vpasolve returns emptyset for all variables, although a solution does exist (proof inside)

I have two sparse NxN matrices A and B, and I want to solve A*B = C for a specific dense NxN matrix C.

A small scale example of this problem (for N = 4) is :

vars = [x0 x1 x2 x3 x4 x5 x6 x7 y0 y1 y2 y3 y4 y5 y6 y7]

A = [[y0 y1  0  0]
     [y2 y3  0  0]
     [ 0  0 y4 y5]
     [ 0  0 y6 y7]]

B = [[x0  0 x1  0]
     [ 0 x2  0 x3]
     [x4  0 x5  0]
     [ 0 x6  0 x7]]

C = [[1.0, 1.0, 1.0, 1.0],
    [0.707106781186548, -0.7071067811865477, 0.707106781186548, -0.7071067811865477],
    [0.9238795325112871, -0.3826834323650893, -0.9238795325112868, 0.38268343236508967],
    [0.38268343236509117, 0.9238795325112865, -0.38268343236509067, -0.9238795325112865]]

when calling

sol = vpasolve(A*B == C, vars)

I get the empty-set [0*1 sym] for all variables.

However, an exact (up to the precision of the given matrix C) solution does exist because we can remplace A and B by

A = [[                1,                   1,                  0,                   0]
    [0.707106781186548,  -0.707106781186548,                  0,                   0]
    [                0,                   0,                  1,                   1]
    [                0,                   0, 0.4142135623730962, -2.4142135623730976]]

B = [[  1,   0,   1,  0]
    [  0,   1,   0,  1]
    [0.9238795325112871,   0,  -0.9238795325112868,  0]
    [  0, -0.3826834323650893,   0, 0.3826834323650893]]

And you can check by yourself that A*B = C.

I must have done something wrong, but I can't find out what and where. Am I using the correct solver?

Answer 1

[Karan Nandankar]

Hi,

The reason why 'vpasolve' returns empty [0x1] structure is that it is not able to find any analytical solution for the system of symbolic equations.

>> A*B == C

[                                 x0*y0 == 1,                                  x2*y1 == 1,                                   x1*y0 == 1,                                  x3*y1 == 1]
[                         x0*y2 == 2^(1/2)/2,                         x2*y3 == -2^(1/2)/2,                           x1*y2 == 2^(1/2)/2,                         x3*y3 == -2^(1/2)/2]
[ x4*y4 == 8321567036706121/9007199254740992, x6*y5 == -3446905926800557/9007199254740992,  x5*y4 == -8321567036706119/9007199254740992, x7*y5 == 6893811853601121/18014398509481984]
[ x4*y6 == 1723452963400287/4503599627370496,  x6*y7 == 2080391759176529/2251799813685248, x5*y6 == -6893811853601139/18014398509481984, x7*y7 == -2080391759176529/2251799813685248]

As you can see, there are 16 equations with as many variables, but there are multiple instances when solving couple of equations together leads to an already existing one, which falls under a case of Infinite Solutions, and the one which you shared earlier belongs to one of them.

You can follow this link for a workaround.

[Jean-Philippe Leonard Bossuat]

Thank you for the answer. Its a shame that vpasolve can't produce parameterized answers. I'll try to find a workaround.