monomial_symmetrize, an Octave 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.
By grouping the equivalent monomials, we have
P = C111 X1 X1 X1
+ C112 X1 X1 X2 + C121 X1 X2 X1 + C211 X2 X1 X1
+ C122 X1 X2 X2 + C212 X2 X1 X2 + C221 X2 X2 X1
+ C222 X2 X2 X2
and taking advantage of the equivalence of the monomials, we can write:
P = C111 X1 X1 X1
+ ( C112 + C121 + C211 ) / 3 * ( X1 X1 X2 + X1 X2 X1 + X2 X1 X1 )
+ ( C122 + C212 + C221 ) / 3 * ( X1 X2 X2 + X2 X1 X2 + X2 X2 X1 )
+ C222 X2 X2 X2
which means that our equivalent polynomial P2 can be written in the form
P2 = C111 * X1 X1 X1
+ ( C112 + C121 + C211 ) / 3 * X1 X1 X2
+ ( C112 + C121 + C211 ) / 3 * X1 X2 X1
+ ( C122 + C212 + C221 ) / 3 * X1 X2 X2
+ ( C112 + C121 + C211 ) / 3 * X2 X1 X1
+ ( C122 + C212 + C221 ) / 3 * X2 X1 X2
+ ( C122 + C212 + C221 ) / 3 * X2 X2 X1
+ C222 * X2 X2 X2
where, now, the equivalent monomials have equal coefficients.
The computer code and data files made available on this web page are distributed under the MIT license
monomial_symmetrize is available in a MATLAB version and an Octave version.
monomial, an Octave 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.
vector, an Octave code which considers a problem involving vectors, which can be considered to belong to equivalence classes, for which an arbitrary collection of coefficients must be gathered, averaged, and then scattered again.