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<html>
<head>
<title>
LEGENDRE_RULE_FAST - Gauss-Legendre Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LEGENDRE_RULE_FAST <br> Gauss-Legendre Quadrature Rules
</h1>
<hr>
<p>
<b>LEGENDRE_RULE_FAST</b>
is a FORTRAN90 program which
implements a fast algorithm for the computation of the points and weights
of the Gauss-Legendre quadrature rule.
</p>
<p>
The standard algorithm for computing the N points and weights of such a rule is
by Golub and Welsch. It sets up and solves an eigenvalue problem, whose
solution requires work of order N*N.
</p>
<p>
By contrast, the fast algorithm, by Glaser, Liu and Rokhlin, can compute
the same information expending work of order N. For quadrature problems
requiring high accuracy, where N might be 100 or more, the fast algorithm
provides a significant improvement in speed.
</p>
<p>
The Gauss-Legendre quadrature rule is designed for the interval [-1,+1].
</p>
<p>
The Gauss-Legendre quadrature assumes that the integrand has the form:
<pre>
Integral ( -1 <= x <= +1 ) f(x) dx
</pre>
</p>
<p>
The <i>standard Gauss-Legendre quadrature rule </i> 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>
This program allows the user to request that the rule be transformed
from the standard interval [-1,+1] to the interval [a,b].
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>legendre_rule_fast</b> <i>n</i> <i>a</i> <i>b</i>
</blockquote>
where
<ul>
<li>
<i>n</i> is the order (number of points);
</li>
<li>
<i>a</i> is the left endpoint (often -1.0 or 0.0);
</li>
<li>
<i>b</i> is the right endpoint (usually 1.0).
</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>LEGENDRE_RULE_FAST</b> is available in
<a href = "../../c_src/legendre_rule_fast/legendre_rule_fast.html">a C version</a> and
<a href = "../../cpp_src/legendre_rule_fast/legendre_rule_fast.html">a C++ version</a> and
<a href = "../../f77_src/legendre_rule_fast/legendre_rule_fast.html">a FORTRAN77 version</a> and
<a href = "../../f_src/legendre_rule_fast/legendre_rule_fast.html">a FORTRAN90 version</a> and
<a href = "../../m_src/legendre_rule_fast/legendre_rule_fast.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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
can compute and print a Gauss-Legendre quadrature rule.
</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_legendre/quadrature_rules_legendre.html">
QUADRATURE_RULES_LEGENDRE</a>,
a dataset directory which
contains triples of files defining standard Gauss-Legendre
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_sinh_rule/tanh_sinh_rule.html">
TANH_SINH_RULE</a>,
a FORTRAN90 program which
computes and writes out a tanh-sinh quadrature rule of given order.
</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>
Andreas Glaser, Xiangtao Liu, Vladimir Rokhlin,<br>
A fast algorithm for the calculation of the roots of special functions,<br>
SIAM Journal on Scientific Computing,<br>
Volume 29, Number 4, pages 1420-1438, 2007.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "legendre_rule_fast.f90">legendre_rule_fast.f90</a>, the source code.
</li>
<li>
<a href = "legendre_rule_fast.sh">legendre_rule_fast.sh</a>,
commands to compile and load the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
The following files were created by the command <b>legendre_rule_fast 15 -1 1</b>:
<ul>
<li>
<a href = "leg_o15_r.txt">leg_o15_r.txt</a>,
the region file.
</li>
<li>
<a href = "leg_o15_w.txt">leg_o15_w.txt</a>,
the weight file.
</li>
<li>
<a href = "leg_o15_x.txt">leg_o15_x.txt</a>,
the abscissa file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for LEGENDRE_RULE_FAST.
</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>DIGIT_TO_CH</b> returns the character representation of a decimal digit.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_TO_S_LEFT</b> converts an I4 to a left-justified string.
</li>
<li>
<b>LEGENDRE_COMPUTE_GLR:</b> Legendre quadrature by the Glaser-Liu-Rokhlin method.
</li>
<li>
<b>LEGENDRE_COMPUTE_GLR0</b> gets a starting value for the fast algorithm.
</li>
<li>
<b>LEGENDRE_COMPUTE_GLR1</b> gets the complete set of Legendre points and weights.
</li>
<li>
<b>LEGENDRE_COMPUTE_GLR2</b> finds the first real root.
</li>
<li>
<b>LEGENDRE_HANDLE</b> computes the requested Gauss-Legendre rule and outputs it.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>RESCALE</b> rescales a Legendre quadrature rule from [-1,+1] to [A,B].
</li>
<li>
<b>RK2_LEG</b> advances the value of X(T) using a Runge-Kutta method.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 value from a string.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TS_MULT...</b>
</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 15 November 2009.
</i>
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