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<html>
<head>
<title>
CHEBYSHEV - Interpolation Using Chebyshev Polynomials
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CHEBYSHEV <br> Interpolation Using Chebyshev Polynomials
</h1>
<hr>
<p>
<b>CHEBYSHEV</b>
is a C++ library which
constructs the Chebyshev interpolant to a function.
</p>
<p>
Note that the user is not free to choose the interpolation points.
Instead, the function f(x) will be evaluated at points chosen by the
algorithm.
In the standard case, in which the interpolation interval is [-1,+1],
these points will be the zeros of the Chebyshev polynomial of order N.
However, the algorithm can also be applied to an interval of the form [a,b],
in which case the evaluation points are linearly mapped from [-1,+1].
</p>
<p>
The resulting interpolant is defined by a set of N coefficients c(),
and has the form:
<pre>
C(f)(x) = sum ( 1 <= i <= n ) c(i) T(i-1,x) - 0.5 * c(1)
</pre>
where T(i-1,x) is the (i-1)-th Chebyshev polynomial.
</p>
<p>
Within the interval [-1,+1], or the generalized interval [a,b], the
interpolant actually remains bounded by the sum of the absolute values
of the coefficients c(). It is therefore common to use Chebyshev
interpolants as approximating functions over a given interval.
</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>CHEBYSHEV</b> is available in
<a href = "../../c_src/chebyshev/chebyshev.html">a C version</a> and
<a href = "../../cpp_src/chebyshev/chebyshev.html">a C++ version</a> and
<a href = "../../f77_src/chebyshev/chebyshev.html">a FORTRAN77 version</a> and
<a href = "../../f_src/chebyshev/chebyshev.html">a FORTRAN90 version</a> and
<a href = "../../m_src/chebyshev/chebyshev.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/bernstein_polynomial/bernstein_polynomial.html">
BERNSTEIN_POLYNOMIAL</a>,
a C++ library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
</p>
<p>
<a href = "../../cpp_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a C++ library which
evaluates the Chebyshev polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/chebyshev_series/chebyshev_series.html">
CHEBYSHEV_SERIES</a>,
a C++ library which
can evaluate a Chebyshev series approximating a function f(x),
while efficiently computing one, two or three derivatives of the
series, which approximate f'(x), f''(x), and f'''(x).
</p>
<p>
<a href = "../../cpp_src/divdif/divdif.html">
DIVDIF</a>,
a C++ library which
computes interpolants by divided differences.
</p>
<p>
<a href = "../../cpp_src/hermite/hermite.html">
HERMITE</a>,
a C++ library which
computes the Hermite interpolant, a polynomial that matches function values
and derivatives.
</p>
<p>
<a href = "../../cpp_src/hermite_cubic/hermite_cubic.html">
HERMITE_CUBIC</a>,
a C++ library which
can compute the value, derivatives or integral of a Hermite cubic polynomial,
or manipulate an interpolating function made up of piecewise Hermite cubic
polynomials.
</p>
<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/rbf_interp/rbf_interp.html">
RBF_INTERP</a>,
a C++ library which
defines and evaluates radial basis function (RBF) interpolants to multidimensional data.
</p>
<p>
<a href = "../../cpp_src/spline/spline.html">
SPLINE</a>,
a C++ library which
includes many routines to construct
and evaluate spline interpolants and approximants.
</p>
<p>
<a href = "../../cpp_src/test_approx/test_approx.html">
TEST_APPROX</a>,
a C++ library which
defines test problems for approximation,
provided as a set of (x,y) data.
</p>
<p>
<a href = "../../cpp_src/test_interp_1d/test_interp_1d.html">
TEST_INTERP_1D</a>,
a C++ library which
defines test problems for interpolation of data y(x),
depending on a 1D argument.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Roger Broucke,<br>
Algorithm 446:
Ten Subroutines for the Manipulation of Chebyshev Series,<br>
Communications of the ACM,<br>
Volume 16, Number 4, April 1973, pages 254-256.
</li>
<li>
Philip Davis,<br>
Interpolation and Approximation,<br>
Dover, 1975,<br>
ISBN: 0-486-62495-1,<br>
LC: QA221.D33
</li>
<li>
William Press, Brian Flannery, Saul Teukolsky, William Vetterling,<br>
Numerical Recipes in C: The Art of Scientific Computing,<br>
Cambridge University Press, 1988,<br>
ISBN: 0-521-35465-X,<br>
LC: QA76.73.C15N865.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "chebyshev.cpp">chebyshev.cpp</a>, the source code.
</li>
<li>
<a href = "chebyshev.hpp">chebyshev.hpp</a>, the include file.
</li>
<li>
<a href = "chebyshev.sh">chebyshev.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "chebyshev_prb.cpp">chebyshev_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "chebyshev_prb.sh">chebyshev_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "chebyshev_prb_output.txt">chebyshev_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CHEBYSHEV_COEFFICIENTS</b> determines Chebyshev interpolation coefficients.
</li>
<li>
<b>CHEBYSHEV_INTERPOLANT</b> evaluates a Chebyshev interpolant.
</li>
<li>
<b>CHEBYSHEV_ZEROS</b> returns zeroes of the Chebyshev polynomial T(N)(X).
</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 14 September 2011.
</i>
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