solve A*X=B containing very large or small elements

Question

hosein Javan 2020-8-9

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评论： Bruno Luong 2020-8-10

hello

i have this linear system of equations A*X=B, which is apparently ill-conditioned with RCOND = 0. the elements in these matrices are either extremely large or small which have been derived using "besseli" and "besselk" functions.

A=
   2.7673e+84  -2.7673e+84  -9.1252e-88            0            0            0            0            0            0            0
   1.1039e+87  -2.2078e+90   7.2433e-82            0            0            0            0            0            0            0
            0  7.7866e+105  2.5944e-109 -7.7866e+105 -2.5944e-109            0            0            0            0            0
            0  6.2089e+111 -2.0604e-103 -6.2089e+111  2.0604e-103            0            0            0            0            0
            0            0            0  1.5218e+110  1.2764e-113 -1.5218e+110 -1.2764e-113            0            0            0
            0            0            0  1.2134e+116 -1.0138e-107 -1.2134e+116  1.0138e-107            0            0            0
            0            0            0            0            0  2.2359e+127  7.5295e-131 -2.2359e+127 -7.5295e-131            0
            0            0            0            0            0  1.7822e+133 -5.9817e-125 -8.9112e+129  2.9909e-128            0
            0            0            0            0            0            0            0  1.9154e+170  6.5919e-174 -6.5919e-174
            0            0            0            0            0            0            0  7.6309e+172 -2.6195e-171   5.239e-168
B=
  -5.3706e+78
  -4.2847e+84
  1.5112e+100
   1.205e+106
            0
            0
            0
            0
            0
            0

the solution is returend using X=A\B :

X=
            0
9407e-06
6374e+150
1617e-89
6374e+150
5549e-107
6374e+150
 -3.1817e-193
2546e+150
2616e+147

but it has realy large error:

A*X-B=
  -8.2275e+63
            0
  -1.9427e+84
            0
   1.8889e+22
            0
            0
  -1.0771e+10
    -2.25e-39
            0

also I am uncertained that in the future I might need to calculate other matrices which could probably contain even larger or smaller values that leads to "inf" or "0" elements.

I know that using "digits" and "vpa" functions in symbolic math toolbox it is really accurate but SPEED plays an important role in my code.

I have searched about "Fixed-point Designer Tollbox" but I did not find an example like mine.

what is the best workaround in this condition?

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Answer 1

Bruno Luong 2020-8-9

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https://ww2.mathworks.cn/matlabcentral/answers/577159-solve-a-x-b-containing-very-large-or-small-elements#answer_477106

编辑：Bruno Luong 2020-8-9

Work with unitless system, scale your matrix, or use automatic scaling if you don't want to bother.

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hosein Javan 2020-8-10

Exactly. I must scale the unknown vector "X" to "X=C.*X1" in which "C" is scaling coefficiets vector. and therefore the equation "A*X=B" must be transfomed into "A*(C.*X1)=B" or "A1*X1=B". in which "A1=A.*repmat(C.',[length(X) 1])". now that "A1" and "X1" are scaled and "A1" is well-conditioned; the solution is "X=C.*X1=C.*(A1\B)". am I rigth?

I think that's the spot.

many thanks.