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<html>
<head>
<title>
KRONROD - Gauss-Kronrod Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
KRONROD <br> Gauss-Kronrod Quadrature Rules
</h1>
<hr>
<p>
<b>KRONROD</b>
is a FORTRAN90 library which
computes both a Gauss quadrature rule of order N, and the
Gauss-Kronrod rule of order 2*N+1.
</p>
<p>
A pair of Gauss and Gauss-Kronrod quadrature rules are typically
used to provide an error estimate for an integral. The integral
is estimated using the Gauss rule, and then the Gauss-Kronrod
rule provides a higher precision estimate. The difference between
the two estimates is taken as an approximation to the level
of error.
</p>
<p>
The advantage of using a Gauss and Gauss-Kronrod pair is that
the second rule, which uses 2*N+1 points, actually includes the
N points in the previous Gauss rule. This means that the function
values from that computation can be reused. This efficiency comes
at the cost of a mild reduction in the degree of polynomial precision of the
Gauss-Kronrod rule.
</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>KRONROD</b> is available in
<a href = "../../c_src/kronrod/kronrod.html">a C version</a> and
<a href = "../../cpp_src/kronrod/kronrod.html">a C++ version</a> and
<a href = "../../f77_src/kronrod/kronrod.html">a FORTRAN77 version</a> and
<a href = "../../f_src/kronrod/kronrod.html">a FORTRAN90 version</a> and
<a href = "../../m_src/kronrod/kronrod.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a FORTRAN90 program which
checks the polynomial exactness
of a 1-dimensional quadrature rule for a finite interval.
</p>
<p>
<a href = "../../f_src/patterson_rule/patterson_rule.html">
PATTERSON_RULE</a>,
a FORTRAN90 program which
computes a Gauss-Patterson quadrature rule.
</p>
<p>
<a href = "../../f_src/quadpack/quadpack.html">
QUADPACK</a>,
a FORTRAN90 library which
contains a variety of routines for
numerical estimation of integrals in 1D.
</p>
<p>
<a href = "../../datasets/quadrature_rules_patterson/quadrature_rules_patterson.html">
QUADRATURE_RULES_PATTERSON</a>,
a dataset directory which
contains Gauss-Patterson quadrature rules for the interval [-1,+1].
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules for 1D domains.
</p>
<p>
<a href = "../../f_src/test_int/test_int.html">
TEST_INT</a>,
a FORTRAN90 library which
contains a number of functions that may be used as test integrands for
quadrature rules in 1D.
</p>
<p>
<a href = "../../f_src/toms672/toms672.html">
TOMS672</a>,
a FORTRAN90 library which
generates an interpolatory quadrature rule of highest possible order,
given a set of preassigned abscissas;<br>
this library is commonly called <b>EXTEND</b>;<br>
this is ACM TOMS algorithm 672.
</p>
<p>
<a href = "../../f77_src/toms699/toms699.html">
TOMS699</a>,
a FORTRAN77 library which
implements a new representation of Patterson's quadrature formula;<br>
this is ACM TOMS algorithm 699.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Robert Piessens, Maria Branders,<br>
A Note on the Optimal Addition of Abscissas to Quadrature Formulas
of Gauss and Lobatto,<br>
Mathematics of Computation,<br>
Volume 28, Number 125, January 1974, pages 135-139.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "kronrod.f90">kronrod.f90</a>, the source code.
</li>
<li>
<a href = "kronrod.sh">kronrod.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "kronrod_prb.f90">kronrod_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "kronrod_prb.sh">kronrod_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "kronrod_prb_output.txt">kronrod_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>KRONROD</b> adds N+1 points to an N-point Gaussian rule.
</li>
<li>
<b>ABWE1</b> calculates a Kronrod abscissa and weight.
</li>
<li>
<b>ABWE2</b> calculates a Gaussian abscissa and two weights.
</li>
<li>
<b>TIMESTAMP</b> prints out the current YMDHMS date as a timestamp.
</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 03 August 2010.
</i>
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