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<html>
<head>
<title>
HERMITE - Hermite polynomial interpolating function and derivative values
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
HERMITE <br> Hermite polynomial interpolating function and derivative values
</h1>
<hr>
<p>
<b>HERMITE</b>
is a C++ library which
constructs the Hermite polynomial which interpolates function
and derivative values at given points.
</p>
<p>
In other words, the user supplies <b>n</b> sets of data,
<b>(x(i),y(i),yp(i))</b>, and the algorithm determines a polynomial <b>p(x)</b>
such that, for 1 <= <b>i</b> <= <b>n</b>
<blockquote>
p(x(i)) = y(i) <br>
p'(x(i)) = yp(i)
</blockquote>
</p>
<p>
Note that <b>p(x)</b> is a "global" polynomial, not a piecewise polynomial.
Given <b>n</b> data points, <b>p(x)</b> will be a polynomial of degree 2<b>n</b>-1.
As the value <b>n</b> increases, the increasing degree of the interpolating
polynomial makes it liable to oscillations between the data, and eventually
to severe inaccuracy even at the data points.
</p>
<p>
Generally, the interpolation problem for a large number of data points should
be handled differently, for instance by piecewise polynomials.
</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>HERMITE</b> is available in
<a href = "../../c_src/hermite/hermite.html">a C version</a> and
<a href = "../../cpp_src/hermite/hermite.html">a C++ version</a> and
<a href = "../../f77_src/hermite/hermite.html">a FORTRAN77 version</a> and
<a href = "../../f_src/hermite/hermite.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hermite/hermite.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/chebyshev.html">
CHEBYSHEV</a>,
a C++ library which
computes the Chebyshev interpolant/approximant to a given function
over an interval.
</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_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/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>
Philip Davis,<br>
Interpolation and Approximation,<br>
Dover, 1975,<br>
ISBN: 0-486-62495-1,<br>
LC: QA221.D33
</li>
<li>
Carl deBoor,<br>
A Practical Guide to Splines,<br>
Springer, 2001,<br>
ISBN: 0387953663,<br>
LC: QA1.A647.v27.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "hermite.cpp">hermite.cpp</a>, the source code.
</li>
<li>
<a href = "hermite.hpp">hermite.hpp</a>, the include file.
</li>
<li>
<a href = "hermite.sh">hermite.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "hermite_prb.cpp">hermite_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "hermite_prb.sh">hermite_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "hermite_prb_output.txt">hermite_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>DIF_DERIV</b> computes the derivative of a polynomial in divided difference form.
</li>
<li>
<b>DIF_SHIFT_X</b> replaces one abscissa of a divided difference table with a new one.
</li>
<li>
<b>DIF_SHIFT_ZERO</b> shifts a divided difference table so that all abscissas are zero.
</li>
<li>
<b>DIF_TO_R8POLY</b> converts a divided difference table to a standard polynomial.
</li>
<li>
<b>DIF_VALS</b> evaluates a divided difference polynomial at a set of points.
</li>
<li>
<b>HERMITE_BASIS_0</b> evaluates a zero-order Hermite interpolation basis function.
</li>
<li>
<b>HERMITE_BASIS_1</b> evaluates a first-order Hermite interpolation basis function.
</li>
<li>
<b>HERMITE_DEMO</b> computes and prints Hermite interpolant information for data.
</li>
<li>
<b>HERMITE_INTERPOLANT</b> sets up a divided difference table from Hermite data.
</li>
<li>
<b>HERMITE_INTERPOLANT_RULE:</b> quadrature rule for a Hermite interpolant.
</li>
<li>
<b>HERMITE_INTERPOLANT_VALUE</b> evaluates the Hermite interpolant polynomial.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8POLY_ANT_VAL</b> evaluates the antiderivative of an R8POLY in standard form.
</li>
<li>
<b>R8POLY_DEGREE</b> returns the degree of a polynomial.
</li>
<li>
<b>R8POLY_PRINT</b> prints out a polynomial.
</li>
<li>
<b>R8VEC_CHEBYSHEV_NEW</b> creates a vector of Chebyshev spaced values.
</li>
<li>
<b>R8VEC_LINSPACE_NEW</b> creates a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_PRODUCT</b> returns the product of the entries of an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom 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 01 November 2011.
</i>
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