monomial_symmetrize

monomial_symmetrize, a MATLAB code which symmetrizes the coefficients of equivalent monomials in a polynomial.

The polynomial is assumed to be written as a sum of monomials.

Each monomial is of total degree D.

Each monomial involves factors selected from a B-dimensional space.

Each monomial has a coefficient C.

Every possible permutation of D factors from a B dimensional space is used to form a separate monomial.

For instance, if D = 3 and B = 2, the polynomial has the form

      P = C111 X1 X1 X1
        + C112 X1 X1 X2
        + C121 X1 X2 X1
        + C122 X1 X2 X2
        + C211 X2 X1 X1
        + C212 X2 X1 X2
        + C221 X2 X2 X1
        + C222 X2 X2 X2

There are 4 equivalence classes of monomials, with representatives X1 X1 X1, X1 X1 X2, X1 X2 X2 and X2 X2 X2. (Two monomial are equivalent if one is simply a permutation of the other.) It is the task of this function to take the values of D, B and the coefficient vector C, and produce a new coefficient vector C2. The C2 vector is computed by averaging the coefficients of equivalent monomials.

Licensing:

The computer code and data files made available on this web page are distributed under the MIT license

Languages:

monomial_symmetrize is available in a MATLAB version.

Related Data and Programs:

monomial_symmetrize_test

monomial, a MATLAB code which enumerates, lists, ranks, unranks and randomizes multivariate monomials in a space of M dimensions, with total degree less than N, equal to N, or lying within a given range.

Source Code:

• monomial_symmetrize.m, symmetrizes the coefficients of equivalent monomials.
• vector_equivalent_next.m, generates equivalent vectors one at a time.
• vector_rank.m, ranks a vector.
• vector_representative_next.m, generates vector representatives one at a time.