,e,N)

n=size(A,l);

M=eye(n)-eye(n)/diag(diag(A))*A;

g=eye(n)/diag(diag(A))*b;

out=[];

fork=l:N

x_l=M*x_0+g;

out=[out;kx_l'];

ifnorm(x_l-x_0)<e

break;

end

x_0=x_l;

end

x=x_l;

disp('kx

disp(out)

end

%G-S迭代算法

function[x]=G_S_zhao(A,b,x_0,e,N)

n=size(A,l);

L=-tril(A,-l);

U=-tril(A',-l)';

D=diag(diag(A));

out=[];

M=eye(n)/(D-L)*U;

g=eye(n)/(D-L)*b;

fork=l:N

x_l=M*x_0+g;

out=[out;kx_l'];

ifnorm(x_l-x_0)<e

break;

end

x_0=x_l;

end

x=x_l;

disp('kx,)

disp(out)

end

求解题目过程：

»A=[101234;19-12-3;2-173-5;32312-1;4-3-5-115]

A=

101234

19-12-3

2-173-5

32312-1

4-3-5-115

»b=[12;-27;14;-17;12]

b=

12

-27

14

-17

12

»xO=[O;O;O;O;O]

xO=

0

0

0

0

0

x=jacobi_zhao(A,b,xO,O.OOOO1,60);

kX

1.00001.2000-3.00002.0000-1.41670.8000

2.00001.2050-2.32962.4071-1.65000.4522

3.00001.2656-2.34902.3531-1.89370.7051

4.00001.2504-2.22332.6181-1.87110.6508

5.00001.1997-2.21532.5919-1.95900.7699

6.00001.1829-2.15342.7302-1.93120.7704

7.00001.1405-2.14212.7323-1.97190.8352

8.00001.1252-2.10652.8098-1.95830.8468

9.00001.0975-2.09542.8217-1.97880.8847

10.00001.0850-2.07382.8671-1.97350.8969

11.00001.0673-2.06452.8802-1.98430.9200

12.00001.0577-2.05092.9077-1.98280.9303

13.00001.0463-2.04372.9191-1.98870.9448

14.00001.0392-2.03502.9363-1.98860.9527

15.00001.0318-2.02972.9451-1.99200.9620

16.00001.0267-2.02412.9561-1.99240.9678

17.00001.0218-2.02032.9627-1.99440.9739

18.00001.0182-2.01652.9699-1.99490.9781

19.00001.0149-2.01382.9746-1.99610.9821

20.00001.0124-2.01132.9793-1.99660.9851

21.00001.0102-2.00942.9827-1.99730.9878

22.00001.0085-2.00772.9858-1.99770.9898

23.00001.0070-2.00642.9882-1.99810.9916

24.00001.0058-2.00532.9903-1.99840.9930

25.00001.0048-2.00442.9919-1.99870.9943

26.00001.0040-2.00362.9934-1.99890.9952

27.00001.0033-2.00302.9945-1.99910.9961

28.00001.0027-2.00252.9955-1.99930.9967

29.00001.0022-2.00212.9962-1.99940.9973

30.00001.0019-2.00172.9969-1.99950.9978

31.00001.0015-2.00142.9974-1.99960.9982

32.00001.0013-2.00122.9979-1.99970.9985

33.00001.0010-2.00102.9982-1.99970.9987

34.00001.0009-2.00082.9985-1.99980.9990

35.00001.0007-2.00072.9988-1.99980.9991

36.00001.0006-2.00052.9990-1.99980.9993

37.00001.0005-2.00042.9992-1.99990.9994

38.00001.0004-2.00042.9993-1.99990.9995

39.00001.0003-2.00032.9994-1.99990.9996

40.00001.0003-2.00032.9995-1.99990.9997

41.00001.0002-2.00022.9996-1.99990.9997

42.00001.0002-2.00022.9997-1.99990.9998

43.00001.0002-2.00012.9997-2.00000.9998

44.00001.0001-2.00012.9998-2.00000.9998

45.00001.0001-2.00012.9998-2.00000.9999

46.00001.0001-2.00012.9999-2.00000.9999

47.00001.0001-2.00012.9999-2.00000.9999

48.00001.0001-2.00012.9999-2.00000.9999

49.00001.0000-2.00002.9999-2.00000.9999

50.00001.0000-2.00002.9999-2.00001.0000

51.00001.0000-2.00002.9999-2.00001.0000

52.00001.0000-2.00003.0000-2.00001.0000

53.00001.0000-2.00003.0000-2.00001.0000

54.00001.0000-2.00003.0000-2.00001.0000

55.00001.0000-2.00003.0000-2.00001.0000

»x=G_S_zhao(A,b,xO,0.00001,60);

k

1.00001.2000-3.13331.2095-1.49680.1567

2.00001.6578-2.66491.8991-1.84870.3347

3.00001.5074-2.43412.2530-1.92320.5340

4.00001.3562-2.29502.4903-1.95130.6794

5.00001.2451-2.20162.6513-1.96720.7803

6.00001.1680-2.13792.7613-1.97760.8496

7.00001.1150-2.09442.8366-1.98470.8970

8.00001.0787-2.06462.8882-1.98950.9295

9.00001.0539-2.04422.9234-1.99280.9517

10.00001.0369-2.03032.9476-1.99510.9670

11.00001.0253-2.02072.9641-1.99660.9774

12.00001.0173-2.01422.9754-1.99770.9845

13.00001.0118-2.00972.9832-1.99840.9894

14.00001.0081-2.00672.9885-1.99890.9927

15.00001.0055-2.00462.9921-1.99930.9950

16.00001.0038-2.00312.9946-1.99950.9966

17.00001.0026-2.00212.9963-1.99970.9977

18.00001.0018-2.00152.9975-1.99980.9984

19.00001.0012-2.00102.9983-1.99980.9989

20.00001.0008-2.00072.9988-1.99990.9993

21.00001.0006-2.00052.9992-1.99990.9995

22.00001.0004-2.00032.9994-1.99990.9997

23.00001.0003-2.00022.9996-2.00000.9998

24.00001.0002-2.00022.9997-2.00000.9998

25.00001.0001-2.00012.9998-2.00000.9999

26.00001.0001-2.00012.9999-2.00000.9999

27.00001.0001-2.00002.9999-2.00000.9999

28.00001.0000-2.00002.9999-2.00001.0000

29.00001.0000-2.00003.0000-2.00001.0000

30.00001.0000-2.00003.0000-2.00001.0000

31.00001.0000-2.00003.0000-2.00001.0000

32.00001.0000-2.00003.0000-2.00001.0000

2.题

formatlong

A=[10,l,2,3,4;l,9,-l,2,-3;2,-l,7,3,-5;3,2,3,12,-l

;4,-3,-5,-l,15];

b=[12,-27,14,-17,12]';

N=200;

%A=[2,-l,0;-l,2,-l;0,-l,2];

%b=[1,0,1.8]';

w=l.4;

n=length(A);

x0=ones(n,1);

x=ones(n,0);

flag=l;

sum=0;

k=0;

whileflag==l

fori=l:n

forj=l:i

ifi-=j

sum=sum+A(i,j)*x(j,1);

else

continue;

end

㊀nd

form=j+1:n

sum=sum+A(i,m)*x0(m,1);

end

x(i,1)=(1-w)*x0(i,l)+w*(b(i,1)-sum)/A(i,i

sum=0;

㊀nd

k=k+l

x

ifk~=N

ifnorm(x-xO,inf)>wc

flag=l;

㊀Is㊀

flag=0;

end

else

fprintf(1ON'ip^xiz6piizu'IEy1);

flag=O;

end

xO=x;

㊀nd

结果：

k=

1

x=

-0.120000000000000

-4.270222222222222

1.993955555555556

-1.926165925925926

0.319874883950618

k=

2

X=

2.397383309432098

-1.806139972249109

2.157810899061290

-2.348433539283285

0.379099041230848

k=

3

x=

1.143765844014580

-2.412267015519451

2.785075064466581

-1.811963710490288

0.996504787835416

k=

4

x=

0.983372103734337

-1.926070677869254

2.991090011275371

-2.083934205973658

1.016314189039255

k=

5

X=

1.024912270897810

-2.001106816938999

3.060052436366378

-1.994002385812657

1.012448082134657

k=

6

x=

0.963885439875658

-1.980654782076844

3.003143406710068

-1.993920749912968

1.015954517083863

k=

x=

0.999369525058159

-2.001596919855795

3.010982410353194

-2.003820902469318

0.998174940186140

k=

8

x=

1.000027496390528

-1.997320103697281

2.996599498230785

-1.998129319911984

1.000057825553756

k=

9

x=

0.999747888509718

-2.002116712222330

2.999973120360280

-2.000149979306557

0.999451770078803

k=

10

x=

1.000774710670985

-1.999487187338363

2.999345187782558

-2.000165589181306

0.999752620171508

k=

11

x=

1.000009749539964

-2.000372428623808

3.000035513026520

-1.999891567193516

1.000017725229458

k=

12

x=

0.999982828636702

-1.999868296078108

2.999971669664657

-2.000056110499740

1.000017739845509

k=

13

X=

1.000009994586556

-2.000032908028784

3.000052158839111

-1.999989561310395

1.000005273570704

k=

14

x=

0.999978667365228

-1.999976191374097

2.999991441600386

-1.999998653709936

1.000008652904539

k=

15

x=

1.000002185129806

-2.000007575489729

3.000008879340478

-2.000001633960826

0.999997593108479

k=

16

X=

0.999999734424105

-1.999996162134237

2.999995895552294

-1.999998993213431

1.000000315065173

k=

17

x=

1.000000118887653

-2.000002358301658

3.000000833556922

-2.000000149042239

0.999999544347364

k=

18

x=

1.000000366974449

-1.999999150369042

2.999999323486351

-2.000000083451083

0.999999959658791

k=

19

x=

0.999999981326240

-2.000000435046371

3.000000200795130

-1.999999933557701

1.000000001200712

k=

20

x=

0.999999983575195

-1.999999812302289

2.999999925126745

-2.000000038278648

1.000000019693475

k=

21

x=

1.000000006305440

-2.000000066607681

3.000000056766253

-1.999999988924269

0.999999998643082

k=

22

x=

0.999999987016416

-1.999999966585965

2.999999981167382

-2.000000001249537

1.000000005840724

k=

23

x=

1.000000003042602

-2.000000013654010

3.000000010175650

-2.000000000259220

0.999999997429126

k=

24

X=

0.999999999393900

-1.999999993980330

2.999999994960771

-1.999999999624972

1.000000000623497

k=

25

x=

1.000000000304000

-2.000000003064747

3.000000001679621

-2.000000000056430

0.999999999557535

k=

26

x=

1.000000000108651

-1.999999998718656

2.999999999132353

-2.000000000062381

1.000000000084475

k=

27

X=

0.999999999998986

-2.000000000588518

3.000000000351665

-1.999999999950599

0.999999999970524

k=

28

x=

0.999999999980090

-1.999999999735918

2.999999999860998

-2.000000000029200

1.000000000025574

k=

29

x=

1.000000000007856

-2.000000000107458

3.000000000074061

-1.999999999988933

0.999999999992344

k=

30

x=

0.999999999990804

-1.999999999951082

2.999999999969541

-2.000000000002855

1.000000000005712

k=

31

x=

1.000000000003359

-2.000000000021274

3.000000000014010

-1.999999999999307

0.999999999997107

第四章

1.塞法

cl㊀ar;

A=[19066-8430;6630342-36;336-168147-112;30

-3628291];

x0=[0001]1;

tol=10A(-6);

nmax=100;

k=l;u=0;lm=0;

whil㊀k<=nmax

xr=norm(xO,inf);

a=xr;

y=x0/a;

x=A*y;

lm=xr;

ifabs(Im-u)<tol;

break;

㊀nd

k=k+l;

u=lm;

x0=x;

end

Im,ul=xO

运行结果：

Im=

343.0000

ul=

-114.3333

-343.0000

0.0000

171.5002

结果分析：

按模最大特征向量为

-114.3333-343.00000.0000171.5002

按模最大特征值为

343.0000

2.题反塞法

formatlong

%A=[19066-8430;

%6630342-36;

%336-168147-112;

%30-3628291];

A=[2-10;02-1;0-12];

ep=lE-8;

N=200;

k=0;

lamda0=0;

x0=[001]';

flag=l;

a=max(abs(xO(:,1)));

whileflag==l

k=k+l

y=xO./a;

x=A*y

a=max(abs(x(:,1)));

lamda=a;

ifk<N

ifabs(lamda-lamdaO)>ep

flag=l;

else

flag=O;

end

else

flag=O;

fprintf('ON'fx6pii"uzIEy');

end

lamdaO=lamda;

xO=x;

end

结果：

k=

1

x=

0

-1

2

k=

2

x=

0.500000000000000

-2.000000000000000

2.500000000000000

k=

3

x=

1.200000000000000

-2.600000000000000

2.800000000000000

k=

4

x=

1.785714285714286

-2.857142857142858

2.928571428571429

k=

5

x=

2.195121951219512

-2.951219512195122

2.975609756097561

k=

6

x=

2.467213114754098

-2.983606557377049

2.991803278688525

k=

7

x=

2.646575342465753

-2.994520547945205

2.997260273972603

k=

8

x=

2.765082266910421

-2.998171846435100

2.999085923217550

k=

9

x=

2.843645230112770

-2.999390429747028

2.999695214873514

k=

10

x=

2.895854501117659

-2.99979678927