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<html>
<head>
<title>
QUADRULE - Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
QUADRULE <br> Quadrature Rules
</h1>
<hr>
<p>
<b>QUADRULE</b>
is a FORTRAN90 library which
sets up a variety of
quadrature rules, used to approximate the integral of a function
over various domains.
</p>
<p>
<b>QUADRULE</b> returns the abscissas and weights for a variety of
one dimensional quadrature rules for approximating the integral
of a function. The best rule is generally Gauss-Legendre quadrature,
but other rules offer special features, including the ability to
handle certain weight functions, to approximate an integral
on an infinite integration region, or to estimate the approximation
error.
</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>QUADRULE</b> is available in
<a href = "../../c_src/quadrule/quadrule.html">a C version</a> and
<a href = "../../cpp_src/quadrule/quadrule.html">a C++ version</a> and
<a href = "../../f77_src/quadrule/quadrule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/quadrule/quadrule.html">a FORTRAN90 version</a> and
<a href = "../../math_src/quadrule/quadrule.html">a MATHEMATICA version</a> and
<a href = "../../m_src/quadrule/quadrule.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<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/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a FORTRAN90 program which
checks the polynomial exactness
of a 1-dimensional quadrature rule for a finite interval.
</p>
<p>
<a href = "../../f_src/int_exactness_laguerre/int_exactness_laguerre.html">
INT_EXACTNESS_LAGUERRE</a>,
a FORTRAN90 program which
checks the polynomial exactness
of a 1-dimensional quadrature rule for a semi-infinite interval.
</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/intlib/intlib.html">
INTLIB</a>,
a FORTRAN90 library which
contains a variety
of routines for numerical estimation of integrals in 1D.
</p>
<p>
<a href = "../../f_src/kronrod/kronrod.html">
KRONROD</a>,
a FORTRAN90 library which
can compute a Gauss and Gauss-Kronrod pair of quadrature rules
of arbitrary order,
by Robert Piessens, Maria Branders.
</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/quadpack/quadpack.html">
QUADPACK</a>,
a FORTRAN90 library which
contains a variety of 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 = "../../f_src/quadrature_test/quadrature_test.html">
QUADRATURE_TEST</a>,
a FORTRAN90 program which
reads the definition of a
multidimensional quadrature rule from three files, applies
the rule to a number of test integrals, and prints the
results.
</p>
<p>
<a href = "../../f_src/quadrature_weights/quadrature_weights.html">
QUADRATURE_WEIGHTS</a>,
a FORTRAN90 library which
illustrates techniques for computing the weights of a quadrature
rule, assuming that the points have been specified.
</p>
<p>
<a href = "../../f_src/r16_hermite_rule/r16_hermite_rule.html">
R16_HERMITE_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Hermite quadrature rule, using
"quadruple precision real" arithmetic.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
contains quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../f_src/tanh_quad/tanh_quad.html">
TANH_QUAD</a>,
a FORTRAN90 library which
sets up the tanh quadrature rule;
</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 a number of functions that may be used as test integrands for
quadrature rules in 1D.
</p>
<p>
<a href = "../../f_src/test_int_hermite/test_int_hermite.html">
TEST_INT_HERMITE</a>,
a FORTRAN90 library which
defines some test integration problems over infinite intervals.
</p>
<p>
<a href = "../../f_src/test_int_laguerre/test_int_laguerre.html">
TEST_INT_LAGUERRE</a>,
a FORTRAN90 library which
defines test integrands for integration over [ALPHA,+Infinity).
</p>
<p>
<a href = "../../f77_src/toms351/toms351.html">
TOMS351</a>,
a FORTRAN77 library which
estimates an integral using Romberg integration.
</p>
<p>
<a href = "../../f77_src/toms379/toms379.html">
TOMS379</a>,
a FORTRAN77 library which
estimates an integral.
</p>
<p>
<a href = "../../f77_src/toms418/toms418.html">
TOMS418</a>,
a FORTRAN77 library which
estimates the integral of a function
with a sine or cosine factor.
</p>
<p>
<a href = "../../f77_src/toms424/toms424.html">
TOMS424</a>,
a FORTRAN77 library which
estimates the integral of a function
using Clenshaw-Curtis quadrature.
</p>
<p>
<a href = "../../f77_src/toms468/toms468.html">
TOMS468</a>,
a FORTRAN77 library which
carries out the "automatic" integration of a function.
</p>
<p>
<a href = "../../f_src/toms655/toms655.html">
TOMS655</a>,
a FORTRAN90 library which
computes the weights for interpolatory quadrature rule;<br>
this library is commonly called <b>IQPACK</b>;<br>
this is a FORTRAN90 version of ACM TOMS algorithm 655.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
Donald Anderson,<br>
Gaussian Quadrature Formulae for the integral from 0 to 1 of
-ln(X) f(X) dx,<br>
Mathematics of Computation,<br>
Volume 19, Number 91, July 1965, pages 477-481.
</li>
<li>
Claudio Canuto, Yousuff Hussaini, Alfio Quarteroni, Thomas Zang,<br>
Spectral Methods in Fluid Dynamics,<br>
Springer, 1993,<br>
ISNB13: 978-3540522058,<br>
LC: QA377.S676.
</li>
<li>
Charles Clenshaw, Alan Curtis,<br>
A Method for Numerical Integration on an Automatic Computer,<br>
Numerische Mathematik,<br>
Volume 2, Number 1, December 1960, pages 197-205.
</li>
<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>
Sylvan Elhay, Jaroslav Kautsky,<br>
Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of
Interpolatory Quadrature,<br>
ACM Transactions on Mathematical Software,<br>
Volume 13, Number 4, December 1987, pages 399-415.
</li>
<li>
Hermann Engels,<br>
Numerical Quadrature and Cubature,<br>
Academic Press, 1980,<br>
ISBN: 012238850X,<br>
LC: QA299.3E5.
</li>
<li>
Gwynne Evans,<br>
Practical Numerical Integration,<br>
Wiley, 1993,<br>
ISBN: 047193898X,<br>
LC: QA299.3E93.
</li>
<li>
Simeon Fatunla,<br>
Numerical Methods for Initial Value Problems in Ordinary
Differential Equations,<br>
Academic Press, 1988,<br>
ISBN: 0122499301,<br>
LC: QA372.F35.
</li>
<li>
Walter Gautschi,<br>
Numerical Quadrature in the Presence of a Singularity,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 4, Number 3, September 1967, pages 357-362.
</li>
<li>
Alan Genz, Bradley Keister,<br>
Fully symmetric interpolatory rules for multiple integrals
over infinite regions with Gaussian weight,<br>
Journal of Computational and Applied Mathematics,<br>
Volume 71, 1996, pages 299-309.
</li>
<li>
Florian Heiss, Viktor Winschel,<br>
Likelihood approximation by numerical integration on sparse grids,<br>
Journal of Econometrics,<br>
Volume 144, 2008, pages 62-80.
</li>
<li>
Francis Hildebrand, <br>
Introduction to Numerical Analysis,<br>
Dover, 1987,<br>
ISBN13: 978-0486653631,<br>
LC: QA300.H5.
</li>
<li>
Zdenek Kopal,<br>
Numerical Analysis,<br>
John Wiley, 1955,<br>
LC: QA297.K6.
</li>
<li>
Vladimir Krylov,<br>
Approximate Calculation of Integrals,<br>
Dover, 2006,<br>
ISBN: 0486445798,<br>
LC: QA311.K713.
</li>
<li>
Prem Kythe, Michael Schaeferkotter,<br>
Handbook of Computational Methods for Integration,<br>
Chapman and Hall, 2004,<br>
ISBN: 1-58488-428-2,<br>
LC: QA299.3.K98.
</li>
<li>
Leon Lapidus, John Seinfeld,<br>
Numerical Solution of Ordinary Differential Equations,<br>
Mathematics in Science and Engineering, Volume 74,<br>
Academic Press, 1971,<br>
ISBN: 0124366503,<br>
LC: QA3.M32.v74
</li>
<li>
Federico Paris, Jose Canas,<br>
Boundary Element Method: Fundamentals and Applications,<br>
Oxford, 1997,<br>
ISBN: 0-19-856543-7<br>
LC: TA347.B69.P34.
</li>
<li>
Thomas Patterson,<br>
The Optimal Addition of Points to Quadrature Formulae,<br>
Mathematics of Computation,<br>
Volume 22, Number 104, October 1968, pages 847-856.
</li>
<li>
Robert Piessens, Elise deDoncker-Kapenga,
Christian Ueberhuber, David Kahaner,<br>
QUADPACK: A Subroutine Package for Automatic Integration,<br>
Springer, 1983,<br>
ISBN: 3540125531,<br>
LC: QA299.3.Q36.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Lloyd Trefethen,<br>
Is Gauss Quadrature Better Than Clenshaw-Curtis?,<br>
SIAM Review,<br>
Volume 50, Number 1, March 2008, pages 67-87.
</li>
<li>
Joerg Waldvogel,<br>
Fast Construction of the Fejer and Clenshaw-Curtis
Quadrature Rules,<br>
BIT Numerical Mathematics,<br>
Volume 43, Number 1, 2003, pages 1-18.
</li>
<li>
Stephen Wolfram,<br>
The Mathematica Book,<br>
Fourth Edition,<br>
Cambridge University Press, 1999,<br>
ISBN: 0-521-64314-7,<br>
LC: QA76.95.W65.
</li>
<li>
Daniel Zwillinger, editor,<br>
CRC Standard Mathematical Tables and Formulae,<br>
30th Edition,<br>
CRC Press, 1996,<br>
ISBN: 0-8493-2479-3,<br>
LC: QA47.M315.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "quadrule.f90">quadrule.f90</a>, the source code;
</li>
<li>
<a href = "quadrule.sh">quadrule.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "quadrule_prb.f90">quadrule_prb.f90</a>, the calling
program;
</li>
<li>
<a href = "quadrule_prb.sh">quadrule_prb.sh</a>,
commands to compile, link and run the calling program;
</li>
<li>
<a href = "quadrule_prb_output.txt">quadrule_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BASHFORTH_SET</b> sets abscissas and weights for Adams-Bashforth quadrature.
</li>
<li>
<b>BDF_SET</b> sets weights for ODE backward differentiation.
</li>
<li>
<b>BDF_SUM:</b> an explicit backward difference quadrature rule for [0,1].
</li>
<li>
<b>BDFC_SET</b> sets weights for backward differentiation corrector quadrature.
</li>
<li>
<b>BDFP_SET</b> sets weights for backward differentiation predictor quadrature.
</li>
<li>
<b>CHEB_SET</b> sets abscissas and weights for Chebyshev quadrature.
</li>
<li>
<b>CHEBYSHEV1_COMPUTE</b> computes a Gauss-Chebyshev type 1 quadrature rule.
</li>
<li>
<b>CHEBYSHEV1_INTEGRAL</b> evaluates a monomial Chebyshev type 1 integral.
</li>
<li>
<b>CHEBYSHEV2_COMPUTE</b> computes a Gauss-Chebyshev type 2 quadrature rule.
</li>
<li>
<b>CHEBYSHEV2_INTEGRAL</b> evaluates a monomial Chebyshev type 2 integral.
</li>
<li>
<b>CHEBYSHEV3_COMPUTE</b> computs a Gauss-Chebyshev type 3 quadrature rule.
</li>
<li>
<b>CLENSHAW_CURTIS_COMPUTE</b> computes a Clenshaw Curtis quadrature rule.
</li>
<li>
<b>CLENSHAW_CURTIS_SET</b> sets a Clenshaw-Curtis quadrature rule.
</li>
<li>
<b>FEJER1_COMPUTE</b> computes a Fejer type 1 quadrature rule.
</li>
<li>
<b>FEJER1_SET</b> sets abscissas and weights for Fejer type 1 quadrature.
</li>
<li>
<b>FEJER2_COMPUTE</b> computes a Fejer type 2 quadrature rule.
</li>
<li>
<b>FEJER2_SET</b> sets abscissas and weights for Fejer type 2 quadrature.
</li>
<li>
<b>GEGENBAUER_COMPUTE</b> computes a Gauss-Gegenbauer quadrature rule.
</li>
<li>
<b>GEGENBAUER_INTEGRAL:</b> integral of a monomial with Gegenbauer weight.
</li>
<li>
<b>GEGENBAUER_RECUR</b> finds the value and derivative of a Gegenbauer polynomial.
</li>
<li>
<b>GEGENBAUER_ROOT</b> improves an approximate root of a Gegenbauer polynomial.
</li>
<li>
<b>GEN_HERMITE_DR_COMPUTE</b> computes a generalized Gauss-Hermite rule.
</li>
<li>
<b>GEN_HERMITE_EK_COMPUTE</b> computes a generalized Gauss-Hermite quadrature rule.
</li>
<li>
<b>GEN_HERMITE_INTEGRAL</b> evaluates a monomial generalized Hermite integral.
</li>
<li>
<b>GEN_LAGUERRE_EK_COMPUTE:</b> generalized Gauss-Laguerre quadrature rule.
</li>
<li>
<b>GEN_LAGUERRE_INTEGRAL</b> evaluates a monomial genearlized Laguerre integral.
</li>
<li>
<b>GEN_LAGUERRE_SS_COMPUTE:</b> generalized Gauss-Laguerre quadrature rule.
</li>
<li>
<b>GEN_LAGUERRE_SS_RECUR</b> evaluates a generalized Laguerre polynomial.
</li>
<li>
<b>GEN_LAGUERRE_SS_ROOT</b> seeks roots of a generalized Laguerre polynomial.
</li>
<li>
<b>HERMITE_EK_COMPUTE</b> computes a Gauss-Hermite quadrature rule.
</li>
<li>
<b>HERMITE_GK16_SET</b> sets a Hermite Genz-Keister 16 rule.
</li>
<li>
<b>HERMITE_GK18_SET</b> sets a Hermite Genz-Keister 18 rule.
</li>
<li>
<b>HERMITE_GK22_SET</b> sets a Hermite Genz-Keister 22 rule.
</li>
<li>
<b>HERMITE_GK24_SET</b> sets a Hermite Genz-Keister 24 rule.
</li>
<li>
<b>HERMITE_INTEGRAL</b> evaluates a monomial Hermite integral.
</li>
<li>
<b>HERMITE_SET</b> sets abscissas and weights for Hermite quadrature.
</li>
<li>
<b>HERMITE_SS_COMPUTE</b> computes a Gauss-Hermite quadrature rule.
</li>
<li>
<b>HERMITE_SS_RECUR</b> finds the value and derivative of a Hermite polynomial.
</li>
<li>
<b>HERMITE_SS_ROOT</b> improves an approximate root of a Hermite polynomial.
</li>
<li>
<b>IMTQL2</b> computes all eigenvalues/vectors of a symmetric tridiagonal matrix.
</li>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>JACOBI_EK_COMPUTE:</b> Elhay-Kautsky method for Gauss-Jacobi quadrature rule.
</li>
<li>
<b>JACOBI_INTEGRAL</b> evaluates the integral of a monomial with Jacobi weight.
</li>
<li>
<b>JACOBI_SS_COMPUTE</b> computes a Gauss-Jacobi quadrature rule.
</li>
<li>
<b>JACOBI_SS_RECUR</b> finds the value and derivative of a Jacobi polynomial.
</li>
<li>
<b>JACOBI_SS_ROOT</b> improves an approximate root of a Jacobi polynomial.
</li>
<li>
<b>KRONROD_SET</b> sets abscissas and weights for Gauss-Kronrod quadrature.
</li>
<li>
<b>LAGUERRE_EK_COMPUTE:</b> Laguerre quadrature rule by the Elhay-Kautsky method.
</li>
<li>
<b>LAGUERRE_INTEGRAL</b> evaluates a monomial Laguerre integral.
</li>
<li>
<b>LAGUERRE_SET</b> sets abscissas and weights for Laguerre quadrature.
</li>
<li>
<b>LAGUERRE_SS_COMPUTE</b> computes a Gauss-Laguerre quadrature rule.
</li>
<li>
<b>LAGUERRE_SS_RECUR</b> finds the value and derivative of a Laguerre polynomial.
</li>
<li>
<b>LAGUERRE_SS_ROOT</b> improves an approximate root of a Laguerre polynomial.
</li>
<li>
<b>LAGUERRE_SUM</b> carries out Laguerre quadrature over [ A, +oo ).
</li>
<li>
<b>LEGENDRE_COS2_SET</b> sets a Gauss-Legendre rule for COS(X) * F(X) on [0,PI/2].
</li>
<li>
<b>LEGENDRE_COS_SET:</b> Gauss-Legendre rule for COS(X) * F(X) on [-PI/2,PI/2].
</li>
<li>
<b>LEGENDRE_DR_COMPUTE:</b> Gauss-Legendre quadrature by Davis-Rabinowitz method.
</li>
<li>
<b>LEGENDRE_EK_COMPUTE:</b> Legendre quadrature rule by the Elhay-Kautsky method.
</li>
<li>
<b>LEGENDRE_GW_COMPUTE:</b> Legendre quadrature rule by the Golub-Welsch method.
</li>
<li>
<b>LEGENDRE_INTEGRAL</b> evaluates a monomial Legendre integral.
</li>
<li>
<b>LEGENDRE_LOG_COMPUTE:</b> Gauss-Legendre rules for - LOG(X) * F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_LOG_ROOTS</b> is a root finder for LEGENDRE_LOG_COMPUTE.
</li>
<li>
<b>LEGENDRE_LOG_SET</b> sets a Gauss-Legendre rule for - LOG(X) * F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_LOG_VALUE</b> evaluates a polynomial for LEGENDRE_LOG_ROOT.
</li>
<li>
<b>LEGENDRE_POLYNOMIAL_VALUE</b> evaluates a Legendre polynomial.
</li>
<li>
<b>LEGENDRE_RECUR</b> finds the value and derivative of a Legendre polynomial.
</li>
<li>
<b>LEGENDRE_ROOT</b> improves an approximate root of a Legendre polynomial.
</li>
<li>
<b>LEGENDRE_SET</b> sets abscissas and weights for Gauss-Legendre quadrature.
</li>
<li>
<b>LEGENDRE_SQRTX2_01_SET:</b> Gauss-Legendre rule for F(X) / SQRT(X) on [0,1].
</li>
<li>
<b>LEGENDRE_SQRTX_01_SET</b> sets a Gauss-Legendre rule for SQRT(X) * F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_SS_COMPUTE:</b> Gauss-Legendre quadrature by Stroud-Secrest method.
</li>
<li>
<b>LEGENDRE_SS_RECUR:</b> value and derivative of a scaled Legendre polynomial.
</li>
<li>
<b>LEGENDRE_SS_ROOT:</b> improve approximate root of scaled Legendre polynomial.
</li>
<li>
<b>LEGENDRE_X0_01_SET</b> sets a Gauss-Legendre rule for F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_X1_01_SET</b> sets a Gauss-Legendre rule for X * F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_X1_SET</b> sets a Gauss-Legendre rule for ( 1 + X ) * F(X) on [-1,1].
</li>
<li>
<b>LEGENDRE_X2_01_SET</b> sets a Gauss-Legendre rule for X * X * F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_X2_SET</b> sets a Gauss-Legendre rule for ( 1 + X )^2 * F(X) on [-1,1].
</li>
<li>
<b>LOBATTO_COMPUTE</b> computes a Lobatto quadrature rule.
</li>
<li>
<b>LOBATTO_SET</b> sets abscissas and weights for Lobatto quadrature.
</li>
<li>
<b>MOULTON_SET</b> sets weights for Adams-Moulton quadrature.
</li>
<li>
<b>NC_COMPUTE</b> computes a Newton-Cotes quadrature rule.
</li>
<li>
<b>NCC_COMPUTE:</b> Newton-Cotes Closed quadrature rule.
</li>
<li>
<b>NCC_COMPUTE_POINTS:</b> Newton-Cotes Closed points
</li>
<li>
<b>NCC_COMPUTE_WEIGHTS:</b> Newton-Cotes Closed weights.
</li>
<li>
<b>NCC_SET</b> sets abscissas and weights for Newton-Cotes closed quadrature.
</li>
<li>
<b>NCO_COMPUTE</b> computes a Newton-Cotes Open quadrature rule.
</li>
<li>
<b>NCO_COMPUTE_POINTS:</b> points for a Newton-Cotes Open quadrature rule.
</li>
<li>
<b>NCO_COMPUTE_WEIGHTS:</b> weights for a Newton-Cotes Open quadrature rule.
</li>
<li>
<b>NCO_SET</b> sets abscissas and weights for open Newton-Cotes quadrature.
</li>
<li>
<b>NCOH_COMPUTE</b> computes a Newton-Cotes Open Half quadrature rule.
</li>
<li>
<b>NCOH_COMPUTE_POINTS:</b> points for a Newton-Cotes Open Half quadrature rule.
</li>
<li>
<b>NCOH_COMPUTE_WEIGHTS:</b> weights for a Newton-Cotes Open Half quadrature rule.
</li>
<li>
<b>NCOH_SET</b> sets abscissas and weights for Newton-Cotes "open half" quadrature.
</li>
<li>
<b>PATTERSON_SET</b> sets abscissas and weights for Gauss-Patterson quadrature.
</li>
<li>
<b>PYTHAG</b> computes SQRT ( A * A + B * B ) carefully.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function of N.
</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>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_REVERSE</b> reverses the elements of an R8VEC.
</li>
<li>
<b>RADAU_COMPUTE</b> computes a Radau quadrature rule.
</li>
<li>
<b>RADAU_SET</b> sets abscissas and weights for Radau quadrature.
</li>
<li>
<b>RULE_ADJUST</b> maps a quadrature rule from [A,B] to [C,D].
</li>
<li>
<b>SUM_SUB</b> carries out a composite quadrature rule.
</li>
<li>
<b>SUM_SUB_GK</b> carries out a composite Gauss-Kronrod rule.
</li>
<li>
<b>SUMMER</b> carries out a quadrature rule over a single interval.
</li>
<li>
<b>SUMMER_GK</b> carries out Gauss-Kronrod quadrature over a single interval.
</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 03 July 2011.
</i>
<!-- John Burkardt -->
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