3数值计算方法上机实习题要点文档格式.docx

3数值计算方法上机实习题要点文档格式.docx

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I（4）=-0.1667；

I=0.182;

forn=1:

20

I=（-5）*I+1/n;

end

故计算结果

=-3.0666e+10

I=0.008;

forn=20:

-1:

1

I=（-1/5）*I+1/（5*n）;

I

=0.1823

假设

的真值为

，误差为

，即

。

对于真值也有

综合2个递推等式，有

，即意味着只要n足够大，按照这种每计算一步误差增长5倍的方式，所得的结果总是不可信的，因此整个算法是数值不稳定的。

而第二种方式的误差会以每计算一步缩小到1/5的方式进行，这样的计算结果和实际是很相近的。

2．求方程

的近似根，要求

，并比较计算量。

（1）在[0，1]上用二分法；

（2）取初值

，并用迭代

（3）加速迭代的结果；

（4）取初值

，并用牛顿迭代法；

（5）分析绝对误差。

程序：

a=0;

b=1.0;

i=0;

whileabs（b-a）>

5*1e-4

c=（b+a）/2;

ifexp（c）+10*c-2>

b=c;

elsea=c;

end

i=i+1;

end

c

方程的近似根为：

x*=（a+b）/2=0.0906。

步长为i=11。

程序:

x=0;

y=0.1;

whileabs（y-x）>

y=x;

x=（2-exp（x））/10;

x=0.0905。

步长为i=4。

x=0;

xx=1;

whileabs（xx-x）>

y=exp（x）+10*x-2;

z=exp（y）+10*y-2;

xx=x;

x=x-（y-x）^2/（z-2*y+x）;

x

x=0.0995。

步长为i=3。

x=x-（exp（x）+10*x-2）/（exp（x）+10）;

%diff（exp（x）+10*x-2）=exp（x）+10

end

i

x=0.0905。

步长为i=2。

（6）分析绝对误差。

solve（'

exp（x）+10*x-2=0'

）

方程的精确解为x=0.0905。

3．钢水包使用次数多以后，钢包的容积增大，数据如下：

2

3

4

5

6

7

8

9

y

6.42

8.2

9.58

9.5

9.7

10

9.93

9.99

11

12

13

14

15

16

10.49

10.59

10.60

10.8

10.6

10.9

10.76

试从中找出使用次数和容积之间的关系，计算均方差。

（注：

增速减少，用何种模型）

解：

（1）设y=f（x）具有指数形式

（a>

0,b<

0）。

对此式两边取对数，得

记A=lna，B=b

x=[2345678910111213141516];

fori=1:

X（i）=1/x（i）;

y=[6.428.29.589.59.7109.939.9910.4910.5910.6010.810.610.910.76];

Y（i）=log（y（i））;

polyfit（X,Y,1）

经计算ans=-1.11072.4578。

故方程为

故原方程的系数为

故原方程为：

（2）计算均方差：

x=[2:

16];

y=[6.428.29.589.59.7109.939.9910.4910.5910.6010.810.610.910.76];

f（x）=11.6791*exp（-1.1107./x）;

c=0;

a=y（i）;

b=x（i）;

c=c+（a-f（b））^2;

averge=c/15

结果：

averge=0.0594

4．设

分析下列迭代法的收敛性，并求

的近似解及相应的迭代次数。

（1）JACOBI迭代；

以文件名math4a.m保存。

functionmath4a

A=[4-10-100;

-14-10-10;

0-14-10-1;

-10-14-10;

0-10-14-1;

00-10-14];

b=[05-25-26];

x0=[000000];

imax=100;

tol=10^-4;

tx=jacobi（A,b,imax,x0,tol）;

forj=1:

size（tx,1）

fprintf（'

%d%f%f%f%f%f%f\n'

j,tx（j,1）,tx（j,2）,tx（j,3）,tx（j,4）,tx（j,5）,tx（j,6））

functiontx=jacobi（A,b,imax,x0,tol）%初始值x0,次数imax,精度tol

del=10^-10;

tx=[x0];

n=length（x0）;

n

dg=A（i,i）;

ifabs（dg）<

del

disp（'

对角元太小'

）;

return

fork=1:

imax%jacobi循环

fori=1:

sm=b（i）;

forj=1:

ifj~=i

sm=sm-A（i,j）*x0（j）;

x（i）=sm/A（i,i）;

tx=[tx;

x];

ifnorm（x-x0）<

tol

else

x0=x;

近似解y为：

10.0000000.0000000.0000000.0000000.0000000.000000

20.0000001.250000-0.5000001.250000-0.5000001.500000

30.6250001.0000000.5000001.0000000.5000001.250000

40.5000001.6562500.3125001.6562500.3125001.750000

50.8281251.5312500.7656251.5312500.7656251.656250

60.7656251.8398440.6796881.8398440.6796881.882813

70.9199221.7812500.8906251.7812500.8906251.839844

80.8906251.9252930.8505861.9252930.8505861.945313

90.9626461.8979490.9489751.8979490.9489751.925293

100.9489751.9651490.9302981.9651490.9302981.974487

110.9825741.9523930.9761961.9523930.9761961.965149

120.9761961.9837420.9674841.9837420.9674841.988098

130.9918711.9777910.9888951.9777910.9888951.983742

140.9888951.9924150.9848311.9924150.9848311.994448

150.9962081.9896390.9948201.9896390.9948201.992415

160.9948201.9964620.9929231.9964620.9929231.997410

170.9982311.9951670.9975831.9951670.9975831.996462

180.9975831.9983490.9966991.9983490.9966991.998792

190.9991751.9977450.9988731.9977450.9988731.998349

200.9988731.9992300.9984601.9992300.9984601.999436

210.9996151.9989480.9994741.9989480.9994741.999230

220.9994741.9996410.9992821.9996410.9992821.999737

230.9998201.9995090.9997551.9995090.9997551.999641

240.9997551.9998320.9996651.9998320.9996651.999877

250.9999161.9997710.9998861.9997710.9998861.999832

260.9998861.9999220.9998441.9999220.9998441.999943

270.9999611.9998930.9999471.9998930.9999471.999922

280.9999471.9999640.9999271.9999640.9999271.999973

290.9999821.9999500.9999751.9999500.9999751.999964

（2）GAUSS-SEIDEL迭代；

以文件名math4b.m保存。

functionmath4b

tx=gs（A,b,imax,x0,tol）;

fprintf（'

functiontx=gs（A,b,imax,x0,tol）%初始值x0,次数imax,精度tol

del=10^-10;

tx=[x0];

fork=1:

imax%gs循环

x=x0;

sm=sm-A（i,j）*x（j）;

20.0000001.250000-0.1875001.2031250.1132811.481445

30.6132811.3847660.5173341.5609740.6067961.781033

40.7364351.7151410.7642871.7768800.8182631.895638

50.8730051.8638890.8841021.8938430.9133421.949361

60.9394331.9342190.9443561.9492830.9582161.975643

70.9708751.9683620.9733221.9756030.9799021.988306

80.9859911.9848040.9871781.9882680.9903441.994381

90.9932681.9926980.9938371.9943620.9953601.997299

100.9967651.9964900.9970381.9972910.9977701.998702

110.9984451.9983130.9985771.9986980.9989281.999376

120.9992531.9991890.9993161.9993740.9994851.999700

130.9996411.9996100.9996711.9996990.9997521.999856

140.9998271.9998130.9998421.9998550.9998811.999931

150.9999171.9999100.9999241.9999310.9999431.999967

160.9999601.9999570.9999641.9999670.9999731.999984

（3）SOR迭代（

）。

以文件名math4c.m保存。

functionmath4c

w0=[1.3341.950.95];

w=w0（i）;

tx=sor（A,b,imax,x0,tol,w）;

\n'

functiontx=sor（A,b,imax,x0,tol,w）%初始值x0,次数imax,精度tol，松弛因子w

imax%SOR循环

x（i）=w*x（i）+（1-w）*x0（i）;

w=1.334时的近似解y：

20.0000001.667500-0.1108891.6305190.4328892.108387

31.0998891.5847551.1454782.0161051.0924492.043147

40.8335242.1625211.0253721.9783941.0305072.004224

51.1025981.9985700.9852522.0466881.0063131.995776

60.9808261.9912701.0161761.9855120.9887402.003050

70.9986612.0041090.9921531.9980201.0054881.998194

81.0011571.9982271.0007672.0031330.9980182.000198

91.0000672.0002101.0009251.9986231.0003392.000355

100.9995882.0002140.9994222.0002431.0001581.999741

111.0002901.9998851.0001492.0001180.9998622.000090

120.9999042.0000101.0000231.9998901.0000431.999992

130.9999992.0000180.9999592.0000371.0000011.999990

141.0000191.9999871.0000182.0000000.9999922.000007

w=1.95时的近似解y：

20.0000002.4375000.2132812.5414751.4522503.736947

32.4272502.1171062.9152023.3355172.1252771.832135

40.3522663.0552130.2642040.605361-0.3163181.159066

51.4498770.216457-0.2603122.2881151.1115262.238853

6-0.1564042.5705852.7323542.0614351.3186012.772930

72.4066943.1435480.3184972.4504851.8512231.348455

80.4407320.7237240.9272271.678891-0.9050161.654796

90.7525802.1276730.8065471.1614332.2949172.864907

100.8884882.3613131.3727613.5552701.1258051.421389

112.0402702.4069361.3203791.2471400.4297742.427880

12-0.1568940.9276250.0144291.3927800.9315031.079656

131.2802472.6415191.5043452.9259561.3805483.305712

141.4979092.0646731.6403381.8607551.2386601.188085

150.4906332.118756-0.0141181.5059300.1944991.884254

161.3009341.1468231.2502032.3453631.4612412.456788

170.4665542.8972931.5907861.9247131.1852331.944361

181.9075021.9682880.3594672.2919710.9237811.703441

190.2647501.3222761.2758791.4615270.3349392.092005

201.1055922.5055880.7667362.1250921.9841172.278636

211.2071441.9867151.4119422.6627240.5175261.700911

221.1198142.0366460.8037921.0979640.8906672.135183

230.4642991.5550810.5956592.3453631.1212351.733562

241.4603832.5090951.6707842.2824511.1408172.648772

250.9485151.8869160.7615981.6590000.9611341.248499

260.8275451.8881900.6393802.0451290.6380592.361678

271.1313251.8179921.4521782.0651391.4631892.102648

280.8182672.5305540.9108722.0318780.8841971.802581

291.4468321.6139000.8157472.0412690.8456662.022487

300.4074051.9128441.1636331.6764370.9573542.037619

311.3627402.3186160.8604762.3954121.4069412.094628

321.0034861.8293811.2882651.9649710.5592831.835783

330.8964352.0372800.6471901.5959451.1598232.061924

340.9