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<html>
<head>
<title>
TOMS655 - Weights for Interpolatory Quadrature
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TOMS655 <br> Weights for Interpolatory Quadrature
</h1>
<hr>
<p>
<b>TOMS655</b>
is a FORTRAN90 library which
computes weights for interpolatory
quadrature schemes.
</p>
<p>
The FORTRAN90 version is a "translation" of the original FORTRAN77 code.
Aside from some standard FORTRAN90 changes, a major change to the code was
the elimination of the work vectors WF and IWF, which have been replaced
by the use of allocatable arrays and automatic arrays. This frees the
user from having to declare and pass workspace arrays of the appropriate
size. It also means it is easier to translate the FORTRAN90 code into
MATLAB or C.
</p>
<p>
The typical use of this library is for the user to specify
a quadrature interval, a weight function, and a sequence of abscissas
(which may be repeated), and to request the corresponding weight
vector so that an interpolatory quadrature rule is produced.
</p>
<p>
Note that when an abscissa is repeated, this indicates that, at this
point, not only the function value but one or more derivatives are
to be used in the quadrature formula.
</p>
<p>
The library is also suitable for the simpler task of computing
both the abscissas and weights for a variety of classical Gaussian
quadrature rules, including
<table border="1">
<tr>
<th>Index</th><th>Name</th><th>Interval</th><th>Weight function</th>
</tr>
<tr>
<td>1</td><td>Legendre</td><td>(a,b)</td><td>1.0</td>
</tr>
<tr>
<td>2</td><td>Chebyshev Type 1</td><td>(a,b)</td><td>((b-x)*(x-a))^(-0.5)</td>
</tr>
<tr>
<td>3</td><td>Gegenbauer</td><td>(a,b)</td><td>((b-x)*(x-a))^alpha</td>
</tr>
<tr>
<td>4</td><td>Jacobi</td><td>(a,b)</td><td>(b-x)^alpha*(x-a)^beta</td>
</tr>
<tr>
<td>5</td><td>Laguerre and Generalized Laguerre</td><td>(a,+oo)</td><td>(x-a)^alpha*exp(-b*(x-a))</td>
</tr>
<tr>
<td>6</td><td>Hermite and Generalized Hermite</td><td>(-oo,+oo)</td><td>|x-a|^alpha*exp(-b*(x-a)^2)</td>
</tr>
<tr>
<td>7</td><td>Exponential</td><td>(a,b)</td><td>|x-(a+b)/2.0|^alpha</td>
</tr>
<tr>
<td>8</td><td>Rational</td><td>(a,+oo)</td><td>(x-a)^alpha*(x+b)^beta</td>
</tr>
<tr>
<td>9</td><td>Chebyshev Type 2</td><td>(a,b)</td><td>((b-x)*(x-a))^(+0.5)</td>
</tr>
</table>
</p>
<p>
The original, true, correct version of ACM TOMS Algorithm 655
is available through ACM:
<a href = "http://www.acm.org/pubs/calgo/">
http://www.acm.org/pubs/calgo</a>
or NETLIB:
<a href = "http://www.netlib.org/toms/index.html">
http://www.netlib.org/toms/index.html</a>.
</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>TOMS655</b> is available in
<a href = "../../c_src/toms655/toms655.html">a C version</a> and
<a href = "../../cpp_src/toms655/toms655.html">a C++ version</a> and
<a href = "../../f77_src/toms655/toms655.html">a FORTRAN77 version</a> and
<a href = "../../f_src/toms655/toms655.html">a FORTRAN90 version</a> and
<a href = "../../m_src/toms655/toms655.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/chebyshev1_rule/chebyshev1_rule.html">
CHEBYSHEV1_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Chebyshev type 1 quadrature rule.
</p>
<p>
<a href = "../../f_src/chebyshev2_rule/chebyshev2_rule.html">
CHEBYSHEV2_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Chebyshev type 2 quadrature rule.
</p>
<p>
<a href = "../../f_src/gegenbauer_rule/gegenbauer_rule.html">
GEGENBAUER_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Gegenbauer quadrature rule.
</p>
<p>
<a href = "../../f_src/gen_hermite_rule/gen_hermite_rule.html">
GEN_HERMITE_RULE</a>,
a FORTRAN90 program which
can compute and print a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../f_src/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a FORTRAN90 program which
can compute and print a generalized Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../f_src/hermite_rule/hermite_rule.html">
HERMITE_RULE</a>,
a FORTRAN90 program which
computes a Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../f_src/jacobi_rule/jacobi_rule.html">
JACOBI_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Jacobi quadrature rule.
</p>
<p>
<a href = "../../f_src/laguerre_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../f_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a FORTRAN90 program which
computes a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../f_src/quadrature_weights/quadrature_weights.html">
QUADRATURE_WEIGHTS</a>,
a FORTRAN90 library which
illustrates techniques for computing the weights of a quadrature
rule, assuming that the points have been specified.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
contains information about quadrature rules, both as tabulated values,
and as computational procedures.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Sylvan Elhay, Jaroslav Kautsky,<br>
Algorithm 655:
IQPACK,
FORTRAN Subroutines for the Weights of Interpolatory Quadrature,<br>
ACM Transactions on Mathematical Software,<br>
Volume 13, Number 4, December 1987, pages 399-415.
</li>
<li>
Jaroslav Kautsky, Sylvan Elhay,<br>
Calculation of the Weights of Interpolatory Quadratures,<br>
Numerische Mathematik,<br>
Volume 40, Number 3, October 1982, pages 407-422.
</li>
<li>
Roger Martin, James Wilkinson,<br>
The Implicit QL Algorithm,<br>
Numerische Mathematik,<br>
Volume 12, Number 5, December 1968, pages 377-383.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "toms655.f90">toms655.f90</a>,
the source code;
</li>
<li>
<a href = "toms655.sh">toms655.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>TOMS655_PRB</b> tests various routines in the package.
<ul>
<li>
<a href = "toms655_prb.f90">toms655_prb.f90</a>,
a sample calling program;
</li>
<li>
<a href = "toms655_prb.sh">toms655_prb.sh</a>,
commands to compile, link and run the test program;
</li>
<li>
<a href = "toms655_prb_output.txt">toms655_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "toms655_prb_check.txt">toms655_prb_check.txt</a>,
the expected output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CAWIQ</b> computes quadrature weights for a given set of knots.
</li>
<li>
<b>CDGQF</b> computes a Gauss quadrature formula with default A, B and simple knots.
</li>
<li>
<b>CEGQF</b> computes a quadrature formula and applies it to a function.
</li>
<li>
<b>CEGQFS</b> estimates an integral using a standard quadrature formula.
</li>
<li>
<b>CEIQF</b> constructs and applies a quadrature formula based on user knots.
</li>
<li>
<b>CEIQFS</b> computes and applies a quadrature formula based on user knots.
</li>
<li>
<b>CGQF</b> computes knots and weights of a Gauss quadrature formula.
</li>
<li>
<b>CGQFS</b> computes knots and weights of a Gauss quadrature formula.
</li>
<li>
<b>CHKQF</b> computes and prints the moments of a quadrature formula.
</li>
<li>
<b>CHKQFS</b> checks the polynomial accuracy of a quadrature formula.
</li>
<li>
<b>CIQF</b> computes weights for a classical weight function and any interval.
</li>
<li>
<b>CIQFS</b> computes some weights of a quadrature formula in the default interval.
</li>
<li>
<b>CLASS</b> computes the Jacobi matrix for a quadrature rule.
</li>
<li>
<b>CLIQF</b> computes a classical quadrature formula, with optional printing.
</li>
<li>
<b>CLIQFS</b> computes the weights of a quadrature formula in the default interval.
</li>
<li>
<b>CWIQD</b> computes all the weights for a given knot.
</li>
<li>
<b>EIQF</b> evaluates an interpolatory quadrature formula.
</li>
<li>
<b>EIQFS</b> evaluates a quadrature formula defined by CLIQF or CLIQFS.
</li>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>PARCHK</b> checks parameters ALPHA and BETA for classical weight functions.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>SCMM</b> computes moments of a classical weight function scaled to [A,B].
</li>
<li>
<b>SCQF</b> scales a quadrature formula to a nonstandard interval.
</li>
<li>
<b>SCT</b> rescales distinct knots to an interval [A,B].
</li>
<li>
<b>SGQF</b> computes knots and weights of a Gauss Quadrature formula.
</li>
<li>
<b>WM</b> evaluates the first M moments of classical weight functions.
</li>
<li>
<b>WTFN</b> evaluates the classical weight functions at given points.
</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 14 February 2010.
</i>
<!-- John Burkardt -->
</body>
</html>