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<html>
<head>
<title>
QUADRATURE_WEIGHTS_VANDERMONDE_2D - 2D Quadrature Weights by Vandermonde Matrix
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
QUADRATURE_WEIGHTS_VANDERMONDE_2D <br>
2D Quadrature Weights by Vandermonde Matrix
</h1>
<hr>
<p>
<b>QUADRATURE_WEIGHTS_VANDERMONDE_2D</b>
is a C++ library which
illustrates a method for computing the weights W of a
2D interpolatory quadrature rule, assuming that the points (X,Y) have been
specified, by setting up a linear system involving the Vandermonde matrix.
</p>
<p>
We assume that the abscissas (quadrature points) have been chosen,
that the interval [A,B]x[C,D] is known, and that the integrals of polynomials
of total degree 0 through T can be computed.
</p>
<h3 align="center">
The Vandermonde Matrix
</h3>
<p>
We assume that the quadrature formula approximates integrals of the form:
<pre>
I(F) = Integral ( C <= Y <= D ) Integral ( A <= X <= B ) F(X,Y) dX dY
</pre>
by specifying N=(T+1)*(T+2)/2 points (X,Y) and weights W such that
<pre>
Q(F) = Sum ( 1 <= I <= N ) W(I) * F(X(I),Y(I))
</pre>
</p>
<p>
Now let us assume that the points (X,Y) 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=(T+1)*(T+2)/2 monomials exactly, then we have N conditions
on the weights W. (This means that we are assuming that N only takes
on appropriate values, namely 1, 3, 6, 10, 15, 21, 28, ...)
</p>
<p>
The K-th condition, for the monomial X^I*Y^J, J = 0 to T, I = 0 to T - J,
has the form:
<pre>
W(1)*X(1)^I*Y(1)^J + W(2)*X(2)^I*Y(2)^j+...+W(N)*X(N)^I*Y(N)^J = (B^(I+1)-A^(I+1))*(D^(J+1)-C(J+1))/(I+1)/(J+1)
</pre>
</p>
<p>
The corresponding matrix is known as a two-dimensional Vandermonde matrix.
It is theoretically guaranteed to be nonsingular as long as the points
(X,Y) are distinct and in "general position". The condition number of
the matrix grows quickly with increasing T.
</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_WEIGHTS_VANDERMONDE_2D</b> is available in
<a href = "../../c_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a C version</a> and
<a href = "../../cpp_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a C++ version</a> and
<a href = "../../f77_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/quadrature_weights_vandermonde_2d/quadrature_weights_vandermonde_2d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/exactness_2d/exactness_2d.html">
EXACTNESS_2D</a>,
a C++ library which
investigates the exactness of 2D quadrature rules that estimate the
integral of a function f(x,y) over a 2D domain.
</p>
<p>
<a href = "../../cpp_src/quadrature_least_squares/quadrature_least_squares.html">
QUADRATURE_LEAST_SQUARES</a>,
a C++ library which
computes weights for "sub-interpolatory" quadrature rules,
that is, it estimates integrals by integrating a polynomial that
approximates the function data in a least squares sense.
</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 1D 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 1-dimensional domains.
</p>
<p>
<a href = "../../cpp_src/toms655/toms655.html">
TOMS655</a>,
a C++ 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>
<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">
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>
<ul>
<li>
<a href = "qwv_2d.cpp">qwv_2d.cpp</a>,
the source code.
</li>
<li>
<a href = "qwv_2d.hpp">qwv_2d.hpp</a>,
the include file.
</li>
<li>
<a href = "qwv_2d.sh">qwv_2d.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "qwv_2d_prb.cpp">qwv_2d_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "qwv_2d_prb.sh">qwv_2d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "qwv_2d_prb_output.txt">qwv_2d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>I4VEC_ZERO_NEW</b> creates and zeroes an I4VEC.
</li>
<li>
<b>QWV_2D</b> computes 2D quadrature weights using the Vandermonde matrix.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8MAT_SOLVE2</b> computes the solution of an N by N linear system.
</li>
<li>
<b>R8VEC_EVEN_NEW</b> returns an R8VEC of values evenly spaced between ALO and AHI.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_PRINT_16</b> prints an R8VEC to 16 decimal places.
</li>
<li>
<b>R8VEC_ZERO_NEW</b> creates and zeroes an R8VEC.
</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 28 May 2014.
</i>
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