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<html>
<head>
<title>
LEGENDRE_PRODUCT_POLYNOMIAL - Multivariate Products of Legendre Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
LEGENDRE_PRODUCT_POLYNOMIAL <br> Multivariate Products of Legendre Polynomials
</h1>
<hr>
<p>
<b>LEGENDRE_PRODUCT_POLYNOMIAL</b>,
a C++ library which
defines a Legendre product polynomial (LPP), creating a multivariate
polynomial as the product of univariate Legendre polynomials.
</p>
<p>
The Legendre polynomials are a polynomial sequence L(I,X),
with polynomial I having degree I.
</p>
<p>
The first few Legendre polynomials are
<pre>
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
</pre>
</p>
<p>
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:
<pre>
L((I1,I2,...IM),X) = L(1,X(1)) * L(2,X(2)) * ... * L(M,X(M)).
</pre>
</p>
<p>
Families of polynomials which are formed in this way can have useful
properties for interpolation, derivable from the properties of the
1D family.
</p>
<p>
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.
</p>
<p>
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.
<ul>
<li>
O is the number of terms in the polynomial.
</li>
<li>
C() is a real vector of length O, containing the coefficients of each term.
</li>
<li>
E() is an integer vector of length O, which defines the index (the
exponents of X(1) through X(M)) of each term.
</li>
</ul>
</p>
<p>
The exponent indexing is done in a natural way, suggested by the
following indexing for the case M = 2:
<pre>
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
...
</pre>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>LEGENDRE_PRODUCT_POLYNOMIAL</b> is available in
<a href = "../../c_src/legendre_product_polynomial/legendre_product_polynomial.html">a C version</a> and
<a href = "../../cpp_src/legendre_product_polynomial/legendre_product_polynomial.html">a C++ version</a> and
<a href = "../../f77_src/legendre_product_polynomial/legendre_product_polynomial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/legendre_product_polynomial/legendre_product_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/legendre_product_polynomial/legendre_product_polynomial.html">a MATLAB version</a> and
<a href = "../../py_src/legendre_product_polynomial/legendre_product_polynomial.html">a Python version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/combo/combo.html">
COMBO</a>,
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.
</p>
<p>
<a href = "../../cpp_src/hermite_product_polynomial/hermite_product_polynomial.html">
HERMITE_PRODUCT_POLYNOMIAL</a>,
a C++ library which
defines Hermite product polynomials, creating a multivariate
polynomial as the product of univariate Hermite polynomials.
</p>
<p>
<a href = "../../cpp_src/legendre_polynomial/legendre_polynomial.html">
LEGENDRE_POLYNOMIAL</a>,
a C library which
evaluates the Legendre polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/monomial/monomial.html">
MONOMIAL</a>,
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.
</p>
<p>
<a href = "../../cpp_src/polpak/polpak.html">
POLPAK</a>,
a C++ library which
evaluates a variety of mathematical functions, including
Chebyshev, Gegenbauer, Hermite, Jacobi, Laguerre, Legendre polynomials,
and the Collatz sequence.
</p>
<p>
<a href = "../../cpp_src/polynomial/polynomial.html">
POLYNOMIAL</a>,
a C++ library which
adds, multiplies, differentiates, evaluates and prints multivariate
polynomials in a space of M dimensions.
</p>
<p>
<a href = "../../cpp_src/subset/subset.html">
SUBSET</a>,
a C++ library which
enumerates, generates, ranks and unranks combinatorial objects
including combinations, compositions, Gray codes, index sets, partitions,
permutations, subsets, and Young tables.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lpp.cpp">lpp.cpp</a>, the source code.
</li>
<li>
<a href = "lpp.hpp">lpp.hpp</a>, the include file.
</li>
<li>
<a href = "lpp.sh">lpp.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lpp_prb.cpp">lpp_prb.cpp</a>
a sample calling program.
</li>
<li>
<a href = "lpp_prb.sh">lpp_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "lpp_prb_output.txt">lpp_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>COMP_ENUM</b> returns the number of compositions of the integer N into K parts.
</li>
<li>
<b>COMP_NEXT_GRLEX</b> returns the next composition in grlex order.
</li>
<li>
<b>COMP_RANDOM_GRLEX:</b> random composition with degree less than or equal to NC.
</li>
<li>
<b>COMP_RANK_GRLEX</b> computes the graded lexicographic rank of a composition.
</li>
<li>
<b>COMP_UNRANK_GRLEX</b> computes the composition of given grlex rank.
</li>
<li>
<b>I4_CHOOSE</b> computes the binomial coefficient C(N,K).
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_UNIFORM_AB</b> returns a scaled pseudorandom I4 between A and B.
</li>
<li>
<b>I4VEC_PERMUTE</b> permutes an I4VEC in place.
</li>
<li>
<b>I4VEC_PRINT</b> prints an I4VEC.
</li>
<li>
<b>I4VEC_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an I4VEC.
</li>
<li>
<b>I4VEC_SUM</b> sums the entries of an I4VEC.
</li>
<li>
<b>I4VEC_UNIFORM_AB_NEW</b> returns a scaled pseudorandom I4VEC.
</li>
<li>
<b>LP_COEFFICIENTS:</b> coefficients of Legendre polynomials P(n,x).
</li>
<li>
<b>LP_VALUE</b> evaluates the Legendre polynomials P(n,x).
</li>
<li>
<b>LP_VALUES</b> returns values of the Legendre polynomials P(n,x).
</li>
<li>
<b>LPP_TO_POLYNOMIAL</b> writes a Legendre Product Polynomial as a polynomial.
</li>
<li>
<b>LPP_VALUE</b> evaluates a Legendre Product Polynomial at several points X.
</li>
<li>
<b>MONO_NEXT_GRLEX</b> returns the next monomial in grlex order.
</li>
<li>
<b>MONO_PRINT</b> prints a monomial.
</li>
<li>
<b>MONO_RANK_GRLEX</b> computes the graded lexicographic rank of a monomial.
</li>
<li>
<b>MONO_UNRANK_GRLEX</b> computes the composition of given grlex rank.
</li>
<li>
<b>MONO_UPTO_ENUM</b> enumerates monomials in M dimensions of degree up to N.
</li>
<li>
<b>MONO_UPTO_NEXT_GRLEX:</b> grlex next monomial with total degree up to N.
</li>
<li>
<b>MONO_UPTO_RANDOM:</b> random monomial with total degree less than or equal to N.
</li>
<li>
<b>MONO_VALUE</b> evaluates a monomial.
</li>
<li>
<b>PERM_CHECK0</b> checks a 0-based permutation.
</li>
<li>
<b>PERM_UNIFORM_NEW</b> selects a random permutation of N objects.
</li>
<li>
<b>POLYNOMIAL_COMPRESS</b> compresses a polynomial.
</li>
<li>
<b>POLYNOMIAL_PRINT</b> prints a polynomial.
</li>
<li>
<b>POLYNOMIAL_SORT</b> sorts the information in a polynomial.
</li>
<li>
<b>POLYNOMIAL_VALUE</b> evaluates a polynomial.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8MAT_UNIFORM_AB_NEW</b> returns a new scaled pseudorandom R8MAT.
</li>
<li>
<b>R8VEC_PERMUTE</b> permutes an R8VEC in place.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC
</li>
<li>
<b>R8VEC_UNIFORM_AB_NEW</b> returns a scaled pseudorandom R8VEC.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 06 November 2014.
</i>
<!-- John Burkardt -->
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