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<html>
<head>
<title>
HERMITE_RULE - Gauss-Hermite Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
HERMITE_RULE <br> Gauss-Hermite Quadrature Rules
</h1>
<hr>
<p>
<b>HERMITE_RULE</b>
is a C++ program which
generates a specific Gauss-Hermite quadrature rule,
based on user input.
</p>
<p>
The rule is written to three files for easy use as input
to other programs.
</p>
<p>
The <i>Gauss-Hermite quadrature rule</i> is used as follows:
<pre>
c * Integral ( -oo < x < +oo ) f(x) exp ( - b * ( x - a )^2 ) dx
</pre>
is to be approximated by
<pre>
Sum ( 1 <= i <= order ) w(i) * f(x(i))
</pre>
Generally, a Gauss-Hermite quadrature rule of <i>n</i> points will
produce the exact integral when f(x) is a polynomial of degree
<i>2n-1</i> or less.
</p>
<p>
The value of C in front of the integral depends on the user's
choice of the SCALE parameter:
<ul>
<li>
<i>scale=0</i>, then C = 1; this is the standard choice for
Gauss-Hermite quadrature.
</li>
<li>
<i>scale=1</i>, then C is a normalization factor so that
f(x)=1 will integrate to 1. This implies in turn that the
weights will sum to 1. This choice is appropriate when
using the rule to compute probabilities.
</li>
</ul>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>hermite_rule</b> <i>order</i> <i>a</i> <i>b</i> <i>scale</i> <i>filename</i>
</blockquote>
where
<ul>
<li>
<i>order</i> is the number of points in the quadrature rule.
</li>
<li>
<i>a</i> is the center point (default 0);
</li>
<li>
<i>b</i> is the scale factor (default 1);
</li>
<li>
<i>scale</i> is the normalization option (0/1). If 1,
then the weights are normalized to have unit sum;
</li>
<li>
<i>filename</i> specifies the output filenames:
<i>filename</i><b>_w.txt</b>,
<i>filename</i><b>_x.txt</b>, and <i>filename</i><b>_r.txt</b>,
containing the weights, abscissas, and interval limits.
</li>
</ul>
</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>HERMITE_RULE</b> is available in
<a href = "../../c_src/hermite_rule/hermite_rule.html">a C version</a> and
<a href = "../../cpp_src/hermite_rule/hermite_rule.html">a C++ version</a> and
<a href = "../../f77_src/hermite_rule/hermite_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/hermite_rule/hermite_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hermite_rule/hermite_rule.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/ccn_rule/ccn_rule.html">
CCN_RULE</a>,
a C++ program which
defines a nested Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../cpp_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a C++ library which
evaluates the Chebyshev polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/chebyshev1_rule/chebyshev1_rule.html">
CHEBYSHEV1_RULE</a>,
a C++ program which
can compute and print a Gauss-Chebyshev type 1 quadrature rule.
</p>
<p>
<a href = "../../cpp_src/chebyshev2_rule/chebyshev2_rule.html">
CHEBYSHEV2_RULE</a>,
a C++ program which
can compute and print a Gauss-Chebyshev type 2 quadrature rule.
</p>
<p>
<a href = "../../cpp_src/clenshaw_curtis_rule/clenshaw_curtis_rule.html">
CLENSHAW_CURTIS_RULE</a>,
a C++ program which
defines a Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../cpp_src/gegenbauer_rule/gegenbauer_rule.html">
GEGENBAUER_RULE</a>,
a C++ program which
can compute and print a Gauss-Gegenbauer quadrature rule.
</p>
<p>
<a href = "../../cpp_src/gen_hermite_rule/gen_hermite_rule.html">
GEN_HERMITE_RULE</a>,
a C++ program which
can compute and print a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../cpp_src/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a C++ program which
can compute and print a generalized Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/hermite_exactness/hermite_exactness.html">
HERMITE_EXACTNESS</a>,
a C++ program which
tests the polynomial exactness of Gauss-Hermite quadrature rules
for estimating the integral of a function with density exp(-x^2)
over the interval (-oo,+oo).
</p>
<p>
<a href = "../../cpp_src/hermite_test_int/hermite_test_int.html">
HERMITE_TEST_INT</a>,
a C++ library which
defines test integrands for Hermite integrals with
interval (-oo,+oo) and density exp(-x^2).
</p>
<p>
<a href = "../../cpp_src/jacobi_rule/jacobi_rule.html">
JACOBI_RULE</a>,
a C++ program which
can compute and print a Gauss-Jacobi quadrature rule.
</p>
<p>
<a href = "../../cpp_src/laguerre_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a C++ program which
can compute and print a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/lattice_rule/lattice_rule.html">
LATTICE_RULE</a>,
a C++ library which
approximates M-dimensional integrals using lattice rules.
</p>
<p>
<a href = "../../cpp_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a C++ program which
computes a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/line_felippa_rule/line_felippa_rule.html">
LINE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../cpp_src/patterson_rule/patterson_rule.html">
PATTERSON_RULE</a>,
a C++ 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 = "../../datasets/quadrature_rules_hermite_physicist/quadrature_rules_hermite_physicist.html">
QUADRATURE_RULES_HERMITE_PHYSICIST</a>,
a dataset directory which
contains Gauss-Hermite quadrature rules, for integration
on the interval (-oo,+oo), with weight function exp(-x^2).
</p>
<p>
<a href = "../../datasets/quadrature_rules_hermite_probabilist/quadrature_rules_hermite_probabilist.html">
QUADRATURE_RULES_HERMITE_PROBABILIST</a>,
a dataset directory which
contains Gauss-Hermite quadrature rules, for integration
on the interval (-oo,+oo), with weight function exp(-x^2/2).
</p>
<p>
<a href = "../../datasets/quadrature_rules_hermite_unweighted/quadrature_rules_hermite_unweighted.html">
QUADRATURE_RULES_HERMITE_UNWEIGHTED</a>,
a dataset directory which
contains Gauss-Hermite quadrature rules, for integration
on the interval (-oo,+oo), with weight function 1.
</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>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<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>
<li>
Roger Martin, James Wilkinson,<br>
The Implicit QL Algorithm,<br>
Numerische Mathematik,<br>
Volume 12, Number 5, December 1968, pages 377-383.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "hermite_rule.cpp">hermite_rule.cpp</a>, the source code.
</li>
<li>
<a href = "hermite_rule.sh">hermite_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
HERM_O4 is a Hermite rule of order 4, created by the command
<pre>
hermite_rule 4 0.0 1.0 0 herm_o4
</pre>
<ul>
<li>
<a href = "herm_o4_r.txt">herm_o4_r.txt</a>,
the region file;
</li>
<li>
<a href = "herm_o4_w.txt">herm_o4_w.txt</a>,
the weight file;
</li>
<li>
<a href = "herm_o4_x.txt">herm_o4_x.txt</a>,
the abscissa file;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for HERMITE_RULE.
</li>
<li>
<b>CDGQF</b> computes a Gauss quadrature formula with default A, B and simple knots.
</li>
<li>
<b>CGQF</b> computes knots and weights of a Gauss quadrature formula.
</li>
<li>
<b>CLASS_MATRIX</b> computes the Jacobi matrix for a quadrature rule.
</li>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>PARCHK</b> checks parameters ALPHA and BETA for classical weight functions.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_SIGN</b> returns the sign of an R8.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file with no header.
</li>
<li>
<b>RULE_WRITE</b> writes a quadrature rule to three files.
</li>
<li>
<b>SCQF</b> scales a quadrature formula to a nonstandard interval.
</li>
<li>
<b>SGQF</b> computes knots and weights of a Gauss Quadrature formula.
</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 23 February 2010.
</i>
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