LEGENDRE_PRODUCT_POLYNOMIAL
Multivariate Products of Legendre Polynomials

LEGENDRE_PRODUCT_POLYNOMIAL, a MATLAB library which can define a Legendre product polynomial (LPP), 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
       ...
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

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.

Related Data and Programs:

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

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

legendre_product_polynomial_test

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

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

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

Source Code:

• comp_enum.m, enumerates the compositions of an integer into K parts.
• comp_next_grlex.m, returns the next composition of an integer into K parts, using grlex order.
• comp_random_grlex.m, returns a random composition of an integer into K parts, with the integer between 0 and N.
• comp_rank_grlex.m, ranks a composition of an integer into K parts, using grlex order.
• comp_unrank_grlex.m, returns the composition of an integer into K parts of a given rank, using grlex order.
• i4_choose.m, computes the binomial coefficient C(N,K) as an I4.
• i4_uniform_ab.m, returns a random I4 in a given range.
• i4vec_print.m, prints an I4VEC.
• i4vec_uniform_ab.m, returns a random I4VEC in a given range.
• lp_coefficients.m, returns the coefficients of a Legendre polynomial
• lp_value.m, evaluates a Legendre polynomial at a point.
• lp_values.m, returns a table of sample values of Legendre polynomials.
• lpp_to_polynomial.m, converts a Legendre Product Polynomial to standard polynomial form.
• lpp_value.m, evaluates a Legendre Product Polynomial at a point.
• mono_next_grlex.m, returns the next monomial in D variables, in grlex order.
• mono_print.m, prints a monomial.
• mono_rank_grlex.m, returns the grlex rank of a monomial in the sequence of all monomials in D dimensions of degree N or less.
• mono_unrank_grlex.m, given the grlex rank, returns the corresponding monomial in the sequence of all monomials in M dimensions.
• mono_upto_enum.m, enumerates the monomials of D variables of total degree up to N.
• mono_upto_next_grlex.m, computes, in grlex order, the monomials of D variables of total degree up to N.
• mono_upto_random.m, randomly selects a monomial of D variables of total degree up to N.
• mono_value.m, evaluates a monomial.
• polynomial_compress.m, "compresses" a polynomial by merging coefficients associated with the same monomial.
• polynomial_print.m, prints a polynomial.
• polynomial_sort.m, sorts the terms in a polynomial.
• polynomial_value.m, evaluates a polynomial.
• r8vec_uniform_ab.m, returns a random real vector in [A,B].
• timestamp.m, prints the YMDHMS date as a timestamp.