Solving a symbolic linear matrix equation

Questions
,
1

Hiya,

I’m trying to solve the linear equation A.x = b for x, where A is a 24 by 24 matrix with symbolic entries and b = {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}. (You can find my notebook in the attachment) My computer seems to be taking quite some time to do this. I tried both the ‘Solve’ and ‘LinearSolve’ functions with no result after waiting for a few hours. I am suspecting that these ‘naive’ methods for solving this equation will not get me anywhere and I was wondering if there would be some method to tackle this, without giving my matrix entries numerical values, or if I should forget about solving my problem in this form.

Best regards,
Maxim

2

I think that the problem might be that the number of terms in an analytical/symbolic solution of such a system increases dramatically fast.

Try an execute such a problem for small matrices and look at the rapid increase of the cpu time needed:

Table[(Solve[
     Table[Subscript[a, i, j], {i, 1, n}, {j, 1, n}].Table[Subscript[
        x, i], {i, 1, n}] == RandomReal[1, n], 
     Table[Subscript[x, i], {i, 1, n}]] // AbsoluteTiming)[[1]], {n,  2, 5}]

This gives:

{0.000691, 0.007968, 0.127200, 326.635776}

The corresponding figure is:

1993CPUTime.jpg
1993CPUTime.jpg360×187 4.04 KB

This is because of the number of permutations growing very fast for larger matrix sizes. The solution of linear equations with real entries is much, much faster:

  Table[(Solve[
     Table[RandomReal[], {i, 1, n}, {j, 1, n}].Table[Subscript[x, 
        i], {i, 1, n}] == RandomReal[1, n], 
     Table[Subscript[x, i], {i, 1, n}]] // AbsoluteTiming)[[1]], {n, 2, 30}]

This gives

{0.000247, 0.000217, 0.000246, 0.000277, 0.000378, 0.000461, \
0.000561, 0.000640, 0.000503, 0.000439, 0.000557, 0.000982, 0.000772, \
0.000691, 0.000738, 0.000811, 0.000891, 0.001443, 0.001112, 0.001240, \
0.001318, 0.001424, 0.001646, 0.001748, 0.002253, 0.002522, 0.002084, \
0.002231, 0.002249}

This is very fast so that even very large problems can be handled. Your problem involves the symbolic solution of a 25 times 25 matrix, which should be quite impossible.

M.

3

Thanks for your reply. It confirms what I already feared, so I guess I’ll need to convert my problem to a numerical one.