数值分析三.docx
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数值分析三.docx
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数值分析三
《数值分析B》
计算实习
题目三
学号:
ZY1207227
姓名:
丁少行
2012-12-08
1.设计方案
对Fredholm积分方程,用迭代法进行求解:
,其中
对于公式中的积分部分用数值积分方法。
复化梯形积分法,取2601个节点,取迭代次数上限为50次。
实际计算迭代次数为18次,最后算得误差为r=0.97E-10。
复化Simpson积分法,取迭代次数上限为50次,取2*41+1,即83个节点时能满足精度要求。
实际计算迭代次数为17次,最后的误差为r=0.97E-10。
Guass积分法选择的Gauss—Legendre法,取迭代次数上限为50次,直接选择8个节点,满足精度要求。
实际计算迭代次数为24次,最后算得误差为r=0.87E-10。
2.全部源程序
moduleintegral
implicitnone
contains
!
//////////复化梯形
subroutinetrapezoid(m)
implicitnone
integer:
:
i,j,k,m
real*8:
:
x(m+1),u(m+1)
real*8:
:
sum,sum1,g,r,h
real*8:
:
e=1.0e-10
h=2./m
doi=1,m+1
x(i)=-1.+(i-1)*h
enddo
u=0.02
dok=1,50
doi=1,m+1
sum1=0.
g=dexp(x(i)*4.)+(dexp(x(i)+4.)-dexp(-4.-x(i)))/(x(i)+4.)
doj=2,m
sum1=sum1+dexp(x(i)*x(j))*u(j)
enddo
sum=h/2.*(dexp(x(i)*-1.)*u
(1)+dexp(x(i)*1.)*u(m+1)+2*sum1)
u(i)=g-sum
enddo
r=h/2.*((dexp(x
(1)*4)-u
(1))**2+(dexp(x(m+1)*4)-u(m+1))**2)
doi=2,m
r=r+h*(dexp(x(i)*4)-u(i))**2
enddo
if(dabs(r)<=e)exit
enddo
write(*,*)k
open(1,file="trapezoid.txt")
doi=1,m+1
write(1,'(3(f18.12))')x(i),u(i),dexp(x(i)*4.)
enddo
write(1,'(4x,a2,e9.2)')"r=",r
close
(1)
return
endsubroutinetrapezoid
!
///////////复化simpson
subroutinesimpson(m)
implicitnone
integer:
:
i,j,k,m
real*8:
:
x(2*m+1),u(2*m+1)
real*8:
:
sum,sum1,sum2,g,r,h
real*8:
:
e=1.0e-10
h=2./(2.*m)
doi=1,2*m+1
x(i)=-1.+(i-1)*h
enddo
u=0.02
dok=1,50
doi=1,2*m+1
sum1=0.
sum2=0.
g=dexp(x(i)*4.)+(dexp(x(i)+4.)-dexp(-4.-x(i)))/(x(i)+4.)
doj=1,m
sum1=sum1+dexp(x(i)*x(2*j))*u(2*j)
enddo
doj=1,m-1
sum2=sum2+dexp(x(i)*x(2*j+1))*u(2*j+1)
enddo
sum=h/3.*(dexp(x(i)*-1.)*u
(1)+dexp(x(i)*1.)*u(2*m+1)+4*sum1+2*sum2)
u(i)=g-sum
enddo
r=h/3.*((dexp(x
(1)*4)-u
(1))**2+(dexp(x(2*m+1)*4)-u(2*m+1))**2)
doi=1,m
r=r+4.*h/3.*(dexp(x(2*i)*4)-u(2*i))**2
enddo
doi=1,m-1
r=r+2.*h/3.*(dexp(x(2*i+1)*4)-u(2*i+1))**2
enddo
if(dabs(r)<=e)exit
enddo
write(*,*)k
open(2,file="simpson.txt")
doi=1,2*m+1
write(2,'(3(f18.12))')x(i),u(i),dexp(x(i)*4.)
enddo
write(2,'(4x,a2,e9.2)')"r=",r
close
(2)
return
endsubroutinesimpson
!
///////////Gauss_Legendre法
subroutineGauss
implicitnone
integer,parameter:
:
m=8
integer:
:
i,j,k
real*8:
:
x(m),u(m),a(m)
real*8:
:
sum,g,r
real*8:
:
e=1.0e-10
datax/-0.9602898565,-0.7966664774,-0.5255324099,-0.1834346425,&
0.1834346425,0.5255324099,0.7966664774,0.9602898565/
dataa/0.1012285363,0.2223810345,0.3137066459,0.3626837834,&
0.3626837834,0.3137066459,0.2223810345,0.1012285363/
u=0.02
dok=1,50
doi=1,m
sum=0.
g=dexp(x(i)*4.)+(dexp(x(i)+4.)-dexp(-4.-x(i)))/(x(i)+4.)
doj=1,m
sum=sum+dexp(x(i)*x(j))*u(j)*a(j)
enddo
u(i)=g-sum
enddo
r=0.
doi=1,m
r=r+a(i)*(dexp(x(i)*4)-u(i))**2
enddo
if(dabs(r)<=e)exit
enddo
write(*,*)k
open(3,file="Gauss.txt")
doi=1,m
write(3,'(3(f18.12))')x(i),u(i),dexp(x(i)*4.)
enddo
write(3,'(4x,a2,e9.2)')"r=",r
close(3)
return
endsubroutineGauss
endmodule
!
//////////主程序
programmain
useintegral
implicitnone
integer:
:
code1=2600
integer:
:
code2=41
calltrapezoid(code1)
callsimpson(code2)
callGauss
endprogram
3.各种积分方法的节点和数值解(由于数据太多,在打印时用了较计算时少的有效数字)
复化梯形法
-1
0.01832
-0.9992
0.01837
-0.9985
0.01843
-0.9977
0.01849
-0.9969
0.01854
-0.9962
0.0186
-0.9954
0.01866
-0.9946
0.01872
-0.9938
0.01877
-0.9931
0.01883
-0.9923
0.01889
-0.9915
0.01895
-0.9908
0.01901
-0.99
0.01906
-0.9892
0.01912
-0.9885
0.01918
-0.9877
0.01924
-0.9869
0.0193
-0.9862
0.01936
-0.9854
0.01942
-0.9846
0.01948
-0.9838
0.01954
-0.9831
0.0196
-0.9823
0.01966
-0.9815
0.01972
-0.9808
0.01978
-0.98
0.01984
-0.9792
0.0199
-0.9785
0.01997
-0.9777
0.02003
-0.9769
0.02009
-0.9762
0.02015
-0.9754
0.02021
-0.9746
0.02027
-0.9738
0.02034
-0.9731
0.0204
-0.9723
0.02046
-0.9715
0.02053
-0.9708
0.02059
-0.97
0.02065
-0.9692
0.02072
-0.9685
0.02078
-0.9677
0.02084
-0.9669
0.02091
-0.9662
0.02097
-0.9654
0.02104
-0.9646
0.0211
-0.9638
0.02117
-0.9631
0.02123
-0.9623
0.0213
-0.9615
0.02136
-0.9608
0.02143
-0.96
0.02149
-0.9592
0.02156
-0.9585
0.02163
-0.9577
0.02169
-0.9569
0.02176
-0.9562
0.02183
-0.9554
0.0219
-0.9546
0.02196
-0.9538
0.02203
-0.9531
0.0221
-0.9523
0.02217
-0.9515
0.02223
-0.9508
0.0223
-0.95
0.02237
-0.9492
0.02244
-0.9485
0.02251
-0.9477
0.02258
-0.9469
0.02265
-0.9462
0.02272
-0.9454
0.02279
-0.9446
0.02286
-0.9438
0.02293
-0.9431
0.023
-0.9423
0.02307
-0.9415
0.02314
-0.9408
0.02321
-0.94
0.02328
-0.9392
0.02336
-0.9385
0.02343
-0.9377
0.0235
-0.9369
0.02357
-0.9362
0.02365
-0.9354
0.02372
-0.9346
0.02379
-0.9338
0.02387
-0.9331
0.02394
-0.9323
0.02401
-0.9315
0.02409
-0.9308
0.02416
-0.93
0.02423
-0.9292
0.02431
-0.9285
0.02438
-0.9277
0.02446
-0.9269
0.02454
-0.9262
0.02461
-0.9254
0.02469
-0.9246
0.02476
-0.9238
0.02484
-0.9231
0.02492
-0.9223
0.02499
-0.9215
0.02507
-0.9208
0.02515
-0.92
0.02522
-0.9192
0.0253
-0.9185
0.02538
-0.9177
0.02546
-0.9169
0.02554
-0.9162
0.02561
-0.9154
0.02569
-0.9146
0.02577
-0.9138
0.02585
-0.9131
0.02593
-0.9123
0.02601
-0.9115
0.02609
-0.9108
0.02617
-0.91
0.02625
-0.9092
0.02633
-0.9085
0.02642
-0.9077
0.0265
-0.9069
0.02658
-0.9062
0.02666
-0.9054
0.02674
-0.9046
0.02682
-0.9038
0.02691
-0.9031
0.02699
-0.9023
0.02707
-0.9015
0.02716
-0.9008
0.02724
-0.9
0.02732
-0.8992
0.02741
-0.8985
0.02749
-0.8977
0.02758
-0.8969
0.02766
-0.8962
0.02775
-0.8954
0.02783
-0.8946
0.02792
-0.8938
0.02801
-0.8931
0.02809
-0.8923
0.02818
-0.8915
0.02827
-0.8908
0.02835
-0.89
0.02844
-0.8892
0.02853
-0.8885
0.02862
-0.8877
0.0287
-0.8869
0.02879
-0.8862
0.02888
-0.8854
0.02897
-0.8846
0.02906
-0.8838
0.02915
-0.8831
0.02924
-0.8823
0.02933
-0.8815
0.02942
-0.8808
0.02951
-0.88
0.0296
-0.8792
0.02969
-0.8785
0.02978
-0.8777
0.02987
-0.8769
0.02997
-0.8762
0.03006
-0.8754
0.03015
-0.8746
0.03024
-0.8738
0.03034
-0.8731
0.03043
-0.8723
0.03052
-0.8715
0.03062
-0.8708
0.03071
-0.87
0.03081
-0.8692
0.0309
-0.8685
0.031
-0.8677
0.03109
-0.8669
0.03119
-0.8662
0.03129
-0.8654
0.03138
-0.8646
0.03148
-0.8638
0.03158
-0.8631
0.03167
-0.8623
0.03177
-0.8615
0.03187
-0.8608
0.03197
-0.86
0.03207
-0.8592
0.03216
-0.8585
0.03226
-0.8577
0.03236
-0.8569
0.03246
-0.8562
0.03256
-0.8554
0.03266
-0.8546
0.03276
-0.8538
0.03286
-0.8531
0.03297
-0.8523
0.03307
-0.8515
0.03317
-0.8508
0.03327
-0.85
0.03337
-0.8492
0.03348
-0.8485
0.03358
-0.8477
0.03368
-0.8469
0.03379
-0.8462
0.03389
-0.8454
0.034
-0.8446
0.0341
-0.8438
0.03421
-0.8431
0.03431
-0.8423
0.03442
-0.8415
0.03452
-0.8408
0.03463
-0.84
0.03474
-0.8392
0.03484
-0.8385
0.03495
-0.8377
0.03506
-0.8369
0.03517
-0.8362
0.03527
-0.8354
0.03538
-0.8346
0.03549
-0.8338
0.0356
-0.8331
0.03571
-0.8323
0.03582
-0.8315
0.03593
-0.8308
0.03604
-0.83
0.03615
-0.8292
0.03626
-0.8285
0.03638
-0.8277
0.03649
-0.8269
0.0366
-0.8262
0.03671
-0.8254
0.03683
-0.8246
0.03694
-0.8238
0.03705
-0.8231
0.03717
-0.8223
0.03728
-0.8215
0.0374
-0.8208
0.03751
-0.82
0.03763
-0.8192
0.03774
-0.8185
0.03786
-0.8177
0.03798
-0.8169
0.03809
-0.8162
0.03821
-0.8154
0.03833
-0.8146
0.03845
-0.8138
0.03857
-0.8131
0.03869
-0.8123
0.0388
-0.8115
0.03892
-0.8108
0.03904
-0.81
0.03916
-0.8092
0.03928
-0.8085
0.03941
-0.8077
0.03953
-0.8069
0.03965
-0.8062
0.03977
-0.8054
0.03989
-0.8046
0.04002
-0.8038
0.04014
-0.8031
0.04026
-0.8023
0.04039
-0.8015
0.04051
-0.8008
0.04064
-0.8
0.04076
-0.7992
0.04089
-0.7985
0.04101
-0.7977
0.04114
-0.7969
0.04127
-0.7962
0.04139
-0.7954
0.04152
-0.7946
0.04165
-0.7938
0.04178
-0.7931
0.04191
-0.7923
0.04204
-0.7915
0.04217
-0.7908
0.0423
-0.79
0.04243
-0.7892
0.04256
-0.7885
0.04269
-0.7877
0.04282
-0.7869
0.04295
-0.7862
0.04308
-0.7854
0.04322
-0.7846
0.04335
-0.7838
0.04348
-0.7831
0.04362
-0.7823
0.04375
-0.7815
0.04389
-0.7808
0.04402
-0.78
0.04416
-0.7792
0.04429
-0.7785
0.04443
-0.7777
0.04457
-0.7769
0.0447
-0.7762
0.04484
-0.7754
0.04498
-0.7746
0.04512
-0.7738
0.04526
-0.7731
0.0454
-0.7723
0.04554
-0.7715
0.04568
-0.7708
0.04582
-0.77
0.04596
-0.7692
0.0461
-0.7685
0.04624
-0.7677
0.04639
-0.7669
0.04653
-0.7662
0.04667
-0.7654
0.04682
-0.7646
0.04696
-0.7638
0.0471
-0.7631
0.04725
-0.7623
0.0474
-0.7615
0.04754
-0.7608
0.04769
-0.76
0.04783
-0.7592
0.04798
-0.7585
0.04813
-0.7577
0.04828
-0.7569
0.04843
-0.7562
0.04858
-0.7554
0.04873
-0.7546
0.04888
-0.7538
0.04903
-0.7531
0.04918
-0.7523
0.04933
-0.7515
0.04948
-0.7508
0.04963
-0.75
0.04979
-0.7492
0.04994
-0.7485
0.05009
-0.7477
0.05025
-0.7469
0.0504
-0.7462
0.05056
-0.7454
0.05071
-0.7446
0.05087
-0.7438
0.05103
-0.7431
0.05119
-0.7423
0.05134
-0.7415
0.0515
-0.7408
0.05166
-0.74
0.05182
-0.7392
0.05198
-0.7385
0.05214
-0.7377
0.0523
-0.7369
0.05246
-0.7362
0.05262
-0.7354
0.05278
-0.7346
0.05295
-0.7338
0.05311
-0.7331
0.05327
-0.7323
0.05344
-0.7315
0.0536
-0.7308
0.05377
-0.73
0.05393
-0.7292
0.0541
-0.7285
0.05427
-0.7277
0.05443
-0.7269
0.0546
-0.7262
0.05477
-0.7254
0.05494
-0.7246
0.05511
-0.7238
0.05528
-0.7231
0.05545
-0.7223
0.05562
-0.7215
0.05579
-0.7208
0.05596
-0.72
0.05613
-0.7192
0.05631
-0.7185
0.05648
-0.7177
0.05666
-0.7169
0.05683
-0.7162
0.05701
-0.7154
0.05718
-0.7146
0.05736
-0.7138
0.05753
-0.7131
0.05771
-0.7123
0.05789
-0.7115
0.05807
-0.7108
0.05825
-0.71
0.05843
-0.7092
0.05861
-0.7085
0.05879
-0.7077
0.05897
-0.7069
0
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