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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 C++ library which
sets up a variety of
quadrature rules, used to approximate the integral of a function
over various domains.
</p>
<p>
QUADRULE 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 = "../../m_src/quadrule/quadrule.html">a MATLAB version</a> and
<a href = "../../py_src/quadrule/quadrule.html">a Python version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../cpp_src/clenshaw_curtis_rule/clenshaw_curtis_rule.html">
CLENSHAW_CURTIS_RULE</a>,
a C++ program which
defines a Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../cpp_src/kronrod/kronrod.html">
KRONROD</a>,
a C++ 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 = "../../cpp_src/line_felippa_rule/line_felippa_rule.html">
LINE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../cpp_src/line_ncc_rule/line_ncc_rule.html">
LINE_NCC_RULE</a>,
a C++ library which
computes a Newton Cotes Closed (NCC) quadrature rule for the line,
that is, for an interval of the form [A,B], using equally spaced points
which include the endpoints.
</p>
<p>
<a href = "../../cpp_src/line_nco_rule/line_nco_rule.html">
LINE_NCO_RULE</a>,
a C++ library which
computes a Newton Cotes Open (NCO) quadrature rule,
using equally spaced points,
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../cpp_src/quadmom/quadmom.html">
QUADMOM</a>,
a C++ library which
computes a Gaussian quadrature rule for a weight function rho(x)
based on the Golub-Welsch procedure that only requires knowledge
of the moments of rho(x).
</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 = "../../m_src/quadrature_test/quadrature_test.html">
QUADRATURE_TEST</a>,
a MATLAB 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 = "../../cpp_src/test_int/test_int.html">
TEST_INT</a>,
a C++ library which
defines test integrands for 1D quadrature rules.
</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>
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>
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.cpp">quadrule.cpp</a>, the source code;
</li>
<li>
<a href = "quadrule.hpp">quadrule.hpp</a>, the include file;
</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.cpp">quadrule_prb.cpp</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 backward differentiation ODE weights.
</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>BDF_SUM</b> carries out explicit backward difference quadrature on [0,1].
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</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> computes 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> evaluates the 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 generalized Laguerre integral.
</li>
<li>
<b>GEN_LAGUERRE_SS_COMPUTE</b> computes a 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> improves a root 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>I4_FACTORIAL2</b> computes the double factorial function.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</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> evaluates a Laguerre polynomial.
</li>
<li>
<b>LAGUERRE_SS_ROOT</b> improves a root of a Laguerre polynomial.
</li>
<li>
<b>LAGUERRE_SUM</b> carries out Laguerre quadrature over [ A, +oo ).
</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_INTEGRAL</b> evaluates a monomial Legendre integral.
</li>
<li>
<b>LEGENDRE_RECUR</b> finds the value and derivative of a Legendre polynomial.
</li>
<li>
<b>LEGENDRE_SET</b> sets abscissas and weights for Gauss-Legendre quadrature.
</li>
<li>
<b>LEGENDRE_SET_COS:</b> Gauss-Legendre rules for COS(X)*F(X) on [-PI/2,PI/2].
</li>
<li>
<b>LEGENDRE_SET_COS2:</b> Gauss-Legendre rules for COS(X)*F(X) on [0,PI/2].
</li>
<li>
<b>LEGENDRE_SET_LOG</b> sets a Gauss-Legendre rule for - LOG(X) * F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_SET_SQRTX_01</b> sets Gauss-Legendre rules for SQRT(X)*F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_SET_SQRTX2_01:</b> Gauss-Legendre rules for F(X)/SQRT(X) on [0,1].
</li>
<li>
<b>LEGENDRE_SET_X0_01</b> sets a Gauss-Legendre rule for F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_SET_X1</b> sets a Gauss-Legendre rule for ( 1 + X ) * F(X) on [-1,1].
</li>
<li>
<b>LEGENDRE_SET_X1_01</b> sets a Gauss-Legendre rule for X * F(X) on [0,1].
</li>
<li>
<b>LEGENDRE_SET_X2</b> sets Gauss-Legendre rules for ( 1 + X )^2*F(X) on [-1,1].
</li>
<li>
<b>LEGENDRE_SET_X2_01</b> sets a Gauss-Legendre rule for X*X * F(X) on [0,1].
</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> computes a Newton-Cotes closed quadrature rule.
</li>
<li>
<b>NCC_COMPUTE_POINTS:</b> points of a Newton-Cotes Closed quadrature rule.
</li>
<li>
<b>NCC_COMPUTE_WEIGHTS:</b> weights of a Newton-Cotes Closed quadrature rule.
</li>
<li>
<b>NCC_SET</b> sets abscissas and weights for closed Newton-Cotes 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> computes points for a Newton-Cotes "open half" quadrature rule.
</li>
<li>
<b>NCOH_COMPUTE_WEIGHTS</b> computes weights for a Newton-Cotes "open half" quadrature rule.
</li>
<li>
<b>NCOH_SET</b> sets abscissas and weights for Newton-Cotes "open half" rules.
</li>
<li>
<b>PATTERSON_SET</b> sets abscissas and weights for Gauss-Patterson quadrature.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</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.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8_GAMMA_LOG</b> calculates the natural logarithm of GAMMA ( X ) for positive X.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_HYPER_2F1</b> evaluates the hypergeometric function 2F1(A,B,C,X).
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_PSI</b> evaluates the function Psi(X).
</li>
<li>
<b>R8_SIGN</b> returns the sign of an R8.
</li>
<li>
<b>R8VEC_COPY</b> copies an R8VEC.
</li>
<li>
<b>R8VEC_DOT_PRODUCT</b> computes the dot product of a pair of R8VEC's.
</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>S_EQI</b> reports whether two strings are equal, ignoring case.
</li>
<li>
<b>SUM_SUB</b> carries out a composite quadrature 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>SUM_SUB_GK</b> carries out a composite Gauss-Kronrod rule.
</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 03 July 2011.
</i>
<!-- John Burkardt -->
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