Multivariate Products of Legendre Polynomials

**LEGENDRE_PRODUCT_POLYNOMIAL**,
a C++ library which
defines a Legendre product polynomial (legendre_product_polynomial), creating a multivariate
polynomial as the product of univariate Legendre polynomials.

The Legendre polynomials are a polynomial sequence L(I,X), with polynomial I having degree I.

The first few Legendre polynomials are

0: 1 1: x 2: 3/2 x^2 - 1/2 3: 5/2 x^3 - 3/2 x 4: 35/8 x^4 - 30/8 x^2 + 3/8 5: 63/8 x^5 - 70/8 x^3 + 15/8 x

A Legendre 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 Legendre polynomials, with each the I-th polynomial evaluated at the I-th coordinate:

L((I1,I2,...IM),X) = L(1,X(1)) * L(2,X(2)) * ... * L(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 Legendre product polynomial from its index set, and it is easy to evaluate it directly, the sum of two Legendre product polynomials cannot be reduced to a single Legendre product polynomial. Thus, it may be useful to generate the Legendre 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.

- O is the number of terms in the polynomial.
- C() is a real vector of length O, containing the coefficients of each term.
- E() is an integer vector of length O, which defines the index (the exponents of X(1) through X(M)) of each term.

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 GNU LGPL license.

**LEGENDRE_PRODUCT_POLYNOMIAL** is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version and
a Python version.

COMBO, a C library 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_PRODUCT_POLYNOMIAL, a C++ library which defines Hermite product polynomials, creating a multivariate polynomial as the product of univariate Hermite polynomials.

LEGENDRE_POLYNOMIAL, a C library which evaluates the Legendre polynomial and associated functions.

legendre_product_polynomial_test

MONOMIAL, a C++ library 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++ library which evaluates a variety of mathematical functions, including Chebyshev, Gegenbauer, Hermite, Jacobi, Laguerre, Legendre polynomials, and the Collatz sequence.

POLYNOMIAL, a C++ library which adds, multiplies, differentiates, evaluates and prints multivariate polynomials in a space of M dimensions.

SUBSET, a C++ library which enumerates, generates, ranks and unranks combinatorial objects including combinations, compositions, Gray codes, index sets, partitions, permutations, subsets, and Young tables.

- legendre_product_polynomial.cpp, the source code.
- legendre_product_polynomial.sh, compiles the source code.
- legendre_product_polynomial.hpp, the include file.