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<html>
<head>
<title>
LAGRANGE_INTERP_2D - Polynomial Interpolation in 2D using Lagrange Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
LAGRANGE_INTERP_2D <br> Polynomial Interpolation in 2D using Lagrange Polynomials
</h1>
<hr>
<p>
<b>LAGRANGE_INTERP_2D</b>
is a C++ library which
defines and evaluates the Lagrange polynomial p(x,y)
which interpolates a set of data depending on a 2D argument
that was evaluated on a product grid,
so that p(x(i),y(j)) = z(i,j).
</p>
<p>
If the data is available on a product grid, then both the LAGRANGE_INTERP_2D
and VANDERMONDE_INTERP_2D libraries will be trying to compute the same
interpolating function. However, especially for higher degree polynomials,
the Lagrange approach will be superior because it avoids the badly conditioned
Vandermonde matrix associated with the usage of monomials as the basis.
The Lagrange approach uses as a basis a set of Lagrange basis polynomials
l(i,j)(x) which are 1 at node (x(i),y(j)) and zero at the other nodes.
</p>
<p>
<b>LAGRANGE_INTERP_2D</b> needs access to the R8LIB library. The test
also needs the TEST_INTERP_2D library.
</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_INTERP_2D</b> is available in
<a href = "../../c_src/lagrange_interp_2d/lagrange_interp_2d.html">a C version</a> and
<a href = "../../cpp_src/lagrange_interp_2d/lagrange_interp_2d.html">a C++ version</a> and
<a href = "../../f77_src/lagrange_interp_2d/lagrange_interp_2d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/lagrange_interp_2d/lagrange_interp_2d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/lagrange_interp_2d/lagrange_interp_2d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/lagrange_interp_1d/lagrange_interp_1d.html">
LAGRANGE_INTERP_1D</a>,
a C++ library which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of data, so that p(x(i)) = y(i).
</p>
<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 data depending on a multidimensional argument x
that was evaluated on a product grid, so that p(x(i)) = z(i).
</p>
<p>
<a href = "../../cpp_src/padua/padua.html">
PADUA</a>,
a C++ library which
returns the points and weights for Padu sets, useful for interpolation
in 2D. GNUPLOT is used to plot the points.
</p>
<p>
<a href = "../../cpp_src/pwl_interp_2d/pwl_interp_2d.html">
PWL_INTERP_2D</a>,
a C++ library which
evaluates a piecewise linear interpolant to data defined on
a regular 2D grid.
</p>
<p>
<a href = "../../cpp_src/r8lib/r8lib.html">
R8LIB</a>,
a C++ library which
contains many utility routines using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../cpp_src/rbf_interp_2d/rbf_interp_2d.html">
RBF_INTERP_2D</a>,
a C++ library which
defines and evaluates radial basis function (RBF) interpolants to 2D data.
</p>
<p>
<a href = "../../cpp_src/shepard_interp_2d/shepard_interp_2d.html">
SHEPARD_INTERP_2D</a>,
a C++ library which
defines and evaluates Shepard interpolants to 2D data,
based on inverse distance weighting.
</p>
<p>
<a href = "../../cpp_src/test_interp_2d/test_interp_2d.html">
TEST_INTERP_2D</a>,
a C++ library which
defines test problems for interpolation of data z(x,y)),
depending on a 2D argument.
</p>
<p>
<a href = "../../cpp_src/toms886/toms886.html">
TOMS886</a>,
a C++ library which
defines the Padua points for interpolation in a 2D region,
including the rectangle, triangle, and ellipse,
by Marco Caliari, Stefano de Marchi, Marco Vianello.
This is a C++ version of ACM TOMS algorithm 886.
</p>
<p>
<a href = "../../cpp_src/vandermonde_interp_2d/vandermonde_interp_2d.html">
VANDERMONDE_INTERP_2D</a>,
a C++ library which
finds a polynomial interpolant to data z(x,y) of a 2D argument
by setting up and solving a linear system for the polynomial coefficients,
involving the Vandermonde matrix.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Kendall Atkinson,<br>
An Introduction to Numerical Analysis,<br>
Prentice Hall, 1989,<br>
ISBN: 0471624896,<br>
LC: QA297.A94.1989.
</li>
<li>
Philip Davis,<br>
Interpolation and Approximation,<br>
Dover, 1975,<br>
ISBN: 0-486-62495-1,<br>
LC: QA221.D33
</li>
<li>
David Kahaner, Cleve Moler, Steven Nash,<br>
Numerical Methods and Software,<br>
Prentice Hall, 1989,<br>
ISBN: 0-13-627258-4,<br>
LC: TA345.K34.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lagrange_interp_2d.cpp">lagrange_interp_2d.cpp</a>, the source code.
</li>
<li>
<a href = "lagrange_interp_2d.hpp">lagrange_interp_2d.hpp</a>, the include file.
</li>
<li>
<a href = "lagrange_interp_2d.sh">lagrange_interp_2d.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_interp_2d_prb.cpp">lagrange_interp_2d_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "lagrange_interp_2d_prb.sh">lagrange_interp_2d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "lagrange_interp_2d_prb_output.txt">lagrange_interp_2d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>LAGRANGE_POLY_1D</b> evaluates a 1D Lagrange basis function.
</li>
<li>
<b>LAGRANGE_VALUE_2D</b> evaluates the Lagrange interpolant for a product grid.
</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 13 September 2012.
</i>
<!-- John Burkardt -->
</body>
</html>