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<html>
<head>
<title>
INT_EXACTNESS_GEN_LAGUERRE - Exactness of Generalized Gauss-Laguerre Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
INT_EXACTNESS_GEN_LAGUERRE <br> Exactness of Generalized Gauss-Laguerre Quadrature Rules
</h1>
<hr>
<p>
<b>INT_EXACTNESS_GEN_LAGUERRE</b>
is a C++ program which
investigates the polynomial exactness of a generalized Gauss-Laguerre
quadrature rule for the semi-infinite interval [0,+oo) or [A,+oo).
</p>
<p>
Standard generalized Gauss-Laguerre quadrature assumes that the integrand we are
considering has a form like:
<pre>
Integral ( A <= x < +oo ) x^alpha * exp(-x) * f(x) dx
</pre>
where the factor <b>x^alpha * exp(-x)</b> is regarded as a weight factor.
</p>
<p>
A <i>standard generalized 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 ) x^alpha * 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>x^alpha * 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 generalized Gauss-Laguerre quadrature rule
to handle this case directly.
</p>
<p>
A <i>modified generalized 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 generalized 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 divided by the weighting function evaluated at the corresponding abscissa.
</p>
<p>
For a standard generalized 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 ) x^alpha * exp(-x) * f(x) dx
</pre>
</p>
<p>
For a modified generalized Gauss-Laguerre rule, polynomial exactness is defined in terms of
the function <b>f(x)</b> divided by the implicit weighting function. That is,
we say a modified generalized 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>f(x)/(x^alpha*exp(-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>x^alpha * exp(-x)</b> when performing the
exactness test.
</p>
<p>
Since
<pre>
Integral ( 0 <= x < +oo ) x^alpha * exp(-x) * x<sup>n</sup> dx = gamma(n+alpha+1)
</pre>
our test monomial functions, in order to integrate to 1, will be normalized to:
<pre>
Integral ( 0 <= x < +oo ) x^alpha * exp(-x) x<sup>n</sup> / gamma(n+alpha+1) 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_gen_laguerre</b> <i>prefix</i> <i>degree_max</i> <i>alpha</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>alpha</i> is the value of the parameter, which should be a real number greater than -1.
Setting <i>alpha</i> to 0.0 results in the basic (non-generalized) Gauss-Laguerre rule.
</li>
<li>
<i>option</i>:<br>
0 indicates a standard rule for integrating x^alpha*exp(-x)*f(x).<br>
1 indicates 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_GEN_LAGUERRE</b> is available in
<a href = "../../cpp_src/int_exactness_gen_laguerre/int_exactness_gen_laguerre.html">a C++ version</a> and
<a href = "../../f_src/int_exactness_gen_laguerre/int_exactness_gen_laguerre.html">a FORTRAN90 version</a> and
<a href = "../../m_src/int_exactness_gen_laguerre/int_exactness_gen_laguerre.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a C++ program which
can generate a generalized Gauss-Laguerre quadrature
rule on request.
</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.
</p>
<p>
<a href = "../../cpp_src/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a C++ program which
tests the polynomial exactness of a quadrature rule for a finite interval.
</p>
<p>
<a href = "../../cpp_src/int_exactness_chebyshev1/int_exactness_chebyshev1.html">
INT_EXACTNESS_CHEBYSHEV1</a>,
a C++ program which
tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.
</p>
<p>
<a href = "../../cpp_src/int_exactness_chebyshev2/int_exactness_chebyshev2.html">
INT_EXACTNESS_CHEBYSHEV2</a>,
a C++ program which
tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.
</p>
<p>
<a href = "../../cpp_src/int_exactness_gegenbauer/int_exactness_gegenbauer.html">
INT_EXACTNESS_GEGENBAUER</a>,
a C++ program which
tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.
</p>
<p>
<a href = "../../cpp_src/int_exactness_gen_hermite/int_exactness_gen_hermite.html">
INT_EXACTNESS_GEN_HERMITE</a>,
a C++ program which
tests the polynomial exactness of a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../cpp_src/int_exactness_jacobi/int_exactness_jacobi.html">
INT_EXACTNESS_JACOBI</a>,
a C++ program which
tests the polynomial exactness of a Gauss-Jacobi quadrature rule.
</p>
<p>
<a href = "../../cpp_src/laguerre_exactness/laguerre_exactness.html">
LAGUERRE_EXACTNESS</a>,
a C++ program which
tests the polynomial exactness of Gauss-Laguerre quadrature rules
for integration over [0,+oo) with density function exp(-x).
</p>
<p>
<a href = "../../cpp_src/legendre_exactness/legendre_exactness.html">
LEGENDRE_EXACTNESS</a>,
a C++ program which
tests the monomial exactness of quadrature rules for the Legendre problem
of integrating a function with density 1 over the interval [-1,+1].
</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_gen_laguerre.cpp">int_exactness_gen_laguerre.cpp</a>, the source code.
</li>
<li>
<a href = "int_exactness_gen_laguerre.sh">int_exactness_gen_laguerre.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>GEN_LAG_O1_A0.5</b> is a standard generalized Gauss-Laguerre order 1 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o1_a0.5_x.txt">
gen_lag_o1_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o1_a0.5_w.txt">
gen_lag_o1_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o1_a0.5_r.txt">
gen_lag_o1_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_lag_o1_a0.5_exact.txt">gen_lag_o1_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o1_a0.5 5 0.5 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_LAG_O2_A0.5</b> is a standard generalized Gauss-Laguerre order 2 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o2_a0.5_x.txt">
gen_lag_o2_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o2_a0.5_w.txt">
gen_lag_o2_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o2_a0.5_r.txt">
gen_lag_o2_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_lag_o2_a0.5_exact.txt">gen_lag_o2_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o2_a0.5 5 0.5 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_LAG_O4_A0.5</b> is a standard generalized Gauss-Laguerre order 4 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o4_a0.5_x.txt">
gen_lag_o4_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o4_a0.5_w.txt">
gen_lag_o4_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o4_a0.5_r.txt">
gen_lag_o4_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_lag_o4_a0.5_exact.txt">gen_lag_o4_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o4_a0.5 10 0.5 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_LAG_O8_A0.5</b> is a standard generalized Gauss-Laguerre order 8 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o8_a0.5_x.txt">
gen_lag_o8_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o8_a0.5_w.txt">
gen_lag_o8_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o8_a0.5_r.txt">
gen_lag_o8_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_lag_o8_a0.5_exact.txt">gen_lag_o8_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o8_a0.5 18 0.5 0
</b></blockquote>
</ul>
</p>
<p>
<b>GEN_LAG_O16_A0.5</b> is a standard generalized Gauss-Laguerre order 16 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o16_a0.5_x.txt">
gen_lag_o16_a0.5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o16_a0.5_w.txt">
gen_lag_o16_a0.5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o16_a0.5_r.txt">
gen_lag_o16_a0.5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_lag_o16_a0.5_exact.txt">gen_lag_o16_a0.5_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o16_a0.5 35 0.5 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_LAG_O1_A0.5_MODIFIED</b> is a modified generalized Gauss-Laguerre order 1 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o1_a0.5_modified_x.txt">
gen_lag_o1_a0.5_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o1_a0.5_modified_w.txt">
gen_lag_o1_a0.5_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o1_a0.5_modified_r.txt">
gen_lag_o1_a0.5_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_lag_o1_a0.5_modified_exact.txt">gen_lag_o1_a0.5_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o1_a0.5_modified 5 0.5 1
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_LAG_O2_A0.5_MODIFIED</b> is a modified generalized Gauss-Laguerre order 2 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o2_a0.5_modified_x.txt">
gen_lag_o2_a0.5_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o2_a0.5_modified_w.txt">
gen_lag_o2_a0.5_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o2_a0.5_modified_r.txt">
gen_lag_o2_a0.5_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_lag_o2_a0.5_modified_exact.txt">gen_lag_o2_a0.5_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o2_a0.5_modified 5 0.5 1
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_LAG_O4_A0.5_MODIFIED</b> is a modified generalized Gauss-Laguerre order 4 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o4_a0.5_modified_x.txt">
gen_lag_o4_a0.5_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o4_a0.5_modified_w.txt">
gen_lag_o4_a0.5_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o4_a0.5_modified_r.txt">
gen_lag_o4_a0.5_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "lag_o4_a0.5_modified_exact.txt">gen_lag_o4_a0.5_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o4_a0.5_modified 10 0.5 1
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_LAG_O8_A0.5_MODIFIED</b> is a modified generalized Gauss-Laguerre order 8 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o8_a0.5_modified_x.txt">
gen_lag_o8_a0.5_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o8_a0.5_modified_w.txt">
gen_lag_o8_a0.5_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o8_a0.5_modified_r.txt">
gen_lag_o8_a0.5_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_lag_o8_a0.5_modified_exact.txt">gen_lag_o8_a0.5_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o8_a0.5_modified 18 0.5 1
</b></blockquote>
</ul>
</p>
<p>
<b>GEN_LAG_O16_A0.5_MODIFIED</b> is a modified generalized Gauss-Laguerre order 16 rule with ALPHA = 0.5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o16_a0.5_modified_x.txt">
gen_lag_o16_a0.5_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o16_a0.5_modified_w.txt">
gen_lag_o16_a0.5_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_laguerre/gen_lag_o16_a0.5_modified_r.txt">
gen_lag_o16_a0.5_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_lag_o16_a0.5_modified_exact.txt">gen_lag_o16_a0.5_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_laguerre gen_lag_o16_a0.5_modified 35 0.5 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_GEN_LAGUERRE.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>DTABLE_DATA_READ</b> reads the data from a DTABLE file.
</li>
<li>
<b>DTABLE_HEADER_READ</b> reads the header from a DTABLE file.
</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>MONOMIAL_QUADRATURE_GEN_LAGUERRE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</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>
<li>
<b>TIMESTRING</b> returns the current YMDHMS date as a string.
</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 11 February 2008.
</i>
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