jburkardt

QUADRATURE_TEST\ Quadrature Rule Applied to Test Integrals {#quadrature_test-quadrature-rule-applied-to-test-integrals align=”center”} =========================================

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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:

1. the “X” file lists the abscissas (N rows, M columns);
2. the “W” file lists the weights (N rows);
3. 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:

1. f(x) = ( sum ( x(1:m) ) )**2;
2. f(x) = ( sum ( 2 * x(1:m) - 1 ) )**4;
3. f(x) = ( sum ( x(1:m) ) )**5;
4. f(x) = ( sum ( 2 * x(1:m) - 1 ) )**6;
5. f(x) = 1 / ( 1 + sum ( 2 * x(1:m) ) );
6. f(x) = product ( 2 * abs ( 2 * x(1:m) - 1 ) );
7. f(x) = product ( pi / 2 ) * sin ( pi * x(1:m) );
8. f(x) = ( sin ( (pi/4) * sum ( x(1:m) ) ) )**2;
9. f(x) = exp ( sum ( c(1:m) * x(1:m) ) );
10. f(x) = sum ( abs ( x(1:m) - 0.5 ) );
11. f(x) = exp ( sum ( abs ( 2 * x(1:m) - 1 ) ) );
12. f(x) = product ( 1 <= i <= m ) ( i * cos ( i * x(i) ) );
13. f(x) = product ( 1 <= i <= m ) t(n(i))(x(i)), t(n(i)) is a Chebyshev polynomial;
14. f(x) = sum ( 1 <= i <= m ) (-1)**i * product ( 1 <= j <= i ) x(j);
15. f(x) = product ( 1 <= i <= order ) x(mod(i-1,m)+1);
16. f(x) = sum ( abs ( x(1:m) - x0(1:m) ) );
17. f(x) = sum ( ( x(1:m) - x0(1:m) )**2 );
18. f(x) = 1 inside an m-dimensional sphere around x0(1:m), 0 outside;
19. f(x) = product ( sqrt ( abs ( x(1:m) - x0(1:m) ) ) );
20. f(x) = ( sum ( x(1:m) ) )**power;
21. f(x) = c * product ( x(1:m)\^e(1:m) ) on the surface of an m-dimensional unit sphere;
22. f(x) = c * product ( x(1:m)\^e(1:m) ) in an m-dimensional ball;
23. f(x) = c * product ( x(1:m)\^e(1:m) ) in the unit m-dimensional simplex;
24. f(x) = product ( abs ( 4 * x(1:m) - 2 ) + c(1:m) ) / ( 1 + c(1:m) ) );
25. f(x) = exp ( c * product ( x(1:m) ) );
26. f(x) = product ( c(1:m) * exp ( - c(1:m) * x(1:m) ) );
27. f(x) = cos ( 2 * pi * r + sum ( c(1:m) * x(1:m) ) ),\ Genz “Oscillatory”;
28. f(x) = 1 / product ( c(1:m)**2 + (x(1:m) - x0(1:m))**2),\ Genz “Product Peak”;
29. f(x) = 1 / ( 1 + sum ( c(1:m) * x(1:m) ) )**(m+r),\ Genz “Corner Peak”;
30. f(x) = exp(-sum(c(1:m)**2 * ( x(1:m) - x0(1:m))**2 ) ),\ Genz “Gaussian”;
31. f(x) = exp ( - sum ( c(1:m) * abs ( x(1:m) - x0(1:m) ) ) ), Genz “Continuous”;
32. f(x) = exp(sum(c(1:m)*x(1:m)) for x(1:m) <= x0(1:m), 0 otherwise,\ Genz “Discontinuous”;

Usage: {#usage align=”center”}

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: {#licensing align=”center”}

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages: {#languages align=”center”}

QUADRATURE_TEST is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs: {#related-data-and-programs align=”center”}

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: {#reference align=”center”}

1. JD Beasley, SG Springer,\ Algorithm AS 111: The Percentage Points of the Normal Distribution,\ Applied Statistics,\ Volume 26, 1977, pages 118-121.
2. 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.
3. Roger Broucke,\ Algorithm 446: Ten Subroutines for the Manipulation of Chebyshev Series,\ Communications of the ACM,\ Volume 16, 1973, pages 254-256.
4. 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.
5. Richard Crandall,\ Projects in Scientific Computing,\ Springer, 2005,\ ISBN: 0387950095,\ LC: Q183.9.C733.
6. Philip Davis, Philip Rabinowitz,\ Methods of Numerical Integration,\ Second Edition,\ Dover, 2007,\ ISBN: 0486453391,\ LC: QA299.3.D28.
7. Gerald Folland,\ How to Integrate a Polynomial Over a Sphere,\ American Mathematical Monthly,\ Volume 108, Number 5, May 2001, pages 446-448.
8. Leslie Fox, Ian Parker,\ Chebyshev Polynomials in Numerical Analysis,\ Oxford Press, 1968,\ LC: QA297.F65.
9. 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.
10. 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.
11. 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.
12. John Hart, Ward Cheney, Charles Lawson, Hans Maehly, Charles Mesztenyi, John Rice, Henry Thatcher, Christoph Witzgall,\ Computer Approximations,\ Wiley, 1968,\ LC: QA297.C64.
13. 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.
14. David Kahaner, Cleve Moler, Steven Nash,\ Numerical Methods and Software,\ Prentice Hall, 1989,\ ISBN: 0-13-627258-4,\ LC: TA345.K34.
15. Bradley Keister,\ Multidimensional Quadrature Algorithms,\ Computers in Physics,\ Volume 10, Number 2, March/April, 1996, pages 119-122.
16. Arnold Krommer, Christoph Ueberhuber,\ Numerical Integration on Advanced Compuer Systems,\ Springer, 1994,\ ISBN: 3540584102,\ LC: QA299.3.K76.
17. Anargyros Papageorgiou, Joseph Traub,\ Faster Evaluation of Multidimensional Integrals,\ Computers in Physics,\ Volume 11, Number 6, November/December 1997, pages 574-578.
18. 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.
19. Arthur Stroud,\ Approximate Calculation of Multiple Integrals,\ Prentice Hall, 1971,\ ISBN: 0130438936,\ LC: QA311.S85.
20. Arthur Stroud, Don Secrest,\ Gaussian Quadrature Formulas,\ Prentice Hall, 1966,\ LC: QA299.4G3S7.
21. Xiaoqun Wang, Kai-Tai Fang,\ The Effective Dimension and quasi-Monte Carlo Integration,\ Journal of Complexity,\ Volume 19, pages 101-124, 2003.

Source Code: {#source-code align=”center”}

• quadrature_test.cpp, the source code.

Examples and Tests: {#examples-and-tests align=”center”}

CC_D2_LEVEL4 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 2 of level 4, 65 points.

• cc_d2_level4_x.txt, the “X” file.
• cc_d2_level4_w.txt, the “W” file.
• cc_d2_level4_r.txt, the “R” file.
• cc_d2_level4_test.txt, the test results.

CC_D2_LEVEL5 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 2 of level 5, 145 points.

• cc_d2_level5_x.txt, the “X” file.
• cc_d2_level5_w.txt, the “W” file.
• cc_d2_level5_r.txt, the “R” file.
• cc_d2_level5_test.txt, the test results.

CC_D6_LEVEL0 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 0, 1 point.

• cc_d6_level0_x.txt, the “X” file.
• cc_d6_level0_w.txt, the “W” file.
• cc_d6_level0_r.txt, the “R” file.
• cc_d6_level0_test.txt, the test results.

CC_D6_LEVEL1 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 1, 13 points.

• cc_d6_level1_x.txt, the “X” file.
• cc_d6_level1_w.txt, the “W” file.
• cc_d6_level1_r.txt, the “R” file.
• cc_d6_level1_test.txt, the test results.

CC_D6_LEVEL2 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 2, 85 points.

• cc_d6_level2_x.txt, the “X” file.
• cc_d6_level2_w.txt, the “W” file.
• cc_d6_level2_r.txt, the “R” file.
• cc_d6_level2_test.txt, the test results.

CC_D6_LEVEL3 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 3, 389 points.

• cc_d6_level3_x.txt, the “X” file.
• cc_d6_level3_w.txt, the “W” file.
• cc_d6_level3_r.txt, the “R” file.
• cc_d6_level3_test.txt, the test results.

CC_D6_LEVEL4 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 4, 1457 points.

• cc_d6_level4_x.txt, the “X” file.
• cc_d6_level4_w.txt, the “W” file.
• cc_d6_level4_r.txt, the “R” file.
• cc_d6_level4_test.txt, the test results.

CC_D6_LEVEL5 is a Clenshaw-Curtis sparse grid quadrature rule in dimension 6 of level 5, 4865 points.

• cc_d6_level5_x.txt, the “X” file.
• cc_d6_level5_w.txt, the “W” file.
• cc_d6_level5_r.txt, the “R” file.
• cc_d6_level5_test.txt, the test results.

List of Routines: {#list-of-routines align=”center”}

• 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.

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Last revised on 06 June 2007.