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<html>
<head>
<title>
HERMITE_PRODUCT_POLYNOMIAL - Multivariate Products of Hermite Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
HERMITE_PRODUCT_POLYNOMIAL <br> Multivariate Products of Hermite Polynomials
</h1>
<hr>
<p>
<b>HERMITE_PRODUCT_POLYNOMIAL</b>,
a C++ library which
defines a Hermite product polynomial (HePP), creating a multivariate
polynomial as the product of univariate Hermite polynomials.
</p>
<p>
The Hermite polynomials are a polynomial sequence He(i,x),
with polynomial I having degree I.
</p>
<p>
The first few Hermite polynomials He(i,x) are
<pre>
0: 1
1: x
2: x^2 - 1
3: x^3 - 3 x
4: x^4 - 6 x^2 + 3
5: x^5 - 10 x^3 + 15 x
</pre>
</p>
<p>
A Hermite 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 Hermite polynomials, with
each the I-th polynomial evaluated at the I-th coordinate:
<pre>
He((I1,I2,...IM),X) = He(1,X(1)) * He(2,X(2)) * ... * He(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 Hermite product polynomial from
its index set, and it is easy to evaluate it directly, the sum of
two Hermite product polynomials cannot be reduced to a single
Hermite product polynomial. Thus, it may be useful to generate
the Hermite 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>HERMITE_PRODUCT_POLYNOMIAL</b> is available in
<a href = "../../c_src/hermite_product_polynomial/hermite_product_polynomial.html">a C version</a> and
<a href = "../../cpp_src/hermite_product_polynomial/hermite_product_polynomial.html">a C++ version</a> and
<a href = "../../f77_src/hermite_product_polynomial/hermite_product_polynomial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/hermite_product_polynomial/hermite_product_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hermite_product_polynomial/hermite_product_polynomial.html">a MATLAB 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_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a C++ library which
evaluates the Hermite polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/legendre_product_polynomial/legendre_product_polynomial.html">
LEGENDRE_PRODUCT_POLYNOMIAL</a>,
a C++ library which
defines Legendre product polynomials, creating a multivariate
polynomial as the product of univariate Legendre polynomials.
</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 = "hpp.cpp">hpp.cpp</a>, the source code.
</li>
<li>
<a href = "hpp.hpp">hpp.hpp</a>, the include file.
</li>
<li>
<a href = "hpp.sh">hpp.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "hpp_prb.cpp">hpp_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "hpp_prb.sh">hpp_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "hpp_prb_output.txt">hpp_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>HEP_COEFFICIENTS:</b> coefficients of He(n,x).
</li>
<li>
<b>HEP_VALUE</b> evaluates the Hermite polynomials He(n,x).
</li>
<li>
<b>HEP_VALUES:</b> tabulated values of He(i,x).
</li>
<li>
<b>HEPP_TO_POLYNOMIAL</b> writes a Hermite Product Polynomial as a polynomial.
</li>
<li>
<b>HEPP_VALUE</b> evaluates a Hermite Product Polynomial at several points X.
</li>
<li>
<b>I4_CHOOSE</b> computes the binomial coefficient C(N,K).
</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> returns the sum of the entries of an I4VEC.
</li>
<li>
<b>MONO_RANK_GRLEX</b> computes the graded lexicographic rank of a monomial.
</li>
<li>
<b>MONO_UNRANK_GRLEX</b> computes the monomial of given grlex rank.
</li>
<li>
<b>MONO_VALUE</b> evaluates a monomial.
</li>
<li>
<b>PERM_CHECK</b> checks that a vector represents a permutation.
</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>R8VEC_PERMUTE</b> permutes an R8VEC in place.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_AB</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 22 October 2014.
</i>
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