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<html>
<head>
<title>
INT_EXACTNESS_GEGENBAUER - Exactness of Gauss-Gegenbauer Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
INT_EXACTNESS_GEGENBAUER <br> Exactness of Gauss-Gegenbauer Quadrature Rules
</h1>
<hr>
<p>
<b>INT_EXACTNESS_GEGENBAUER</b>
is a FORTRAN90 program which
investigates the polynomial exactness of a Gauss-Gegenbauer
quadrature rule for the interval [-1,1] with a weight function.
</p>
<p>
The Gauss-Gegenbauer quadrature rule is designed
to approximate integrals on the interval [-1,1], with a weight function of the
form <b>(1-x^2)<sup>ALPHA</sup></b>. ALPHA is a real parameter that must be
greater than -1.
</p>
<p>
Gauss-Gegenbauer quadrature assumes that the integrand we are
considering has a form like:
<pre>
Integral ( -1 <= x <= +1 ) (1-x^2)^alpha f(x) dx
</pre>
</p>
<p>
For a Gauss-Gegenbauer rule, polynomial exactness is defined in terms of
the function <b>f(x)</b>. That is, we say the rule is exact for polynomials
up to degree DEGREE_MAX if, for any polynomial <b>f(x)</b> of that degree or
less, the quadrature rule will produce the exact value of
<pre>
Integral ( -1 <= x <= +1 ) (1-x^2)^alpha f(x) dx
</pre>
</p>
<p>
The program starts at <b>DEGREE</b> = 0, and then
proceeds to <b>DEGREE</b> = 1, 2, and so on up to a maximum degree
<b>DEGREE_MAX</b> specified by the user. At each value of <b>DEGREE</b>,
the program generates the corresponding monomial term, applies the
quadrature rule to it, and determines the quadrature error.
</p>
<p>
The program is very flexible and interactive. The quadrature rule
is defined by three files, to be read at input, and the
maximum degree top be checked is specified by the user as well.
</p>
<p>
Note that the three files that define the quadrature rule
are assumed to have related names, of the form
<ul>
<li>
<i>prefix</i>_<b>x.txt</b>
</li>
<li>
<i>prefix</i>_<b>w.txt</b>
</li>
<li>
<i>prefix</i>_<b>r.txt</b>
</li>
</ul>
When running the program, the user only enters the common <i>prefix</i>
part of the file names, which is enough information for the program
to find all three files.
</p>
<p>
For information on the form of these files, see the
<b>QUADRATURE_RULES</b> directory listed below.
</p>
<p>
The exactness results are written to an output file with the
corresponding name:
<ul>
<li>
<i>prefix</i>_<b>exact.txt</b>
</li>
</ul>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>int_exactness_gegenbauer</b> <i>prefix</i> <i>degree_max</i> <i>alpha</i>
</blockquote>
where
<ul>
<li>
<i>prefix</i> is the common prefix for the files containing the abscissa, weight
and region information of the quadrature rule;
</li>
<li>
<i>degree_max</i> is the maximum monomial degree to check. This would normally be
a relatively small nonnegative number, such as 5, 10 or 15.
</li>
<li>
<i>alpha</i> is the value of the exponent of (1-x^2) in the weight function;
<b>alpha</b> should be a real number greater than -1.0.
</li>
</ul>
</p>
<p>
If the arguments are not supplied on the command line, the
program will prompt for them.
</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>INT_EXACTNESS_GEGENBAUER</b> is available in
<a href = "../../cpp_src/int_exactness_gegenbauer/int_exactness_gegenbauer.html">a C++ version</a> and
<a href = "../../f_src/int_exactness_gegenbauer/int_exactness_gegenbauer.html">a FORTRAN90 version</a> and
<a href = "../../m_src/int_exactness_gegenbauer/int_exactness_gegenbauer.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/gegenbauer_rule/gegenbauer_rule.html">
GEGENBAUER_RULE</a>,
a FORTRAN90 program which
can generate a Gauss-Gegenbauer quadrature
rule on request.
</p>
<p>
<a href = "../../f_src/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a FORTRAN90 program which
tests the polynomial exactness of a quadrature rule for a finite interval.
</p>
<p>
<a href = "../../f_src/int_exactness_chebyshev1/int_exactness_chebyshev1.html">
INT_EXACTNESS_CHEBYSHEV1</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_chebyshev2/int_exactness_chebyshev2.html">
INT_EXACTNESS_CHEBYSHEV2</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_gen_hermite/int_exactness_gen_hermite.html">
INT_EXACTNESS_GEN_HERMITE</a>,
a FORTRAN90 program which
tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_gen_laguerre/int_exactness_gen_laguerre.html">
INT_EXACTNESS_GEN_LAGUERRE</a>,
a FORTRAN90 program which
tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_hermite/int_exactness_hermite.html">
INT_EXACTNESS_HERMITE</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Hermite quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_jacobi/int_exactness_jacobi.html">
INT_EXACTNESS_JACOBI</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Jacobi quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_laguerre/int_exactness_laguerre.html">
INT_EXACTNESS_LAGUERRE</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Laguerre quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_legendre/int_exactness_legendre.html">
INT_EXACTNESS_LEGENDRE</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Legendre quadrature rules.
</p>
<p>
<a href = "../../f_src/integral_test/integral_test.html">
INTEGRAL_TEST</a>,
a FORTRAN90 program which
uses test integrals to measure the effectiveness of
certain sets of quadrature rules.
</p>
<p>
<a href = "../../datasets/quadrature_rules_gegenbauer/quadrature_rules_gegenbauer.html">
QUADRATURE_RULES_GEGENBAUER</a>,
a dataset directory which
contains sets of files that define Gauss-Gegenbauer quadrature rules.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules on a
variety of intervals with different weight functions.
</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>
Shanjie Zhang, Jianming Jin,<br>
Computation of Special Functions,<br>
Wiley, 1996,<br>
ISBN: 0-471-11963-6,<br>
LC: QA351.C45.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "int_exactness_gegenbauer.f90">int_exactness_gegenbauer.f90</a>, the source code.
</li>
<li>
<a href = "int_exactness_gegenbauer.sh">int_exactness_gegenbauer.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>GEGEN_O1_A0.5</b> is a Gauss-Gegenbauer order 1 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o1_a0.5_x.txt">
gegen_o1_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o1_a0.5_w.txt">
gegen_o1_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o1_a0.5_r.txt">
gegen_o1_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gegen_o1_a0.5_exact.txt">gegen_o1_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gegenbauer gegen_o1_a0.5 5 0.5
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEGEN_O2_A0.5</b> is a Gauss-Gegenbauer order 2 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o2_a0.5_x.txt">
gegen_o2_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o2_a0.5_w.txt">
gegen_o2_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o2_a0.5_r.txt">
gegen_o2_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gegen_o2_a0.5_exact.txt">gegen_o2_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gegenbauer gegen_o2_a0.5 5 0.5
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEGEN_O4_A0.5</b> is a Gauss-Gegenbauer order 4 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o4_a0.5_x.txt">
gegen_o4_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o4_a0.5_w.txt">
gegen_o4_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o4_a0.5_r.txt">
gegen_o4_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gegen_o4_a0.5_exact.txt">gegen_o4_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gegenbauer gegen_o4_a0.5 10 0.5
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEGEN_O8_A0.5</b> is a Gauss-Gegenbauer order 8 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o8_a0.5_x.txt">
gegen_o8_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o8_a0.5_w.txt">
gegen_o8_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o8_a0.5_r.txt">
gegen_o8_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gegen_o8_a0.5_exact.txt">gegen_o8_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gegenbauer gegen_o8_a0.5 18 0.5
</b></blockquote>
</ul>
</p>
<p>
<b>GEGEN_O16_A0.5</b> is a Gauss-Gegenbauer order 16 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o16_a0.5_x.txt">
gegen_o16_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o16_a0.5_w.txt">
gegen_o16_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_jacuerre/gegen_o16_a0.5_r.txt">
gegen_o16_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gegen_o16_a0.5_exact.txt">gegen_o16_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gegenbauer gegen_o16_a0.5 35 0.5
</b></blockquote>
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for INT_EXACTNESS_GEGENBAUER.
</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>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GEGENBAUER_INTEGRAL</b> evaluates the integral of a monomial with Gegenbauer weight.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>MONOMIAL_QUADRATURE_GEGENBAUER</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8_HYPER_2F1</b> evaluates the hypergeometric function F(A,B,C,X).
</li>
<li>
<b>R8_PSI</b> evaluates the function Psi(X).
</li>
<li>
<b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</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 28 January 2010.
</i>
<!-- John Burkardt -->
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