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

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Jean-Philippe Leonard Bossuat
评论： Jean-Philippe Leonard Bossuat 2021-1-1
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?

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采纳的回答

Karan Nandankar
Karan Nandankar 2020-12-30
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.

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