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<html>
<head>
<title>
POLYNOMIAL - Multivariate Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
POLYNOMIAL <br> Multivariate Polynomials
</h1>
<hr>
<p>
<b>POLYNOMIAL</b>,
a C++ library which
adds, multiplies, differentiates, evaluates and prints multivariate
polynomials in a space of M dimensions.
</p>
<p>
Any polynomial in M variables can be written as a linear
combination of monomials in M variables. The "total degree" of the
polynomial is the maximum of the degrees of the monomials that it
comprises. For instance, a polynomial in M = 2 variables of total
degree 3 might have the form:
<pre>
p(x,y) = c(0,0) x^0 y^0
+ c(1,0) x^1 y^0 + c(0,1) x^0 y^1
+ c(2,0) x^2 y^0 + c(1,1) x^1 y^1 + c(0,2) x^0 y^2
+ c(3,0) x^3 y^0 + c(2,1) x^2 y^1 + c(1,2) x^1 y^2 + c(0,3) x^0 y^3
</pre>
The monomials in M variables can be regarded as
a natural basis for the polynomials in M variables.
</p>
<p>
When listing the monomials that form a polynomial, it is useful to have
an ordering that organizes them. One natural ordering can be illustrated
for the 3-dimensional case:
<pre>
1
x, y, z
x^2, xy, xz, y^2, yz, z^2
x^3, x^2y, x^2z, xy^2, xyz, xz^2, y^3, y^2z, yz^2, z^3
x^4, x^3y, ...
</pre>
Here, a monomial precedes another if it has a lower degree.
If the monomials have the same degree, then a monomial precedes another
if its exponent vector lexically precedes that of the other. In other
words, we compare the two exponent vectors, reading from left to right,
looking for the first location where the exponents differ. The monomial
with the higher exponent at this point precedes the other.
</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>POLYNOMIAL</b> is available in
<a href = "../../c_src/polynomial/polynomial.html">a C version</a> and
<a href = "../../cpp_src/polynomial/polynomial.html">a C++ version</a> and
<a href = "../../f77_src/polynomial/polynomial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/polynomial/polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/polynomial/polynomial.html">a MATLAB version</a> and
<a href = "../../py_src/polynomial/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_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/set_theory/set_theory.html">
SET_THEORY</a>,
a C++ library which
demonstrates MATLAB commands that implement various
set theoretic operations.
</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 = "polynomial.cpp">polynomial.cpp</a>, the source code.
</li>
<li>
<a href = "polynomial.hpp">polynomial.hpp</a>, the source code.
</li>
<li>
<a href = "polynomial.sh">polynomial.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "polynomial_prb.cpp">polynomial_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "polynomial_prb.sh">polynomial_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "polynomial_prb_output.txt">polynomial_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>I4_CHOOSE</b> computes the binomial coefficient C(N,K).
</li>
<li>
<b>I4_FALL</b> computes the falling factorial function [X]_N.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</li>
<li>
<b>I4VEC_CONCATENATE</b> concatenates two I4VEC's.
</li>
<li>
<b>I4VEC_PERMUTE</b> permutes an I4VEC in place.
</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>MONO_NEXT_GRLEX</b> returns the next monomial in grlex order.
</li>
<li>
<b>MONO_RANK_GRLEX</b> computes the graded lexicographic rank of a monomial.
</li>
<li>
<b>MONO_TOTAL_NEXT_GRLEX:</b> grlex next monomial with total degree equal to N.
</li>
<li>
<b>MONO_UNRANK_GRLEX</b> computes the composition of given grlex rank.
</li>
<li>
<b>MONO_UPTO_ENUM</b> enumerates monomials in D dimensions of degree up to N.
</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_ADD</b> adds two polynomials.
</li>
<li>
<b>POLYNOMIAL_AXPY</b> adds a multiple of one polynomial to another.
</li>
<li>
<b>POLYNOMIAL_COMPRESS</b> compresses a polynomial.
</li>
<li>
<b>POLYNOMIAL_DIF</b> differentiates a polynomial.
</li>
<li>
<b>POLYNOMIAL_MUL</b> multiplies two polynomials.
</li>
<li>
<b>POLYNOMIAL_PRINT</b> prints a polynomial.
</li>
<li>
<b>POLYNOMIAL_SCALE</b> scales 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_CONCATENATE</b> concatenates two R8VEC's.
</li>
<li>
<b>R8VEC_PERMUTE</b> permutes an R8VEC in place.
</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 modified on 01 January 2014.
</i>
<!-- John Burkardt -->
</body>
</html>