LEGENDRE_PRODUCT_POLYNOMIAL, a FORTRAN90 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.
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 FORTRAN90 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 FORTRAN90 library which defines Hermite product polynomials, creating a multivariate polynomial as the product of univariate Hermite polynomials.
LEGENDRE_POLYNOMIAL, a FORTRAN90 library which evaluates the Legendre polynomial and associated functions.
MONOMIAL, a FORTRAN90 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 FORTRAN90 library which evaluates a variety of mathematical functions, including Chebyshev, Gegenbauer, Hermite, Jacobi, Laguerre, Legendre polynomials, and the Collatz sequence.
POLYNOMIAL, a FORTRAN90 library which adds, multiplies, differentiates, evaluates and prints multivariate polynomials in a space of M dimensions.
SUBSET, a FORTRAN90 library which enumerates, generates, ranks and unranks combinatorial objects including combinations, compositions, Gray codes, index sets, partitions, permutations, subsets, and Young tables.
You can go up one level to the FORTRAN90 source codes.