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<html>
<head>
<title>
DIVDIF - Divided Difference Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
DIVDIF <br> Divided Difference Polynomials
</h1>
<hr>
<p>
<b>DIVDIF</b>
is a FORTRAN90 library which
creates, prints and manipulates divided difference polynomials
based on data tabulated at evenly spaced or unevenly spaced argument values.
</p>
<p>
Divided difference polynomials are a
systematic method of computing polynomial approximations to scattered
data. The representations are compact, and may easily be updated with
new data, rebased at zero, or analyzed to produce the standard form
polynomial, integral or derivative polynomials.
</p>
<p>
Other routines are available to convert the divided difference
representation to standard polynomial format. This is a natural
way to determine the coefficients of the polynomial that interpolates
a given set of data, for instance.
</p>
<p>
One surprisingly simple but useful routine is available to take
a set of roots and compute the divided difference or standard form
polynomial that passes through those roots.
</p>
<p>
Finally, the Newton-Cotes quadrature formulas can be derived using
divided difference methods, so a few routines are given which can
compute the weights and abscissas of open or closed rules for an
arbitrary number of nodes.
</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>DIVDIF</b> is available in
<a href = "../../c_src/divdif/divdif.html">a C version</a> and
<a href = "../../cpp_src/divdif/divdif.html">a C++ version</a> and
<a href = "../../f77_src/divdif/divdif.html">a FORTRAN77 version</a> and
<a href = "../../f_src/divdif/divdif.html">a FORTRAN90 version</a> and
<a href = "../../m_src/divdif/divdif.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/bernstein/bernstein.html">
BERNSTEIN</a>,
a FORTRAN90 library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
</p>
<p>
<a href = "../../f_src/chebyshev/chebyshev.html">
CHEBYSHEV</a>,
a FORTRAN90 library which
computes the Chebyshev interpolant/approximant to a given function
over an interval.
</p>
<p>
<a href = "../../f_src/hermite/hermite.html">
HERMITE</a>,
a FORTRAN90 library which
computes the Hermite interpolant, a polynomial that matches function values
and derivatives.
</p>
<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/pppack/pppack.html">
PPPACK</a>,
a FORTRAN90 library which
computes piecewise polynomial functions, including cubic splines.
</p>
<p>
<a href = "../../f_src/rbf_interp/rbf_interp.html">
RBF_INTERP</a>,
a FORTRAN90 library which
defines and evaluates radial basis interpolants to multidimensional data.
</p>
<p>
<a href = "../../f_src/spline/spline.html">
SPLINE</a>,
a FORTRAN90 library which
can construct and evaluate spline interpolants and approximants.
</p>
<p>
<a href = "../../f_src/test_approx/test_approx.html">
TEST_APPROX</a>,
a FORTRAN90 library which
defines test functions for approximation and interpolation.
</p>
<p>
<a href = "../../f_src/toms446/toms446.html">
TOMS446</a>,
a FORTRAN90 library which
manipulates Chebyshev series for interpolation and approximation;<br>
this is a version of ACM TOMS algorithm 446,
by Roger Broucke.
</p>
<p>
<a href = "../../f_src/vandermonde_interp_1d/vandermonde_interp_1d.html">
VANDERMONDE_INTERP_1D</a>,
a FORTRAN90 library which
finds a polynomial interpolant to data y(x) of a 1D 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>
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>
<li>
Jean-Paul Berrut, Lloyd Trefethen,<br>
Barycentric Lagrange Interpolation,<br>
SIAM Review,<br>
Volume 46, Number 3, September 2004, pages 501-517.
</li>
<li>
FM Larkin,<br>
Root Finding by Divided Differences,<br>
Numerische Mathematik,<br>
Volume 37, pages 93-104, 1981.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "divdif.f90">divdif.f90</a>, the source code;
</li>
<li>
<a href = "divdif.sh">divdif.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "divdif_prb.f90">divdif_prb.f90</a>, the calling program;
</li>
<li>
<a href = "divdif_prb.sh">divdif_prb.sh</a>, commands to compile,
link and run the calling program;
</li>
<li>
<a href = "divdif_prb_output.txt">divdif_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CHEBY_T_ZERO</b> returns zeroes of the Chebyshev polynomial T(N)(X).
</li>
<li>
<b>CHEBY_U_ZERO</b> returns zeroes of the Chebyshev polynomial U(N)(X).
</li>
<li>
<b>DATA_TO_DIF</b> sets up a divided difference table from raw data.
</li>
<li>
<b>DATA_TO_DIF_DISPLAY</b> computes a divided difference table and shows how.
</li>
<li>
<b>DATA_TO_R8POLY</b> computes the coefficients of a polynomial interpolating data.
</li>
<li>
<b>DIF_ANTIDERIV</b> computes the antiderivative of a divided difference polynomial.
</li>
<li>
<b>DIF_APPEND</b> adds a pair of data values to a divided difference table.
</li>
<li>
<b>DIF_BASIS</b> computes all Lagrange basis polynomials in divided difference form.
</li>
<li>
<b>DIF_BASIS_I:</b> I-th Lagrange basis polynomial in divided difference form.
</li>
<li>
<b>DIF_DERIV</b> computes the derivative of a polynomial in divided difference form.
</li>
<li>
<b>DIF_PRINT</b> prints the polynomial represented by a divided difference table.
</li>
<li>
<b>DIF_ROOT</b> seeks a zero of F(X) using divided difference techniques.
</li>
<li>
<b>DIF_SHIFT_X</b> replaces one abscissa of a divided difference table.
</li>
<li>
<b>DIF_SHIFT_ZERO</b> shifts a divided difference table so all abscissas are zero.
</li>
<li>
<b>DIF_TO_R8POLY</b> converts a divided difference table to a standard polynomial.
</li>
<li>
<b>DIF_VAL</b> evaluates a divided difference polynomial at a point.
</li>
<li>
<b>DIF_VALS</b> evaluates a divided difference polynomial at a set of points.
</li>
<li>
<b>LAGRANGE_RULE</b> computes the weights of a Lagrange interpolation rule.
</li>
<li>
<b>LAGRANGE_SUM</b> carries out a Lagrange interpolation rule.
</li>
<li>
<b>LAGRANGE_VAL</b> applies a naive form of Lagrange interpolation.
</li>
<li>
<b>NC_RULE</b> computes the weights of a Newton-Cotes quadrature rule.
</li>
<li>
<b>NCC_RULE</b> computes the coefficients of a Newton-Cotes closed quadrature rule.
</li>
<li>
<b>NCO_RULE</b> computes the coefficients of a Newton-Cotes open quadrature rule.
</li>
<li>
<b>R8_SWAP</b> swaps two R8's.
</li>
<li>
<b>R8POLY_ANT_COF</b> integrates a polynomial in standard form.
</li>
<li>
<b>R8POLY_ANT_VAL</b> evaluates the antiderivative of a polynomial in standard form.
</li>
<li>
<b>R8POLY_BASIS</b> computes all Lagrange basis polynomials in standard form.
</li>
<li>
<b>R8POLY_BASIS_1</b> computes the I-th Lagrange basis polynomial in standard form.
</li>
<li>
<b>R8POLY_DER_COF</b> computes the coefficients of the derivative of a polynomial.
</li>
<li>
<b>R8POLY_DER_VAL</b> evaluates the derivative of a polynomial in standard form.
</li>
<li>
<b>R8POLY_ORDER</b> returns the order of a polynomial.
</li>
<li>
<b>R8POLY_PRINT</b> prints out a polynomial.
</li>
<li>
<b>R8POLY_SHIFT</b> adjusts the coefficients of a polynomial for a new argument.
</li>
<li>
<b>R8POLY_VAL_HORNER</b> evaluates a polynomial in standard form.
</li>
<li>
<b>R8VEC_DISTINCT</b> is true if the entries in an R8VEC are distinct.
</li>
<li>
<b>R8VEC_EVEN</b> returns N values, evenly spaced between ALO and AHI.
</li>
<li>
<b>R8VEC_EVEN_SELECT</b> returns the I-th of N evenly spaced values in [ XLO, XHI ].
</li>
<li>
<b>R8VEC_INDICATOR</b> sets an R8VEC to the indicator vector A(I)=I.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>ROOTS_TO_DIF</b> sets a divided difference table for a polynomial from its roots.
</li>
<li>
<b>ROOTS_TO_R8POLY</b> converts polynomial roots to polynomial coefficients.
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 25 May 2011.
</i>
<!-- John Burkardt -->
</body>
</html>