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<html>
<head>
<title>
TANH_SINH_RULE - Tanh-Sinh Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TANH_SINH_RULE <br> Tanh-Sinh Quadrature Rules
</h1>
<hr>
<p>
<b>TANH_SINH_RULE</b>
is a FORTRAN90 program which
generates a specific tanh-sinh quadrature rule,
based on user input.
</p>
<p>
The rule is output as three files for easy use as input
to other programs.
</p>
<p>
The tanh-sinh quadrature rule is designed for the interval [-1,+1].
</p>
<p>
The tanh-sinh quadrature assumes that the integrand has the form:
<pre>
Integral ( -1 <= x <= +1 ) f(x) dx
</pre>
</p>
<p>
The tanh-sinh quadrature rule is used as follows:
<pre>
Integral ( -1 <= x <= +1 ) f(x) dx
</pre>
is to be approximated by
<pre>
Sum ( 1 <= i <= order ) w(i) * f(x(i))
</pre>
</p>
<p>
A tanh-sinh quadrature rule has two parameters, the order N and the
stepsize H. Various choices are available to relate these quantities
when making a family of rules. One choice, which results in a nested family,
is to take the K-th rule to have order N = (2^K)-1 and parameter H = 4.0/2^K.
</p>
<p>
Another issue with tanh-sinh quadrature is that the weights don't add up
to 2. Particularly for low order rules, the discrepancy is large. Since
these rules are used as families, and we're looking for asymptotic accuracy,
the errors in the early rules might not matter; in that case, there is a
simple relationship between the weights used in successive elements of the
family. We will take a different view here, and force the weights to add
up to 2 by normalizing them.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>tanh_sinh_rule</b> <i>order</i> <i>prefix</i>
</blockquote>
where
<ul>
<li>
<i>order</i> is the number of points in the quadrature rule. A typical value might
be 1, 3, 7, 15, 31, 63, 127 or (2^K)-1.
</li>
<li>
<i>prefix</i> is the common prefix for the three output files:
<ul>
<li>
<b>prefix_x.txt</b>, will contain the abscissas;
</li>
<li>
<b>prefix_w.txt</b>, will contain the weights;
</li>
<li>
<b>prefix_r.txt</b>, contains the region definition.
</li>
</ul>
</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>TANH_SINH_RULE</b> is available in
<a href = "../../cpp_src/tanh_sinh_rule/tanh_sinh_rule.html">a C++ version</a> and
<a href = "../../f_src/tanh_sinh_rule/tanh_sinh_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/tanh_sinh_rule/tanh_sinh_rule.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/ccn_rule/ccn_rule.html">
CCN_RULE</a>,
a FORTRAN90 program which
defines a nested Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../f_src/chebyshev1_rule/chebyshev1_rule.html">
CHEBYSHEV1_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Chebyshev type 1 quadrature rule.
</p>
<p>
<a href = "../../f_src/chebyshev2_rule/chebyshev2_rule.html">
CHEBYSHEV2_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Chebyshev type 2 quadrature rule.
</p>
<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/gegenbauer_rule/gegenbauer_rule.html">
GEGENBAUER_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Gegenbauer quadrature rule.
</p>
<p>
<a href = "../../f_src/gen_hermite_rule/gen_hermite_rule.html">
GEN_HERMITE_RULE</a>,
a FORTRAN90 program which
can compute and print a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../f_src/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a FORTRAN90 program which
can compute and print a generalized Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../f_src/hermite_rule/hermite_rule.html">
HERMITE_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../f_src/int_exactness_legendre/int_exactness_legendre.html">
INT_EXACTNESS_LEGENDRE</a>,
a FORTRAN90 program which
checks the polynomial exactness
of a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../f_src/intlib/intlib.html">
INTLIB</a>,
a FORTRAN90 library which
contains routines for numerical estimation of integrals in 1D.
</p>
<p>
<a href = "../../f_src/jacobi_rule/jacobi_rule.html">
JACOBI_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Jacobi quadrature rule.
</p>
<p>
<a href = "../../f_src/laguerre_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../f_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a FORTRAN90 program which
computes a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../f_src/legendre_rule_fast/legendre_rule_fast.html">
LEGENDRE_RULE_FAST</a>,
a FORTRAN90 program which
uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.
</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 = "../../f_src/product_rule/product_rule.html">
PRODUCT_RULE</a>,
a FORTRAN90 program which
constructs a product rule
from 1D factor rules.
</p>
<p>
<a href = "../../f_src/quadpack/quadpack.html">
QUADPACK</a>,
a FORTRAN90 library which
contains routines for
numerical estimation of integrals in 1D.
</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_tanh_sinh/quadrature_rules_tanh_sinh.html">
QUADRATURE_RULES_TANH_SINH</a>,
a dataset directory which
contains triples of files defining tanh-sinh quadrature rules.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines 1-dimensional quadrature rules.
</p>
<p>
<a href = "../../f_src/tanh_quad/tanh_quad.html">
TANH_QUAD</a>,
a FORTRAN90 library which
sets up the tanh quadrature rule;
</p>
<p>
<a href = "../../f_src/test_int/test_int.html">
TEST_INT</a>,
a FORTRAN90 library which
contains
number of functions that may be used as test integrands for
quadrature rules in 1D.
</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>
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 = "tanh_sinh_rule.f90">tanh_sinh_rule.f90</a>, the source code.
</li>
<li>
<a href = "tanh_sinh_rule.sh">tanh_sinh_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
The directory QUADRATURE_RULES_TANH_SINH contains a number of tanh-sinh
quadrature rules created by this program. Here is a pointer to the
rule of order 31:
<ul>
<li>
<a href = "../../datasets/quadrature_rules_tanh_sinh/ts_o31_x.txt">
ts_o31_x.txt</a>,
the abscissa file;
</li>
<li>
<a href = "../../datasets/quadrature_rules_tanh_sinh/ts_o31_w.txt">
ts_o31_w.txt</a>,
the weight file;
</li>
<li>
<a href = "../../datasets/quadrature_rules_tanh_sinh/ts_o31_r.txt">
ts_o31_r.txt</a>,
the region file;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TANH_SINH_RULE.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>DTABLE_WRITE0</b> writes a DTABLE file with no headers.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>TANH_SINH_HANDLE</b> computes the requested tanh-sinh rule and outputs it.
</li>
<li>
<b>TANH_SINH_COMPUTE</b> computes a tanh-sinh quadrature rule.
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 27 June 2009.
</i>
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