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quadrature_weights.html
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<html>
<head>
<title>
QUADRATURE_WEIGHTS - Computation of Weights for Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
QUADRATURE_WEIGHTS <br> Computation of Weights for Quadrature Rules
</h1>
<hr>
<p>
<b>QUADRATURE_WEIGHTS</b>
is a FORTRAN90 library which
illustrates techniques for computing the weights of a quadrature
rule, assuming that the points have been specified.
</p>
<h3 align="center">
The Vandermonde Matrix
</h3>
<p>
We assume that the quadrature formula approximates integrals of the form:
<pre>
I(F) = Integral ( A <= X <= B ) F(X) dX
</pre>
by specifying N points X and weights W such that
<pre>
Q(F) = Sum ( 1 <= I <= N ) W(I) * F(X(I))
</pre>
</p>
<p>
Now let us assume that the points X have been specified, but that the
corresponding values W remain to be determined.
</p>
<p>
If we require that the quadrature rule with N points integrates the first
N monomials exactly, then we have N conditions on the weights W.
</p>
<p>
The I-th condition, for the monomial X^(I-1), has the form:
<pre>
W(1)*X(1)^(I-1) + W(2)*X(2)^(I-1)+...+W(N)*X(N)^(I-1) = (B^I-A^I)/I.
</pre>
</p>
<p>
The corresponding matrix is known as the Vandermonde matrix. It is
theoretically guaranteed to be nonsingular as long as the X's are
distinct, but its condition number grows quickly with N. Therefore,
this simple, direct approach is often abandoned when more accuracy
or high order rules are needed.
</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">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/clenshaw_curtis_rule/clenshaw_curtis_rule.html">
CLENSHAW_CURTIS_RULE</a>,
a FORTRAN90 program which
defines a Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules for 1-dimensional domains.
</p>
<p>
<a href = "../../f_src/toms655/toms655.html">
TOMS655</a>,
a FORTRAN90 library which
computes the weights for interpolatory quadrature rule;<br>
this library is commonly called <b>IQPACK</b>;<br>
this is a FORTRAN90 version of ACM TOMS algorithm 655.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
<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, 1982, pages 407-422.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<b>QW_VANDERMONDE</b> uses the Vandermonde matrix approach. It assumes
that the abscissas have been chosen, that the interval [A,B] is known,
and that the integrals of polynomials of degree 0 through N-1 can be
computed. (The examples here use a finite interval and a unit weight function,
but the method can easily be extended to non-finite intervals and non-unit
weight functions.)
<ul>
<li>
<a href = "qw_vandermonde.f90">qw_vandermonde.f90</a>,
the source code for the Vandermonde matrix approach.
</li>
<li>
<a href = "qw_vandermonde.sh">qw_vandermonde.sh</a>,
BASH commands to compile the source code.
</li>
<li>
<a href = "qw_vandermonde_prb.f90">qw_vandermonde_prb.f90</a>,
a sample calling program for the Vandermonde matrix approach.
</li>
<li>
<a href = "qw_vandermonde_prb.sh">qw_vandermonde_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "qw_vandermonde_prb_output.txt">qw_vandermonde_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>QW_GOLUB_WELSCH</b> uses the Golub-Welsch approach to compute
a Gaussian quadrature rule given information about the Jacobi matrix
associated with the polynomials that are orthonormal with respect
to the given weight function. Note that this approach computes
both the abscissas and the weights.
<ul>
<li>
<a href = "qw_golub_welsch.f90">qw_golub_welsch.f90</a>,
the source code for the Golub-Welsch approach.
</li>
<li>
<a href = "qw_golub_welsch.sh">qw_golub_welsch.sh</a>,
BASH commands to compile the source code.
</li>
<li>
<a href = "qw_golub_welsch_prb.f90">qw_golub_welsch_prb.f90</a>,
a sample calling program for the Golub-Welsch approach.
</li>
<li>
<a href = "qw_golub_welsch_prb.sh">qw_golub_welsch_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "qw_golub_welsch_prb_output.txt">qw_golub_welsch_prb_output.txt</a>,
the output file.
</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 05 April 2011.
</i>
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