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<html>
<head>
<title>
TOMS672 - Maximally Accurate Extensions of Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TOMS672 <br> Maximally Accurate Extensions of Quadrature Rules
</h1>
<hr>
<p>
<b>TOMS672</b>
is a FORTRAN90 library which
computes a quadrature rule which is a maximally accurate extension of a
given quadrature rule.
</p>
<p>
That is, we suppose we have a quadrature rule that uses N points, and
that we wish to compute a new quadrature rule, which uses N+M points.
The new rule is required to include the original N points of the old rule.
</p>
<p>
This kind of procedure is analogous to the process by which the
nested Gauss-Patterson rules were developed. Thus, one use of the
software is to try to develop nested families of rules for other
weight functions, or starting from other initial rules.
</p>
<p>
In certain cases, this algorithm <i>may</i> be able to extend the given
rule in a way which produces a rule with the maximum possible precision.
(However, it is also possible that the extension cannot be made.)
</p>
<p>
By calling this algorithm repeatedly, it is possible to compute a family
of nested quadrature rules of any order. Nested rules can be efficient when
function evaluations are expensive, and a series of integral estimates must
be made to estimate accuracy, or to detect convergence.
</p>
<p>
The original, true, correct version of ACM TOMS Algorithm 672
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">
Languages:
</h3>
<p>
<b>TOMS672</b> is available in
<a href = "../../f77_src/toms672/toms672.html">a FORTRAN77 version</a> and
<a href = "../../f_src/toms672/toms672.html">a FORTRAN90 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/kronrod/kronrod.html">
KRONROD</a>,
a FORTRAN90 library which
can compute a Gauss and Gauss-Kronrod pair of quadrature rules
of arbitrary order,
by Robert Piessens, Maria Branders.
</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 = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>
a dataset directory which
contains sets of files that define quadrature
rules over various 1D intervals or multidimensional hypercubes.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>
a FORTRAN90 library which
defines quadrature rules on a
variety of intervals with different weight functions.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Gene Golub, Thomas Robertson,<br>
A generalized Bairstow Algorithm,<br>
Communications of the ACM,<br>
Volume 10, Number 6, June 1967, pages 371-373.
</li>
<li>
Thomas Patterson,<br>
The Optimal Addition of Points to Quadrature Formulae,<br>
Mathematics of Computation,<br>
Volume 22, Number 104, October 1968, pages 847-856.
</li>
<li>
Thomas Patterson,<br>
An algorithm for generating interpolatory quadrature rules of the highest degree
of precision with preassigned nodes for general weight functions,<br>
Transactions on Mathematical Software,<br>
Volume 15, Number 2, June 1989, pages 123-136.
</li>
<li>
Thomas Patterson,<br>
Algorithm 672:
EXTEND: generation of interpolatory quadrature rules of the highest degree
of precision with preassigned nodes for general weight functions,<br>
Transactions on Mathematical Software,<br>
Volume 15, Number 2, June 1989, pages 137-143.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "toms672.f90">toms672.f90</a>, the source code.
</li>
<li>
<a href = "toms672.sh">toms672.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "toms672_prb.f90">toms672_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "toms672_prb.sh">toms672_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "toms672_prb_input.txt">toms672_prb_input.txt</a>,
the input file.
</li>
<li>
<a href = "toms672_prb_output.txt">toms672_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ASSIGN</b> generates the polynomial whose roots are the preassigned nodes.
</li>
<li>
<b>BAIR</b> seeks roots of a polynomial.
</li>
<li>
<b>CHECK</b> tests a computed quadrature rule.
</li>
<li>
<b>DAXPY</b> computes constant times a vector plus a vector.
</li>
<li>
<b>DDOT</b> forms the dot product of two vectors.
</li>
<li>
<b>DGEFA</b> factors a real general matrix.
</li>
<li>
<b>DGESL</b> solves a real general linear system A * X = B.
</li>
<li>
<b>DSCAL</b> scales a vector by a constant.
</li>
<li>
<b>EPROD</b> expands a product of two orthogonal polynomials.
</li>
<li>
<b>EXTEND</b> extends a quadrature rule by adding new nodes.
</li>
<li>
<b>GENER</b> calculates the polynomial defining the optimal new nodes.
</li>
<li>
<b>IDAMAX</b> indexes the array element of maximum absolute value.
</li>
<li>
<b>LFACT</b> removes a linear factor from a polynomial expansion.
</li>
<li>
<b>NEWTON</b> applies Newton's method for a single root of a polynomial.
</li>
<li>
<b>QFACT</b> divides a polynomial by a quadratic factor.
</li>
<li>
<b>ROOTS</b> calculates roots of a quadratic factor.
</li>
<li>
<b>RSORT</b> carries out a simple ripple sort.
</li>
<li>
<b>SOLVE</b> calculates roots of an orthogonal polynomial expansion.
</li>
<li>
<b>TRANSF</b> scales a polynomial expansion with respect to the moments.
</li>
<li>
<b>WEIGHT</b> calculates quadrature weights.
</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 16 February 2011.
</i>
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