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Best way for manipulating multivariate polynomials matlab

I am looking for a good way to manipulate multivariate polynomials in Matlab. The purpose of this question is very global. Currently, I am manipulating some matrices of polynomials in MATLAB (with currently 2 variables). To simplify this manipulation I split each matrix into four new matrices:

  • One for defining the coefficient: C
  • One for defining an exponent: E (see below)
  • One for defining the power of the first variable X: MX
  • One for defining the poser of the second variable Y: MY

Thus you can evaluate polynomial matrix by this way C./h^E.*X.^MX.*Y.^MY. For some reason, sometimes each component of the polynomial matrix can be a sum of some monomials. In this case, I use some nD-arrays (and sum(.,3)).

For my work, I need also to defined the derivatives of the polynomial matrices with respect to Xor Y. Using the previous formulation, the derivatives can be easily obtained by subtracting 1 to the associated matrix MX or MY and by multiplying C by the right matrix MXor MY.

Currently this approach works fine for lower degrees but I need also to multiplying some polynomial matrices and this is the big problem of this approach. To deal with this problem I write manually the full matrix product (compute using Mathematica).

I want to extend my code for higher degrees and to manipulate more easily the polynomial matrices. So if you have any idea to do this.

I can use any toolbox in Matlab but at the end I need to have the matrices MX, MY, E and C (I need this separated matrices for doing some specific computations). I tried to use the Symbolic Toolbox but it seems to be very difficult to extract these four matrices when the polynomial matrix is complicated.

Example:

    H=[
    1 0 Y/h 10*Y^2/h^2 5X*Y/h^2 0
    0 1 -X/h X/h 50*X^2/h^2 60*X*Y/h^2
    ]

    C=[
    1 0 1 10 5 0
    0 1 -1 1 50 60
    ]

    E=[
    0 0 1 2 2 0
    0 1 1 1 2 2
    ]

    MX=[
    0 0 0 0 1 0
    0 0 1 1 2 1
    ]

    MY=[
    0 0 1 2 1 0
    0 0 0 0 0 1
    ]

Problem: Compute H*D' and extract C, E, MX and MY (with H define above) and

    D=[
    Y/h Y^2/h^2 X/h 
    X/h Y/h X*Y/h
    ]
Solution

  • Finally I have found one solution. Unfortunately (or not) I use the Symbolic Toolbox. In my case I consider 7 symbolic variables and the variables of the polynomial are X and Y. The matlab variable poly is a struct which include the field p. p is a symfun matrix of polynomial.

    For the output arguments, C, E, MX and MY are respectively included in polyOut.mult, polyOut.exph, polyOut.expoX and polyOut.expoY. 

      %% Build the polynomial matrices
  function polyOut=buildPolyMat(poly)
  syms X Y real
  syms c11 c12 c21 c22 c33 real
  P=poly.p(X,Y);
  sizP=size(P);
  polyOut.mult=sym(zeros(sizP));
  polyOut.expoX=zeros(sizP);
  polyOut.expoY=zeros(sizP);
  %for each term of the polynomial matrix
  for it=1:prod(sizP)
      %polynomial
      pT=P(it);
      %read coeff and exponents
      [C,M]=coeffs(pT,[X,Y]);
      if ~isempty(C)
          %subscripts in matrix
          [sC,sR]=ind2sub(sizP,it);
          %exponents
          for iE=1:numel(C)
              eX=feval(symengine,'degree',M(iE),X);
              eY=feval(symengine,'degree',M(iE),Y);
              polyOut.mult(sC,sR,iE)=C(iE);
              polyOut.expoX(sC,sR,iE)=eX;
              polyOut.expoY(sC,sR,iE)=eY;
          end
      end
  end
  polyOut.exph=poly.eH*ones(size(polyOut.mult));
  %clean matrices with zeros multiplier
  Ind=(polyOut.mult==0);
  polyOut.expoX(Ind)=0;
  polyOut.expoY(Ind)=0;
  polyOut.exph(Ind)=0;
  end