@jan123 Is there any software that tackles the problem of multivariate recurrences? The complexity of the problem compared to univariate recurrences is akin to PDEs compared to ODEs.

Your differential order is two in both independent variables. So, you need another layer of boundary/initial conditions.

Suppose we try to compute M(2,2) from your definitions above. We get

M(2,2) = M(2,1) + M(1,1) + M(0,0) = 1 + 1 + ???

@testht06 The only coefficients in the field GF(2^8) are 0 and 1. Yet you show coefficients 27, 37, 217, 213, 30. Are you talking about the ring Z(2^8) = Z(256)? And I don't know what 256^256 = 487 means even if you are talking about rings.

You should ask this as a separate Question.

@testht06 Ah, you must be using an older version of Maple. I retrofitted the code in the Answer at the head of this subthread, and it should now work in current and older Maple. Please download the new code and let me know if it works (I don't have the older Maple to test it myself).

@In-Jee Jeong 

You need to post a worksheet showing exactly what commands you're using and showing the error message.

@testht06 

Like I showed above, your x needs to be a field extension represented as a root of an irreducible polynomial. This is done with alias and RootOf like this:

alias(x= RootOf(y^8+y^4+y^3+y+1)):

Do this before entering the matrix.

@testht06 

That would be a lot of work, if it is possible at all. The procedure GFM seems to me to be a crude and minimal implementation of matrix algebra over finite fields. The machinery provided by mod (see all the subcommands listed at ?mod) seems much more complete. In particular, my two procedures depend on the commands Roots and Linsolve in the mod package. Please try to convert your application to use mod.

@hooshmand 

Certainly the last command was intended to be implicitplot, not plots. But for the plot to work, ellips would need to be a function of x and y. However, ellips is defined on the second-to-last line and it doesn't depend on any variable.

Perhaps we could help you if you could describe in words what you're trying to do.

@testht06 

There are several ways in Maple to represent the field. The way that I used uses RootOf instead of GF.

alias(x= RootOf(y^8+y^4+y^3+y+1)):
M:= Matrix([[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,x,1,x^2]]):
eig:= Eigenvalues(M) mod 2;

Eigenvector(M, eig[1,1]) mod 2;

Eigenvector(M, eig[2,1]) mod 2;

Where does this code come from? It requires significant knowledge of Maple to write code this bad.

@tomleslie 

The fact that polynomials of degree greater than four over fields of characteristic zero are in general insoluble is irrelevant. All roots of any polynomial over any finite field can be computed, and there are efficient algorithms to do it. Indeed, the complete factorization into irreducibles of any polynomial over any finite field can be computed, and this is the basis for algorithms that factor polynomials over the rationals.

Newton's method works in modular arithmetic, and it converges to the exact answer.

@reinhardsiegfried 

The {x,y} is the second argument to coeffs. If map is called with three arguments, as in map(f, a, b), then b is used as the second argument to f. See ?coeffs and ?map.

@zia9206314 The values of the roots computed by NeztZero in Maple 17 and Maple 18 are identical down to the last digit. It is only the computation of the residuals that is different. This is probably because of a bug in Maple 17 that was fixed in Maple 18.

No, I can't send you Maple 18.

evaluation.mw

@ecterrab I didn't say that there were several bugs; I said that I'd seen several bug reports. All the reports may have been about the same bug.

It must've been Maple 18 where that bug was fixed. Executing the worksheet evaluation.mw (from the head of this subthread) in Maple 18 and Maple 17 shows huge differences in evaluation of a HeunB function such that the residuals from both root finders (NextZero and fsolve) make no sense in Maple 17. Since the OP is using Maple 17, I think that the Heun functions can be blamed.

@zia9206314 

I could not duplicate your results. I executed your worksheet in Maple 18, and all 14 roots found by NextZero have residuals less than 5*10^(-Digits). Try executing the worksheet again. And what version of Maple are your using?

That being said, I have seen several bug reports about numerical evaluation of Heun functions. So, if there is actually a discrepancy with the residuals, I'd blame that instead of blaming NextZero or fsolve.

By the way, guardDigits= 22 is excessive. At Digits=15, guardDigits= 4 is enough to get the first 14 roots.