QUADRATURE_TEST
Quadrature Rule Applied to Test Integrals
QUADRATURE_TEST
is a C++ program which
reads three files that define
a quadrature rule, applies the quadrature rule to a set of
test integrals, and reports the results.
The quadrature rule is defined by three text files:
-
the "X" file lists the abscissas (N rows, M columns);
-
the "W" file lists the weights (N rows);
-
the "R" file lists the integration region corners
(2 rows, M columns);
For more on quadrature rules, see the QUADRATURE_RULES
listing below.
The test integrals come from the TEST_NINT library.
The list of integrand functions includes:
-
f(x) = ( sum ( x(1:m) ) )^2;
-
f(x) = ( sum ( 2 * x(1:m) - 1 ) )^4;
-
f(x) = ( sum ( x(1:m) ) )^5;
-
f(x) = ( sum ( 2 * x(1:m) - 1 ) )^6;
-
f(x) = 1 / ( 1 + sum ( 2 * x(1:m) ) );
-
f(x) = product ( 2 * abs ( 2 * x(1:m) - 1 ) );
-
f(x) = product ( pi / 2 ) * sin ( pi * x(1:m) );
-
f(x) = ( sin ( (pi/4) * sum ( x(1:m) ) ) )^2;
-
f(x) = exp ( sum ( c(1:m) * x(1:m) ) );
-
f(x) = sum ( abs ( x(1:m) - 0.5 ) );
-
f(x) = exp ( sum ( abs ( 2 * x(1:m) - 1 ) ) );
-
f(x) = product ( 1 <= i <= m ) ( i * cos ( i * x(i) ) );
-
f(x) = product ( 1 <= i <= m ) t(n(i))(x(i)), t(n(i))
is a Chebyshev polynomial;
-
f(x) = sum ( 1 <= i <= m ) (-1)^i * product ( 1 <= j <= i ) x(j);
-
f(x) = product ( 1 <= i <= order ) x(mod(i-1,m)+1);
-
f(x) = sum ( abs ( x(1:m) - x0(1:m) ) );
-
f(x) = sum ( ( x(1:m) - x0(1:m) )^2 );
-
f(x) = 1 inside an m-dimensional sphere around x0(1:m), 0 outside;
-
f(x) = product ( sqrt ( abs ( x(1:m) - x0(1:m) ) ) );
-
f(x) = ( sum ( x(1:m) ) )^power;
-
f(x) = c * product ( x(1:m)^e(1:m) ) on the surface of
an m-dimensional unit sphere;
-
f(x) = c * product ( x(1:m)^e(1:m) ) in an m-dimensional ball;
-
f(x) = c * product ( x(1:m)^e(1:m) ) in the unit m-dimensional simplex;
-
f(x) = product ( abs ( 4 * x(1:m) - 2 ) + c(1:m) )
/ ( 1 + c(1:m) ) );
-
f(x) = exp ( c * product ( x(1:m) ) );
-
f(x) = product ( c(1:m) * exp ( - c(1:m) * x(1:m) ) );
-
f(x) = cos ( 2 * pi * r + sum ( c(1:m) * x(1:m) ) ),
Genz "Oscillatory";
-
f(x) = 1 / product ( c(1:m)^2 + (x(1:m) - x0(1:m))^2),
Genz "Product Peak";
-
f(x) = 1 / ( 1 + sum ( c(1:m) * x(1:m) ) )^(m+r),
Genz "Corner Peak";
-
f(x) = exp(-sum(c(1:m)^2 * ( x(1:m) - x0(1:m))^2 ) ),
Genz "Gaussian";
-
f(x) = exp ( - sum ( c(1:m) * abs ( x(1:m) - x0(1:m) ) ) ),
Genz "Continuous";
-
f(x) = exp(sum(c(1:m)*x(1:m)) for x(1:m) <= x0(1:m), 0 otherwise,
Genz "Discontinuous";
Usage:
quadrature_test prefix
-
prefix
-
the common prefix for the files containing the abscissa (X),
weight (W) and region (R) information of the quadrature rule;
If the arguments are not supplied on the command line, the
program will prompt for them.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
QUADRATURE_TEST is available in
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
GSL,
a C++ library which
includes routines for estimating multidimensional integrals.
INTEGRAL_TEST,
a FORTRAN90 program which
uses test integrals to evaluate sets of quadrature points.
NINT_EXACTNESS,
a C++ program which
demonstrates how to measure the
polynomial exactness of a multidimensional quadrature rule.
NINTLIB,
a C++ library which
numerically estimates integrals
in multiple dimensions.
PRODUCT_RULE,
a C++ program which
creates a multidimensional quadrature rule as a product of
one dimensional rules.
QUADRATURE_RULES,
a dataset directory which
contains a description and examples of quadrature rules defined
by a set of "X", "W" and "R" files.
STROUD,
a C++ library which
contains quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
TEST_NINT,
a C++ library which
defines a set of integrand functions to be used for testing
multidimensional quadrature rules and routines.
TESTPACK,
a C++ library which
defines a set of integrands used to test multidimensional quadrature.
Reference:
-
JD Beasley, SG Springer,
Algorithm AS 111:
The Percentage Points of the Normal Distribution,
Applied Statistics,
Volume 26, 1977, pages 118-121.
-
Paul Bratley, Bennett Fox, Harald Niederreiter,
Implementation and Tests of Low-Discrepancy Sequences,
ACM Transactions on Modeling and Computer Simulation,
Volume 2, Number 3, July 1992, pages 195-213.
-
Roger Broucke,
Algorithm 446:
Ten Subroutines for the Manipulation of Chebyshev Series,
Communications of the ACM,
Volume 16, 1973, pages 254-256.
-
William Cody, Kenneth Hillstrom,
Chebyshev Approximations for the Natural Logarithm of the
Gamma Function,
Mathematics of Computation,
Volume 21, Number 98, April 1967, pages 198-203.
-
Richard Crandall,
Projects in Scientific Computing,
Springer, 2005,
ISBN: 0387950095,
LC: Q183.9.C733.
-
Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
-
Gerald Folland,
How to Integrate a Polynomial Over a Sphere,
American Mathematical Monthly,
Volume 108, Number 5, May 2001, pages 446-448.
-
Leslie Fox, Ian Parker,
Chebyshev Polynomials in Numerical Analysis,
Oxford Press, 1968,
LC: QA297.F65.
-
Alan Genz,
Testing Multidimensional Integration Routines,
in Tools, Methods, and Languages for Scientific and
Engineering Computation,
edited by B Ford, JC Rault, F Thomasset,
North-Holland, 1984, pages 81-94,
ISBN: 0444875700,
LC: Q183.9.I53.
-
Alan Genz,
A Package for Testing Multiple Integration Subroutines,
in Numerical Integration:
Recent Developments, Software and Applications,
edited by Patrick Keast, Graeme Fairweather,
Reidel, 1987, pages 337-340,
ISBN: 9027725144,
LC: QA299.3.N38.
-
Kenneth Hanson,
Quasi-Monte Carlo: halftoning in high dimensions?
in Computatinal Imaging,
Edited by CA Bouman, RL Stevenson,
Proceedings SPIE,
Volume 5016, 2003, pages 161-172.
-
John Hart, Ward Cheney, Charles Lawson, Hans Maehly,
Charles Mesztenyi, John Rice, Henry Thatcher,
Christoph Witzgall,
Computer Approximations,
Wiley, 1968,
LC: QA297.C64.
-
Stephen Joe, Frances Kuo
Remark on Algorithm 659:
Implementing Sobol's Quasirandom Sequence Generator,
ACM Transactions on Mathematical Software,
Volume 29, Number 1, March 2003, pages 49-57.
-
David Kahaner, Cleve Moler, Steven Nash,
Numerical Methods and Software,
Prentice Hall, 1989,
ISBN: 0-13-627258-4,
LC: TA345.K34.
-
Bradley Keister,
Multidimensional Quadrature Algorithms,
Computers in Physics,
Volume 10, Number 2, March/April, 1996, pages 119-122.
-
Arnold Krommer, Christoph Ueberhuber,
Numerical Integration on Advanced Compuer Systems,
Springer, 1994,
ISBN: 3540584102,
LC: QA299.3.K76.
-
Anargyros Papageorgiou, Joseph Traub,
Faster Evaluation of Multidimensional Integrals,
Computers in Physics,
Volume 11, Number 6, November/December 1997, pages 574-578.
-
Thomas Patterson,
On the Construction of a Practical Ermakov-Zolotukhin
Multiple Integrator,
in Numerical Integration:
Recent Developments, Software and Applications,
edited by Patrick Keast and Graeme Fairweather,
D. Reidel, 1987, pages 269-290.
-
Arthur Stroud,
Approximate Calculation of Multiple Integrals,
Prentice Hall, 1971,
ISBN: 0130438936,
LC: QA311.S85.
-
Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.
-
Xiaoqun Wang, Kai-Tai Fang,
The Effective Dimension and quasi-Monte Carlo Integration,
Journal of Complexity,
Volume 19, pages 101-124, 2003.
Source Code:
Examples and Tests:
CC_D2_LEVEL4 is a Clenshaw-Curtis sparse grid quadrature
rule in dimension 2 of level 4, 65 points.
CC_D2_LEVEL5 is a Clenshaw-Curtis sparse grid quadrature
rule in dimension 2 of level 5, 145 points.
CC_D6_LEVEL0 is a Clenshaw-Curtis sparse grid quadrature
rule in dimension 6 of level 0, 1 point.
CC_D6_LEVEL1 is a Clenshaw-Curtis sparse grid quadrature
rule in dimension 6 of level 1, 13 points.
CC_D6_LEVEL2 is a Clenshaw-Curtis sparse grid quadrature
rule in dimension 6 of level 2, 85 points.
CC_D6_LEVEL3 is a Clenshaw-Curtis sparse grid quadrature
rule in dimension 6 of level 3, 389 points.
CC_D6_LEVEL4 is a Clenshaw-Curtis sparse grid quadrature
rule in dimension 6 of level 4, 1457 points.
CC_D6_LEVEL5 is a Clenshaw-Curtis sparse grid quadrature
rule in dimension 6 of level 5, 4865 points.
List of Routines:
-
MAIN is the main program for QUADRATURE_TEST.
-
CH_EQI is true if two characters are equal, disregarding case.
-
CH_TO_DIGIT returns the integer value of a base 10 digit.
-
DTABLE_DATA_READ reads the data from a DTABLE file.
-
DTABLE_HEADER_READ reads the header from a DTABLE file.
-
FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
-
FILE_ROW_COUNT counts the number of row records in a file.
-
S_CAT concatenates two strings to make a third string.
-
S_TO_I4 reads an I4 from a string.
-
S_TO_R8 reads an R8 value from a string.
-
S_TO_R8VEC reads an R8VEC from a string.
-
S_WORD_COUNT counts the number of "words" in a string.
You can go up one level to
the C++ source codes.
Last revised on 06 June 2007.