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<html>
<head>
<title>
LAGRANGE_INTERP_ND - M-dimensional Lagrange Interpolant
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
LAGRANGE_INTERP_ND <br> M-dimensional Lagrange Interpolant
</h1>
<hr>
<p>
<b>LAGRANGE_INTERP_ND</b>
is a FORTRAN90 library which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of data depending on a M-dimensional argument
that was evaluated on a product grid,
so that p(x(i)) = z(i).
</p>
<p>
The interpolation function requires that the data points defining the interpolant
lie on a product grid [A1,B1]x[A2,B2]x...x[Am,Bm], to be defined
by a vector AB of dimension (M,2).
</p>
<p>
The interpolation function requires that the user supply a vector N_1D of length M,
which specifies the number or "order" of data points in each dimension. The number
of points in the product grid will then be the product of the entries
in N_1D.
</p>
<p>
(A second version of the interpolation function uses instead a vector IND of length M,
which is interpreted as a set of "levels". Each level corresponds in
a simple way to the number of "order" of data points. In particular,
levels 0, 1, 2, 3, 4 correspond to 1, 3, 5, 9 and 17 points. This
version is useful when a nested rule is desired.)
</p>
<p>
The interpolation function sets the location of the data points in each dimension
using the Clenshaw Curtis rule, that is, using the N extrema of
the Chebyshev polynomial of the first kind of order N-1. Those
polynomials are defined on [-1,+1], but a simple linear mapping
is used to adjust the points to the interval specified by the user.
</p>
<p>
The interpolation function needs data at the data points. It is assumed that this
will be supplied by a user specified function of the form
<pre>
v = f ( m, n, x )
</pre>
where M is the spatial dimension, N is the number of points to be
evaluated, X is a vector of dimension (M,N) containing the points,
and the result is the vector V of dimension (N) containing the function
values.
</p>
<p>
Typical usage involves several steps.
The size of the interpolant grid is determined by a call like:
<pre>
call lagrange_interp_nd_size ( m, ind, nd )
</pre>
Then the interpolant grid is determined by
<pre>
call lagrange_interp_nd_grid ( m, ind, ab, nd, xd )
</pre>
and the interpolant data is evaluated by
<pre>
call f ( m, nd, xd, zd )
</pre>
Once the interpolant has been defined, the user is free to evaluate
it repeatedly, by specifying NI points XI, and requesting the interpolated
values ZI by:
<pre>
call lagrange_interp_nd_value ( m, ind, ab, nd, zd, ni, xi, zi );
</pre>
</p>
<p>
<b>LAGRANGE_INTERP_ND</b> needs the R8LIB 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_ND</b> is available in
<a href = "../../c_src/lagrange_interp_nd/lagrange_interp_nd.html">a C version</a> and
<a href = "../../cpp_src/lagrange_interp_nd/lagrange_interp_nd.html">a C++ version</a> and
<a href = "../../f77_src/lagrange_interp_nd/lagrange_interp_nd.html">a FORTRAN77 version</a> and
<a href = "../../f_src/lagrange_interp_nd/lagrange_interp_nd.html">a FORTRAN90 version</a> and
<a href = "../../m_src/lagrange_interp_nd/lagrange_interp_nd.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/lagrange_interp_1d/lagrange_interp_1d.html">
LAGRANGE_INTERP_1D</a>,
a FORTRAN90 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 = "../../f_src/lagrange_interp_2d/lagrange_interp_2d.html">
LAGRANGE_INTERP_2D</a>,
a FORTRAN90 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>
<a href = "../../f_src/r8lib/r8lib.html">
R8LIB</a>,
a FORTRAN90 library which
contains many utility routines using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../f_src/rbf_interp_nd/rbf_interp_nd.html">
RBF_INTERP_ND</a>,
a FORTRAN90 library which
defines and evaluates radial basis function (RBF) interpolants to multidimensional data.
</p>
<p>
<a href = "../../f_src/shepard_interp_nd/shepard_interp_nd.html">
SHEPARD_INTERP_ND</a>,
a FORTRAN90 library which
defines and evaluates Shepard interpolants to multidimensional data,
based on inverse distance weighting.
</p>
<p>
<a href = "../../f_src/sparse_interp_nd/sparse_interp_nd.html">
SPARSE_INTERP_ND</a>
a FORTRAN90 library which
can be used to define a sparse interpolant to a function f(x) of a
multidimensional argument.
</p>
<p>
<a href = "../../m_src/spinterp/spinterp.html">
SPINTERP</a>,
a MATLAB library which
carries out piecewise multilinear hierarchical sparse grid interpolation;
an earlier version of this software is ACM TOMS Algorithm 847,
by Andreas Klimke;
</p>
<p>
<a href = "../../f_src/test_interp_nd/test_interp_nd.html">
TEST_INTERP_ND</a>,
a FORTRAN90 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>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lagrange_interp_nd.f90">lagrange_interp_nd.f90</a>, the source code.
</li>
<li>
<a href = "lagrange_interp_nd.sh">lagrange_interp_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_interp_nd_prb.f90">lagrange_interp_nd_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "lagrange_interp_nd_prb.sh">lagrange_interp_nd_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "lagrange_interp_nd_prb_output.txt">lagrange_interp_nd_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CC_COMPUTE_POINTS:</b> abscissas of a Clenshaw Curtis rule.
</li>
<li>
<b>LAGRANGE_BASIS_1D</b> evaluates the Lagrange basis polynomials.
</li>
<li>
<b>LAGRANGE_INTERP_ND_GRID</b> sets an M-dimensional Lagrange interpolant grid.
</li>
<li>
<b>LAGRANGE_INTERP_ND_GRID2</b> sets an M-dimensional Lagrange interpolant grid.
</li>
<li>
<b>LAGRANGE_INTERP_ND_SIZE</b> sizes an M-dimensional Lagrange interpolant.
</li>
<li>
<b>LAGRANGE_INTERP_ND_SIZE2</b> sizes an M-dimensional Lagrange interpolant.
</li>
<li>
<b>LAGRANGE_INTERP_ND_VALUE</b> evaluates an ND Lagrange interpolant.
</li>
<li>
<b>LAGRANGE_INTERP_ND_VALUE2</b> evaluates an ND Lagrange interpolant.
</li>
<li>
<b>ORDER_FROM_LEVEL_135</b> evaluates the 135 level-to-order relationship.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 28 September 2012.
</i>
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