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<html>
<head>
<title>
QUADRATURE_LEAST_SQUARES - Least Squares Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
QUADRATURE_LEAST_SQUARES <br> Least Squares Quadrature Rules
</h1>
<hr>
<p>
<b>QUADRATURE_LEAST_SQUARES</b>
is a C++ library which
computes weights for "sub-interpolatory" quadrature rules.
</p>
<p>
A large class of quadrature rules may be computed by specifying
a set of N abscissas, or sample points, X(1:N), determining the
Lagrange interpolation basis functions L(1:N), and then setting
a weight vector W by
<pre>
W(i) = I(L(i))
</pre>
after which, the integral of any function f(x) is estimated by
<pre>
I(f) \approx Q(f) = sum ( 1 <= i <= N ) W(i) * f(X(i))
</pre>
</p>
<p>
We call this an interpolatory rule because the function f(x)
has first been interpolated by
<pre>
f(x) \approx sum ( 1 <= i <= N ) L(i) * f(X(i))
</pre>
after which, we apply the integration operator:
<pre>
I(f) \approx I(sum ( 1 <= i <= N ) L(i) * f(X(i)))
= sum ( 1 <= i <= N ) I(L(i)) * f(X(i))
= sum ( 1 <= i <= N ) W(i) * f(X(i)).
</pre>
</p>
<p>
For badly chosen sets of X, or high values of N, or unruly functions f(x),
interpolation may be a bad way to approximate the function. An
alternative is to seek a polynomial interpolant of degree D < N-1,
and then integrate that. We might call this a "sub-interpolatory" rule.
</p>
<p>
As it turns out, a natural way to seek such a rule is to write out
the N by D+1 Vandermonde matrix and use a least squares solver.
Even though the N by N Vandermonde matrix is ill-conditioned for
Gauss elimination, a least squares approach can produce usable solutions
from the N by D+1 matrix.
</p>
<p>
The outline of this procedure was devised by Professor Mac Hyman
of Tulane University.
</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>QUADRATURE_LEAST_SQUARES</b> is available in
<a href = "../../c_src/quadrature_least_squares/quadrature_least_squares.html">a C version</a> and
<a href = "../../cpp_src/quadrature_least_squares/quadrature_least_squares.html">a C++ version</a> and
<a href = "../../f77_src/quadrature_least_squares/quadrature_least_squares.html">a FORTRAN77 version</a> and
<a href = "../../f_src/quadrature_least_squares/quadrature_least_squares.html">a FORTRAN90 version</a> and
<a href = "../../m_src/quadrature_least_squares/quadrature_least_squares.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/clenshaw_curtis_rule/clenshaw_curtis_rule.html">
CLENSHAW_CURTIS_RULE</a>,
a C++ library which
defines a multiple dimension Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../cpp_src/qr_solve/qr_solve.html">
QR_SOLVE</a>,
a C++ library which
computes the least squares solution of a rectangular linear system A*x=b.
</p>
<p>
<a href = "../../cpp_src/quadmom/quadmom.html">
QUADMOM</a>,
a C++ library which
computes a Gaussian quadrature rule for a weight function rho(x)
based on the Golub-Welsch procedure that only requires knowledge
of the moments of rho(x).
</p>
<p>
<a href = "../../cpp_src/quadrature_golub_welsch/quadrature_golub_welsch.html">
QUADRATURE_GOLUB_WELSCH</a>,
a C++ library which
computes the points and weights of a Gaussian quadrature rule using the
Golub-Welsch procedure, assuming that the points have been specified.
</p>
<p>
<a href = "../../cpp_src/quadrature_weights_vandermonde/quadrature_weights_vandermonde.html">
QUADRATURE_WEIGHTS_VANDERMONDE</a>,
a C++ library which
computes the weights of a quadrature rule using the Vandermonde
matrix, assuming that the points have been specified.
</p>
<p>
<a href = "../../cpp_src/quadrule/quadrule.html">
QUADRULE</a>,
a C++ library which
defines quadrature rules for approximating an integral over a 1D domain.
</p>
<p>
<a href = "../../cpp_src/quadrule_fast/quadrule_fast.html">
QUADRULE_FAST</a>,
a C++ library which
defines efficient versions of a few 1D quadrature rules.
</p>
<p>
<a href = "../../cpp_src/test_int/test_int.html">
TEST_INT</a>,
a C++ library which
defines test integrands for 1D quadrature rules.
</p>
<p>
<a href = "../../cpp_src/truncated_normal_rule/truncated_normal_rule.html">
TRUNCATED_NORMAL_RULE</a>,
a C++ program which
computes a quadrature rule for a normal probability density function (PDF),
also called a Gaussian distribution, that has been
truncated to [A,+oo), (-oo,B] or [A,B].
</p>
<p>
<a href = "../../cpp_src/vandermonde/vandermonde.html">
VANDERMONDE</a>,
a C++ library which carries out certain operations associated
with the Vandermonde matrix.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "qls.cpp">qls.cpp</a>, the source code.
</li>
<li>
<a href = "qls.hpp">qls.hpp</a>, the include file.
</li>
<li>
<a href = "qls.sh">qls.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "qls_prb.cpp">qls_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "qls_prb.sh">qls_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "qls_prb_output.txt">qls_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>NCC_SET</b> sets abscissas and weights for Newton-Cotes closed quadrature.
</li>
<li>
<b>R8MAT_MV</b> multiplies a matrix times a vector.
</li>
<li>
<b>R8VEC_UNIFORM_AB</b> returns a scaled pseudorandom R8VEC.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>WEIGHTS_LS</b> computes weights for a least squares quadrature rule.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 15 April 2014.
</i>
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