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<html>
<head>
<title>
INT_EXACTNESS_LAGUERRE - Exactness of Gauss-Laguerre Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
INT_EXACTNESS_LAGUERRE <br> Exactness of Gauss-Laguerre Quadrature Rules
</h1>
<hr>
<p>
<b>INT_EXACTNESS_LAGUERRE</b>
is a FORTRAN90 program which
investigates the polynomial exactness of a Gauss-Laguerre
quadrature rule for the semi-infinite interval [0,+oo) or [A,+oo).
</p>
<p>
Standard Gauss-Laguerre quadrature assumes that the integrand we are
considering has a form like:
<pre>
Integral ( A <= x < +oo ) exp(-x) f(x) dx
</pre>
where the factor <b>exp(-x)</b> is regarded as a weight factor.
</p>
<p>
A <i>standard Gauss-Laguerre quadrature rule</i> is a set of <b>n</b>
positive weights <b>w</b> and abscissas <b>x</b> so that
<pre>
Integral ( A <= x < +oo ) exp(-x) f(x) dx
</pre>
may be approximated by
<pre>
Sum ( 1 <= I <= N ) w(i) * f(x(i))
</pre>
</p>
<p>
It is often convenient to consider approximating integrals in which
the weighting factor <b>exp(-x)</b> is implicit. In that case, we
are looking at approximating
<pre>
Integral ( A <= x < +oo ) f(x) dx
</pre>
and it is easy to modify a standard Gauss-Laguerre quadrature rule
to handle this case directly.
</p>
<p>
A <i>modified Gauss-Laguerre quadrature rule</i> is a set of <b>n</b>
positive weights <b>w</b> and abscissas <b>x</b> so that
<pre>
Integral ( A <= x < +oo ) f(x) dx
</pre>
may be approximated by
<pre>
Sum ( 1 <= I <= N ) w(i) * f(x(i))
</pre>
</p>
<p>
When using a Gauss-Laguerre quadrature rule, it's important to know whether
the rule has been developed for the standard or modified cases.
Basically, the only change is that the weights of the modified rule have
been multiplied by an exponential factor evaluated at the corresponding abscissa.
</p>
<p>
For a standard Gauss-Laguerre 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 ( 0 <= x < +oo ) exp(-x) f(x) dx
</pre>
</p>
<p>
For a modified Gauss-Laguerre rule, polynomial exactness is defined in terms of
the function <b>f(x)</b> divided by the implicit weight function. That is,
we say a modified Gauss-Laguerre rule is exact for polynomials up to degree
DEGREE_MAX if, for any integrand <b>f(x)</b> with the property that
<b>exp(+x) * f(x)</b> is a polynomial of degree no more than DEGREE_MAX,
the quadrature rule will product the exact value of:
<pre>
Integral ( 0 <= x < +oo ) 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. The program
uses a scaling factor on each monomial so that the exact integral
should always be 1; therefore, each reported error can be compared
on a fixed scale.
</p>
<p>
If the program understands that the rule being considered is a modified rule,
then the monomials are multiplied by <b>exp(-x)</b> when performing the
exactness test.
</p>
<p>
Since
<pre>
Integral ( 0 <= x < +oo ) exp(-x) x<sup>n</sup> dx = n!
</pre>
our test monomial functions, in order to integrate to 1, will be normalized to:
<pre>
Integral ( 0 <= x < +oo ) exp(-x) x<sup>n</sup> / n! dx
</pre>
It should be clear that accuracy will be rapidly lost as <b>n</b> increases.
</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_laguerre</b> <i>prefix</i> <i>degree_max</i> <i>option</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>option</i> is:<br>
0: a standard rule for integrating exp(-x)*f(x).<br>
1: a modified rule for integrating f(x).
</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_LAGUERRE</b> is available in
<a href = "../../cpp_src/int_exactness_laguerre/int_exactness_laguerre.html">a C++ version</a> and
<a href = "../../f_src/int_exactness_laguerre/int_exactness_laguerre.html">a FORTRAN90 version</a> and
<a href = "../../m_src/int_exactness_laguerre/int_exactness_laguerre.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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_gegenbauer/int_exactness_gegenbauer.html">
INT_EXACTNESS_GEGENBAUER</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Gegenbauer 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_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 = "../../f_src/intlib/intlib.html">
INTLIB</a>,
a FORTRAN90 library which
numerically estimate integrals in one dimension.
</p>
<p>
<a href = "../../f_src/laguerre_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a FORTRAN90 program which
generates a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../f_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a FORTRAN90 program which
tests the polynomial exactness of multidimensional integration rules.
</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_laguerre/quadrature_rules_laguerre.html">
QUADRATURE_RULES_LAGUERRE</a>,
a dataset directory which
contains sets of files that define Gauss-Laguerre 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>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
defines quadrature rules for a variety of unusual areas, surfaces
and volumes in 2D, 3D and multiple dimensions.
</p>
<p>
<a href = "../../f_src/test_int_laguerre/test_int_laguerre.html">
TEST_INT_LAGUERRE</a>,
a FORTRAN90 library which
defines integrand functions that can be approximately integrated by a Gauss-Laguerre rule.
</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>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "int_exactness_laguerre.f90">int_exactness_laguerre.f90</a>, the source code.
</li>
<li>
<a href = "int_exactness_laguerre.sh">int_exactness_laguerre.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>LAG_O1</b> is a standard Gauss-Laguerre order 1 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o1_x.txt">
lag_o1_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o1_w.txt">
lag_o1_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o1_r.txt">
lag_o1_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o1_exact.txt">lag_o1_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o1 5 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LAG_O2</b> is a standard Gauss-Laguerre order 2 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o2_x.txt">
lag_o2_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o2_w.txt">
lag_o2_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o2_r.txt">
lag_o2_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o2_exact.txt">lag_o2_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o2 5 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LAG_O4</b> is a standard Gauss-Laguerre order 4 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o4_x.txt">
lag_o4_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o4_w.txt">
lag_o4_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o4_r.txt">
lag_o4_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o4_exact.txt">lag_o4_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o4 10 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LAG_O8</b> is a standard Gauss-Laguerre order 8 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o8_x.txt">
lag_o8_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o8_w.txt">
lag_o8_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o8_r.txt">
lag_o8_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o8_exact.txt">lag_o8_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o8 18 0
</b></blockquote>
</ul>
</p>
<p>
<b>LAG_O16</b> is a standard Gauss-Laguerre order 16 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o16_x.txt">
lag_o16_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o16_w.txt">
lag_o16_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o16_r.txt">
lag_o16_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o16_exact.txt">lag_o16_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o16 35 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LAG_O1_MODIFIED</b> is a modified Gauss-Laguerre order 1 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o1_modified_x.txt">
lag_o1_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o1_modified_w.txt">
lag_o1_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o1_modified_r.txt">
lag_o1_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o1_modified_exact.txt">lag_o1_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o1_modified 5 1
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LAG_O2_MODIFIED</b> is a modified Gauss-Laguerre order 2 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o2_modified_x.txt">
lag_o2_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o2_modified_w.txt">
lag_o2_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o2_modified_r.txt">
lag_o2_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o2_modified_exact.txt">lag_o2_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o2_modified 5 1
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LAG_O4_MODIFIED</b> is a modified Gauss-Laguerre order 4 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o4_modified_x.txt">
lag_o4_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o4_modified_w.txt">
lag_o4_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o4_modified_r.txt">
lag_o4_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o4_modified_exact.txt">lag_o4_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o4_modified 10 1
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LAG_O8_MODIFIED</b> is a modified Gauss-Laguerre order 8 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o8_modified_x.txt">
lag_o8_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o8_modified_w.txt">
lag_o8_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o8_modified_r.txt">
lag_o8_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o8_modified_exact.txt">lag_o8_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o8_modified 18 1
</b></blockquote>
</ul>
</p>
<p>
<b>LAG_O16_MODIFIED</b> is a modified Gauss-Laguerre order 16 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o16_modified_x.txt">
lag_o16_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o16_modified_w.txt">
lag_o16_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_laguerre/lag_o16_modified_r.txt">
lag_o16_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o16_modified_exact.txt">lag_o16_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_laguerre lag_o16_modified 35 1
</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_LAGUERRE.
</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>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>LAGUERRE_INTEGRAL</b> evaluates a monomial Laguerre integral.
</li>
<li>
<b>MONOMIAL_QUADRATURE_LAGUERRE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N, also denoted "N!".
</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 27 January 2010.
</i>
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