hermite_product_polynomial, a C code which defines a Hermite product polynomial (HePP), creating a multivariate polynomial as the product of univariate Hermite polynomials.
The Hermite polynomials are a polynomial sequence He(i,x), with polynomial I having degree I.
The first few Hermite polynomials He(i,x) are
0: 1
1: x
2: x^2 - 1
3: x^3 - 3 x
4: x^4 - 6 x^2 + 3
5: x^5 - 10 x^3 + 15 x
A Hermite product polynomial may be defined in a space of M dimensions by choosing M indices. To evaluate the polynomial at a point X, compute the product of the corresponding Hermite polynomials, with each the I-th polynomial evaluated at the I-th coordinate:
He((I1,I2,...IM),X) = He(1,X(1)) * He(2,X(2)) * ... * He(M,X(M)).
Families of polynomials which are formed in this way can have useful properties for interpolation, derivable from the properties of the 1D family.
While it is useful to generate a Hermite product polynomial from its index set, and it is easy to evaluate it directly, the sum of two Hermite product polynomials cannot be reduced to a single Hermite product polynomial. Thus, it may be useful to generate the Hermite product polynomial from its indices, but then to convert it to a standard polynomial form.
The representation of arbitrary multivariate polynomials can be complicated. In this library, we have chosen a representation involving the spatial dimension M, and three pieces of data, O, C and E.
The exponent indexing is done in a natural way, suggested by the following indexing for the case M = 2:
1: x^0 y^0
2: x^0 y^1
3: x^1 y^0
4: x^0 y^2
5: x^1 y^1
6; x^2 y^0
7: x^0 y^3
8: x^1 y^2
9: x^2 y^1
10: x^3 y^0
...
The computer code and data files described and made available on this web page are distributed under the MIT license
hermite_product_polynomial is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
COMBO, a C code which includes routines for ranking, unranking, enumerating and randomly selecting balanced sequences, cycles, graphs, Gray codes, subsets, partitions, permutations, restricted growth functions, Pruefer codes and trees.
HERMITE_POLYNOMIAL, a C code which evaluates the Hermite polynomial and associated functions.
hermite_product_polynomial_test
LEGENDRE_PRODUCT_POLYNOMIAL, a C code which defines Legendre product polynomials, creating a multivariate polynomial as the product of univariate Legendre polynomials.
MONOMIAL, a C 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.
POLPAK, a C code which evaluates a variety of mathematical functions, including Chebyshev, Gegenbauer, Hermite, Jacobi, Laguerre, Legendre polynomials, and the Collatz sequence.
POLYNOMIAL, a C code which adds, multiplies, differentiates, evaluates and prints multivariate polynomials in a space of M dimensions.
SUBSET, a C code which enumerates, generates, ranks and unranks combinatorial objects including combinations, compositions, Gray codes, index sets, partitions, permutations, subsets, and Young tables.