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i have this problem, need to solve with conditions and cannot find yet how to describe in maxima (or maple)

This describe a finite state machine (n variable).

S is a matrix of equations.

S[i][j] = a * S[i][j-1] + b * S[i-1][j]
i-1<0 or j-1<0 -> S[i][j] = 0
i=n-1 and j=n-1 S[i][j] = 0

And the last step.

S[n-1][n-1] = a * S[n-1][n-2] + b * S[n-2][n-1] + b * Sa1 + a Sb1
Sa1 = a * S[n-1][n-1]
Sb1 = b * S[n-1][n-1]

Saf  = a (Sa1 + S[0][n-1]+ S[1][n-1] ... S[n-2][n-1])
Sbf  = b (Sb1 + S[n-1][0]+ S[n-1][1] ... S[n-1][n-2])

How can i write this in wxmaxima and solve? There are a specific concept?

Best regards

DAVID ALEJANDRO PINEDA
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  • Try rewriting the equations in terms of a transition matrix, let's call it `T`, such that `S[k] = T . S[k - 1]`. I think `T` is going to be mostly `0` with some `a` and some `b`. The final equation is going to be something like `S[n - 1] = T . S[n - 2] + something . S[n - 1]`. I think you'll find that `S[k] = T^k . S[k - 1]` and you can maybe substitute that into the final equation to get an equation relating `S[n - 1]` to `S[0]` which you can try to solve. – Robert Dodier Oct 21 '22 at 17:37
  • So far all that stuff is not specific to Maxima. My advice to try to work, in any system you end up using, with matrix algebra (i.e. variables which represent a whole vector or matrix) instead of element by element. – Robert Dodier Oct 21 '22 at 17:39

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