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<html>
<head>
<title>
INT_EXACTNESS_LEGENDRE - Exactness of Gauss-Legendre Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
INT_EXACTNESS_LEGENDRE <br> Exactness of Gauss-Legendre Quadrature Rules
</h1>
<hr>
<p>
<b>INT_EXACTNESS_LEGENDRE</b>
is a FORTRAN90 program which
investigates the polynomial exactness of Gauss-Legendre
quadrature rules for the interval [-1,+1].
</p>
<p>
This program is actually appropriate for <i>any</i> quadrature
rule that estimates integrals on [-1,+1], and which does not including
a weighting function <b>w(x)</b> in the integral. This includes:
<ul>
<li>
<i>Clenshaw-Curtis</i> rules;
</li>
<li>
<i>Fejer</i> rules of Type 1 or 2;
</li>
<li>
<i>Gauss-Legendre</i> rules;
</li>
<li>
<i>Gauss-Lobatto</i> rules (Gauss rule including both endpoints);
</li>
<li>
<i>Gauss-Patterson</i> rules;
</li>
<li>
<i>Gauss-Radau</i> rules (Gauss rule including one endpoint);
</li>
<li>
<i>Newton-Cotes</i> rules, open and closed forms;
</li>
</ul>
</p>
<p>
Standard Gauss-Legendre quadrature assumes that the integrand we are
considering has a form like:
<pre>
Integral ( -1 <= x <= +1 ) f(x) dx
</pre>
</p>
<p>
A <i>standard Gauss-Legendre quadrature rule</i> is a set of <b>n</b>
positive weights <b>w</b> and abscissas <b>x</b> so that
<pre>
Integral ( -1 <= x <= +1 ) f(x) dx
</pre>
may be approximated by
<pre>
Sum ( 1 <= I <= N ) w(i) * f(x(i))
</pre>
</p>
<p>
For a standard Gauss-Legendre 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 ) 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. For comparison,
it also computes and prints the error for the trapezoid rule of the same order.
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>
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_legendre</b> <i>prefix</i> <i>degree_max</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>
</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_LEGENDRE</b> is available in
<a href = "../../cpp_src/int_exactness_legendre/int_exactness_legendre.html">a C++ version</a> and
<a href = "../../f_src/int_exactness_legendre/int_exactness_legendre.html">a FORTRAN90 version</a> and
<a href = "../../m_src/int_exactness_legendre/int_exactness_legendre.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_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/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 estimates integrals in one dimension.
</p>
<p>
<a href = "../../f_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a FORTRAN90 program which
generates a Gauss-Legendre quadrature
rule on request.
</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 = "../../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_legendre/quadrature_rules_legendre.html">
QUADRATURE_RULES_LEGENDRE</a>,
a dataset directory which
contains sets of files that define Gauss-Legendre 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/test_int.html">
TEST_INT</a>,
a FORTRAN90 library which
defines integrand functions that can be approximately integrated by a Gauss-Legendre 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_legendre.f90">int_exactness_legendre.f90</a>, the source code.
</li>
<li>
<a href = "int_exactness_legendre.sh">int_exactness_legendre.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>LEG_O1</b> is a standard Gauss-Legendre order 1 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o1_x.txt">
leg_o1_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o1_w.txt">
leg_o1_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o1_r.txt">
leg_o1_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o1_exact.txt">leg_o1_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_legendre leg_o1 5
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LEG_O2</b> is a standard Gauss-Legendre order 2 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o2_x.txt">
leg_o2_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o2_w.txt">
leg_o2_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o2_r.txt">
leg_o2_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o2_exact.txt">leg_o2_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_legendre leg_o2 5
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LEG_O4</b> is a standard Gauss-Legendre order 4 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o4_x.txt">
leg_o4_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o4_w.txt">
leg_o4_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o4_r.txt">
leg_o4_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o4_exact.txt">leg_o4_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_legendre leg_o4 10
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LEG_O8</b> is a standard Gauss-Legendre order 8 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o8_x.txt">
leg_o8_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o8_w.txt">
leg_o8_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o8_r.txt">
leg_o8_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o8_exact.txt">leg_o8_exact.txt</a>,
the results of the exactness test.
</li>
the results of the command
<blockquote><b>
int_exactness_legendre leg_o8 18
</b></blockquote>
</ul>
</p>
<p>
<b>LEG_O16</b> is a standard Gauss-Legendre order 16 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o16_x.txt">
leg_o16_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o16_w.txt">
leg_o16_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o16_r.txt">
leg_o16_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o16_exact.txt">leg_o16_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_legendre leg_o16 35
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LEG_O32</b> is a standard Gauss-Legendre order 32 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o32_x.txt">
leg_o32_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o32_w.txt">
leg_o32_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o32_r.txt">
leg_o32_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o32_exact.txt">leg_o32_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_legendre leg_o32 70
</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_LEGENDRE.
</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>LEGENDRE_INTEGRAL</b> evaluates a monomial Legendre integral.
</li>
<li>
<b>MONOMIAL_QUADRATURE_LEGENDRE</b> applies a quadrature rule to a monomial.
</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 25 January 2008.
</i>
<!-- John Burkardt -->
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