On a small size inverse problem
hari
chandra_hari18 at yahoo.com
Sun Aug 22 07:32:03 CEST 2010
SCI LAB
dev at lists.scilab.org
This inquiry is from India. I have a small size inverse problem of
the kind: Ax = b, the SVD analysis of which is given below: I have not used any
Matlab routine or advanced software. I have been working out details on excel
spreadsheet. Can I have any advantage in tackling the problems of the kind with
any SCILAB tools?
Given the structure and dynamics of the problem as illustrated
below, to what extent the tools may help to make a regularized solution to
approach reality?
Can the error in the solution be minimized by any routine? Any publications
available in this regard?
A
x
b
2.65
2.71
2.12
2.53
2.42
2.77
1.09
0.30
2.265
5
0
180
12
44
2.04
0
0.05
22.784
3
0
12
3.5
14
16
0
0.07
6.000
-0.06
0
0.44
0.3
0.37
0.52
1
0.12
0.360
2
5.5
2.04
3.45
1.83
6.37
0.8
0.16
2.522
52
47
120
87
77
100
189
0.10
96.910
1
1
1
1
1
1
1
0.20
1.000
Can the inverse problem help me to retrieve the vector x with
precision with any numerical linear algebra tools? I have xT =
[0.30, 0.05, 0.07, 0.12, 0.16, 0.10, 0.20]. Is there any mathematics that can
help me to retrieve these elements x11, x21…x51 with
precision? Data vector b has the maximum error defined as
1.5%. Kernel of the problem A is assumed to be precise but
that too shall in have the same error of 1.5% at least.
Singular Value
Decomposition
U:
-0.017
0.005
-0.136
0.404
0.833
-0.064
-0.346
-0.424
-0.904
0.049
0.018
-0.004
0.001
-0.001
-0.059
-0.03
-0.95
-0.303
-0.004
-0.01
0.013
-0.004
0.003
0.001
-0.05
-0.127
0.842
-0.522
-0.021
0.014
-0.269
0.853
-0.446
0.001
0.029
-0.903
0.426
0.048
-0.016
-0.005
-0.007
0.001
-0.008
0.003
-0.034
0.12
0.301
0.535
0.779
S:
302.25
0
0
0
0
0
0
0
148.658
0
0
0
0
0
0
0
17.475
0
0
0
0
0
0
0
6.26
0
0
0
0
0
0
0
1.866
0
0
0
0
0
0
0
0.112
0
0
0
0
0
0
0
0.073
VT
-0.163
-0.141
-0.614
-0.278
-0.295
-0.305
-0.565
0.118
0.135
-0.753
0.176
-0.049
0.272
0.542
-0.059
0.022
0.13
0.008
-0.475
-0.71
0.498
0.196
0.82
0.057
0.287
-0.331
0.029
-0.307
0.715
-0.068
-0.137
0.181
0.45
-0.466
-0.113
-0.589
0.4
-0.131
0.063
0.601
-0.331
0.047
0.244
0.354
0.025
-0.879
0.107
0.034
0.172
U^T*b and singular values: Discrete Picard condition not
satisfied.
U^transpose
b
U^T*b
Sing value
-0.01772
-0.61485
-0.07008
-0.0042
-0.01969
-0.78502
-0.00829
2.3153
-95.5104
298.505
0.00266
-0.78646
-0.01828
0.00235
0.00883
0.61729
0.00204
51.3332
9.346925
194.1835
-0.13151
0.05757
-0.94948
0.00289
-0.27283
0.04979
-0.03075
7.085
-0.72217
17.17149
0.46477
0.00877
-0.30443
-0.04839
0.82074
-0.01181
0.12306
0.3513
0.430871
3.52994
0.81003
-0.00672
0.02132
-0.1165
-0.50001
-0.00475
0.28241
2.3254
0.376967
1.79652
0.31991
-0.00003
-0.0076
0.58797
-0.03808
-0.00088
-0.74192
80.705
-0.00964
0.05186
-0.08879
-0.00156
-0.00661
0.79897
0.00568
-0.00688
0.59468
1
0.000829
0.01536
Z = U^T*b/S. value
Z
-0.320
0.048
-0.042
0.122
0.210
-0.186
0.054
Computing the errors – (norm) ZV
z
V
Z*V
-0.320
-0.15202
0.14968
-0.04783
0.42298
0.66837
-0.32286
0.47152
-0.82
0.048
-0.46941
-0.41758
0.07247
0.11346
0.03357
0.69704
0.31648
1.08
-0.042
-0.53166
-0.53927
0.0636
-0.006
-0.17228
-0.60614
-0.15923
-0.80
0.122
-0.25474
0.22784
0.02306
0.60297
0.08508
0.17523
-0.69364
-0.19
0.210
-0.29671
0.06539
-0.45265
-0.58171
0.51325
0.0942
-0.30352
0.05
-0.186
-0.27156
0.30847
-0.71201
0.1642
-0.4824
-0.04129
0.25052
0.02
0.054
-0.4972
0.60089
0.52539
-0.28152
-0.13823
-0.03466
0.12701
0.21
error is computed as below:
solution error that can
be arrived at by comparing the inverse solution with the prior x used
for deriving b of Ax = b.
b
U^T*b
Sing
values
Z
v
Error
Norm
Vz = x
Prior used for
getting b
Diff
% error
2.31
-95.51
298.51
-0.32
V
-0.82
0.31
0.35
0.04
10.57
51.33
9.35
194.18
0.05
1.08
0.01
0.05
0.04
72.88
7.08
-0.72
17.17
-0.04
-0.80
0.22
0.20
-0.02
-10.59
0.35
0.43
3.53
0.12
-0.19
0.15
0.09
-0.06
-65.74
2.32
0.38
1.80
0.21
0.05
0.14
0.11
-0.03
-24.18
80.70
-0.01
0.05
-0.19
0.02
0.06
0.08
0.02
28.10
1.00
0.00
0.02
0.00
0.09
0.11
0.12
0.01
9.05
Can this method work for some reasonably good solution?
Regularization can work when discrete Picard condition is not satisfied with
any SCILAB tools?
Is there any MATLAB regularization tools which can help given the
above structure and dynamics evident from SVD?
Small values have undergone severe modification. Can such errors
be minimized? Last but not the least, is there anything wrong in the error
estimated above?
hari
Aum Namah Sivaya
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