LEGENDRE_PRODUCT_POLYNOMIAL
Multivariate Products of Legendre Polynomials

LEGENDRE_PRODUCT_POLYNOMIAL, a C++ library which defines 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 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.

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.

Source Code:

• lpp.cpp, the source code.
• lpp.sh, BASH commands to compile the source code.

Examples and Tests:

• lpp_test.cpp a sample calling program.
• lpp_test.txt, the output file.

List of Routines:

• COMP_ENUM returns the number of compositions of the integer N into K parts.
• COMP_NEXT_GRLEX returns the next composition in grlex order.
• COMP_RANDOM_GRLEX: random composition with degree less than or equal to NC.
• COMP_RANK_GRLEX computes the graded lexicographic rank of a composition.
• COMP_UNRANK_GRLEX computes the composition of given grlex rank.
• I4_CHOOSE computes the binomial coefficient C(N,K).
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_UNIFORM_AB returns a scaled pseudorandom I4 between A and B.
• I4VEC_PERMUTE permutes an I4VEC in place.
• I4VEC_PRINT prints an I4VEC.
• I4VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an I4VEC.
• I4VEC_SUM sums the entries of an I4VEC.
• I4VEC_UNIFORM_AB_NEW returns a scaled pseudorandom I4VEC.
• LP_COEFFICIENTS: coefficients of Legendre polynomials P(n,x).
• LP_VALUE evaluates the Legendre polynomials P(n,x).
• LP_VALUES returns values of the Legendre polynomials P(n,x).
• LPP_TO_POLYNOMIAL writes a Legendre Product Polynomial as a polynomial.
• LPP_VALUE evaluates a Legendre Product Polynomial at several points X.
• MONO_NEXT_GRLEX returns the next monomial in grlex order.
• MONO_PRINT prints a monomial.
• MONO_RANK_GRLEX computes the graded lexicographic rank of a monomial.
• MONO_UNRANK_GRLEX computes the composition of given grlex rank.
• MONO_UPTO_ENUM enumerates monomials in M dimensions of degree up to N.
• MONO_UPTO_NEXT_GRLEX: grlex next monomial with total degree up to N.
• MONO_UPTO_RANDOM: random monomial with total degree less than or equal to N.
• MONO_VALUE evaluates a monomial.
• PERM_CHECK0 checks a 0-based permutation.
• PERM_UNIFORM_NEW selects a random permutation of N objects.
• POLYNOMIAL_COMPRESS compresses a polynomial.
• POLYNOMIAL_PRINT prints a polynomial.
• POLYNOMIAL_SORT sorts the information in a polynomial.
• POLYNOMIAL_VALUE evaluates a polynomial.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8MAT_UNIFORM_AB_NEW returns a new scaled pseudorandom R8MAT.
• R8VEC_PERMUTE permutes an R8VEC in place.
• R8VEC_PRINT prints an R8VEC
• R8VEC_UNIFORM_AB_NEW returns a scaled pseudorandom R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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