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lagrange_nd.html
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<html>
<head>
<title>
LAGRANGE_ND - Multivariate Lagrange Interpolation
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
LAGRANGE_ND <br> Multivariate Lagrange Interpolation
</h1>
<hr>
<p>
<b>LAGRANGE_ND</b>
is a C++ library which
is given a set of ND points X(*) in D-dimensional space, and constructs
a family of ND Lagrange polynomials P(*)(X), associating polynomial P(i)
with point X(i), such that, for 1 <= i <= ND,
<pre>
P(i)(X(i)) = 1
</pre>
but, if i =/= j
<pre>
P(i)(X(j)) = 0
</pre>
</p>
<p>
The library currently includes the following primary routines:
<ul>
<li>
<b>LAGRANGE_COMPLETE</b> requires that the number of data points
ND is exactly <b>equal</b> to R, the number of monomials in D dimensions
of total degree N or less;
</li>
<li>
<b>LAGRANGE_COMPLETE2</b>, a version of LAGRANGE_COMPLETE with
improved "pivoting";
</li>
<li>
<b>LAGRANGE_PARTIAL</b> allows the number of data points
ND to be <b>less than or equal</b> to R, the number of monomials
in D dimensions of total degree N or less;
</li>
<li>
<b>LAGRANGE_PARTIAL2</b>, a version of LAGRANGE_PARTIAL
with improved "pivoting".
</li>
</ul>
</p>
<p>
The set of ND polynomials P(*)(X) are returned as a set of three arrays:
<ul>
<li>
<b>PO(i)</b> contains the order, the number of nonzero coefficients,
for polynomial i;
</li>
<li>
<b>PC(i,j)</b> contains the coefficient of the j-th term in
polynomial i;
</li>
<li>
<b>PE(i,j)</b> contains a code for the exponents of the monomial
associated with the j-th term in polynomial i.
</li>
</ul>
</p>
<p>
Each value of PE(i,j) is an exponent codes which can be converted
to a vector of exponents that define a monomial. For example,
if we are working in spatial dimension D=3, then if PE(i,j)=13,
the corresponding exponent vector is (0,2,1), so
this means that the j-th term in polynomial i is
<pre>
PC(i,j) * x^0 y^2 z^1
</pre>
An exponent code can be converted to an exponent vector by calling
mono_unrank_grlex().
</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>LAGRANGE_ND</b> is available in
<a href = "../../cpp_src/lagrange_nd/lagrange_nd.html">a C++ version</a> and
<a href = "../../f_src/lagrange_nd/lagrange_nd.html">a FORTRAN90 version</a> and
<a href = "../../m_src/lagrange_nd/lagrange_nd.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/lagrange_interp_nd/lagrange_interp_nd.html">
LAGRANGE_INTERP_ND</a>,
a C++ library which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of multivariate data, so that p(x(i)) = y(i).
</p>
<p>
<a href = "../../cpp_src/sparse_interp_nd/sparse_interp_nd.html">
SPARSE_INTERP_ND</a>
a C++ library which
can be used to define a sparse interpolant to a function f(x) of a
multidimensional argument.
</p>
<p>
<a href = "../../cpp_src/test_interp_nd/test_interp_nd.html">
TEST_INTERP_ND</a>,
a C++ library which
defines test problems for interpolation of data z(x),
depending on an M-dimensional argument.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Philip Davis,<br>
Interpolation and Approximation,<br>
Dover, 1975,<br>
ISBN: 0-486-62495-1,<br>
LC: QA221.D33
</li>
<li>
Tomas Sauer, Yuan Xu,<br>
On multivariate Lagrange interpolation,<br>
Mathematics of Computation,<br>
Volume 64, Number 211, July 1995, pages 1147-1170.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lagrange_nd.cpp">lagrange_nd.cpp</a>, the source code.
</li>
<li>
<a href = "lagrange_nd.hpp">lagrange_nd.hpp</a>, the include file.
</li>
<li>
<a href = "lagrange_nd.sh">lagrange_nd.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lagrange_nd_prb.cpp">lagrange_nd_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "lagrange_nd_prb.sh">lagrange_nd_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "lagrange_nd_prb_output.txt">lagrange_nd_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_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>I4MAT_PRINT</b> prints an I4MAT.
</li>
<li>
<b>I4MAT_PRINT_SOME</b> prints some of an I4MAT.
</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_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>LAGRANGE_COMPLETE:</b> Complete Lagrange polynomial basis from data.
</li>
<li>
<b>LAGRANGE_COMPLETE2:</b> Complete Lagrange polynomial basis from data.
</li>
<li>
<b>LAGRANGE_PARTIAL:</b> Partial Lagrange polynomial basis from data.
</li>
<li>
<b>LAGRANGE_PARTIAL2:</b> Partial Lagrange polynomial basis from data.
</li>
<li>
<b>MONO_BETWEEN_ENUM</b> enumerates monomials in D dimensions of degrees in a range.
</li>
<li>
<b>MONO_BETWEEN_NEXT_GRLEX:</b> grlex next monomial, degree between N1 and N2.
</li>
<li>
<b>MONO_NEXT_GRLEX</b> returns the next monomial in grlex order.
</li>
<li>
<b>MONO_TOTAL_ENUM</b> enumerates monomials in D dimensions of degree equal to N.
</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_AXPY</b> adds a multiple of one polynomial to another.
</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_IS_IDENTITY</b> determines if an R8MAT is the identity.
</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_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8VEC_CONCATENATE</b> concatenates two R8VEC's.
</li>
<li>
<b>R8VEC_PERMUTE</b> permutes an R8VEC in place.
</li>
<li>
<b>R8VEC_PRINT</b> prints an 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 25 January 2014.
</i>
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