QUADRATURE_TEST
Quadrature Rule Applied to Test Integrals
QUADRATURE_TEST
is a MATLAB 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' )
where
-
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:
INTEGRAL_TEST,
a FORTRAN90 program which
uses test integrals to evaluate
sets of quadrature points.
NINT_EXACTNESS,
a MATLAB program which
demonstrates how to measure the
polynomial exactness of a multidimensional quadrature rule.
QUADRATURE_RULES,
a dataset directory which
contains a description and examples of quadrature rules defined
by a set of "X", "W" and "R" files.
QUADRATURE_TEST_2D,
a MATLAB program which
reads files defining a 2D quadrature rule, and
applies them to all the test integrals defined by TEST_INT_2D.
STROUD,
a MATLAB library which
contains quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
TEST_NINT,
a MATLAB library which
defines a set of integrand functions to be used for testing
multidimensional quadrature rules and routines.
TESTPACK,
a MATLAB 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:
-
quadrature_test.m,
the main program, which gets the user's input, reads
the quadrature rule from the files, and applies it
to the test integrand functions.
-
csevl.m,
evaluates an N-term Chebyshev series.
-
error_f.m,
evaluates the Error function.
-
error_fc.m,
evaluates the complementary Error function.
-
file_column_count.m,
counts the columns in a file
-
file_row_count.m,
counts the rows in a file
-
get_problem_num.m,
returns the number of test integration problems.
-
inits.m,
estimates the order of an orthogonal series for a given accuracy.
-
normal_01_cdf_inv.m,
inverts the Normal 01 CDF.
-
p00_box_gl05.m,
uses Gauss-Legendre quadrature in an N-dimensional box.
-
p00_box_mc.m,
integrates over an multi-dimensional box using Monte Carlo.
-
p00_default.m,
sets a problem to a default state.
-
p00_exact.m,
returns the exact integral for any problem.
-
p00_f.m,
evaluates the integrand for any problem.
-
p00_i4.m,
sets or gets integer scalar parameters for any problem.
-
p00_lim.m,
returns the integration limits for any problem.
-
p00_name.m,
returns the name of any problem.
-
p00_r8vec.m,
sets or gets real vector parameters for any problem.
-
p00_region.m,
returns the name of the integration region for any problem.
-
p00_remap01.m,
remaps points in [0,1]^DIM_NUM into [A(1:DIM_NUM),B(1:DIM_NUM)].
-
p00_title.m,
prints a title for any problem.
-
p00_volume.m,
returns the volume of the integration region.
-
p01_default.m,
sets default values for problem 01.
-
p01_exact.m,
sets the exact integral for problem 01.
-
p01_f.m,
evaluates the integrand for problem 01.
-
p01_i4.m,
sets or gets I4 parameters for problem 01.
-
p01_lim.m,
returns the integration limits for problem 01.
-
p01_name.m,
returns a name for problem 01.
-
p01_region.m,
returns the region type for problem 01.
-
p01_title.m,
prints a title for problem 01.
-
p02_default.m,
sets default values for problem 02.
-
p02_exact.m,
sets the exact integral for problem 02.
-
p02_f.m,
evaluates the integrand for problem 02.
-
p02_i4.m,
sets or gets I4 parameters for problem 02.
-
p02_lim.m,
returns the integration limits for problem 02.
-
p02_name.m,
returns a name for problem 02.
-
p02_region.m,
returns the region type for problem 02.
-
p02_title.m,
prints a title for problem 02.
-
p03_default.m,
sets default values for problem 03.
-
p03_exact.m,
sets the exact integral for problem 03.
-
p03_f.m,
evaluates the integrand for problem 03.
-
p03_i4.m,
sets or gets I4 parameters for problem 03.
-
p03_lim.m,
returns the integration limits for problem 03.
-
p03_name.m,
returns a name for problem 03.
-
p03_region.m,
returns the region type for problem 03.
-
p03_title.m,
prints a title for problem 03.
-
p04_default.m,
sets default values for problem 04.
-
p04_exact.m,
sets the exact integral for problem 04.
-
p04_f.m,
evaluates the integrand for problem 04.
-
p04_i4.m,
sets or gets I4 parameters for problem 04.
-
p04_lim.m,
returns the integration limits for problem 04.
-
p04_name.m,
returns a name for problem 04.
-
p04_region.m,
returns the region type for problem 04.
-
p04_title.m,
prints a title for problem 04.
-
p05_default.m,
sets default values for problem 05.
-
p05_exact.m,
sets the exact integral for problem 05.
-
p05_f.m,
evaluates the integrand for problem 05.
-
p05_i4.m,
sets or gets I4 parameters for problem 05
-
p05_lim.m,
returns the integration limits for problem 05.
-
p05_name.m,
returns a name for problem 05.
-
p05_region.m,
returns the region type for problem 05.
-
p05_title.m,
prints a title for problem 05.
-
p06_default.m,
sets default values for problem 06.
-
p06_exact.m,
sets the exact integral for problem 06.
-
p06_f.m,
evaluates the integrand for problem 06.
-
p06_i4.m,
sets or gets I4 parameters for problem 06.
-
p06_lim.m,
returns the integration limits for problem 06.
-
p06_name.m,
returns a name for problem 06.
-
p06_region.m,
returns the region type for problem 06.
-
p06_title.m,
prints a title for problem 06.
-
p07_default.m,
sets default values for problem 07.
-
p07_exact.m,
sets the exact integral for problem 07.
-
p07_f.m,
evaluates the integrand for problem 07.
-
p07_i4.m,
sets or gets I4 parameters for problem 07.
-
p07_lim.m,
returns the integration limits for problem 07.
-
p07_name.m,
returns a name for problem 07.
-
p07_region.m,
returns the region type for problem 07.
-
p07_title.m,
prints a title for problem 07.
-
p08_default.m,
sets default values for problem 08.
-
p08_exact.m,
sets the exact integral for problem 08.
-
p08_f.m,
evaluates the integrand for problem 08.
-
p08_i4.m,
sets or gets I4 parameters for problem 08.
-
p08_lim.m,
returns the integration limits for problem 08.
-
p08_name.m,
returns a name for problem 08.
-
p08_region.m,
returns the region type for problem 08.
-
p08_title.m,
prints a title for problem 08.
-
p09_default.m,
sets default values for problem 09.
-
p09_exact.m,
sets the exact integral for problem 09.
-
p09_f.m,
evaluates the integrand for problem 09.
-
p09_i4.m,
sets or gets I4 parameters for problem 09.
-
p09_lim.m,
returns the integration limits for problem 09.
-
p09_name.m,
returns a name for problem 09.
-
p09_r8vec.m,
sets or gets R8VEC parameters for problem 09.
-
p09_region.m,
returns the region type for problem 09.
-
p09_title.m,
prints a title for problem 09.
-
p10_default.m,
sets default values for problem 10.
-
p10_exact.m,
sets the exact integral for problem 10.
-
p10_f.m,
evaluates the integrand for problem 10.
-
p10_i4.m,
sets or gets I4 parameters for problem 10.
-
p10_lim.m,
returns the integration limits for problem 10.
-
p10_name.m,
returns a name for problem 10.
-
p10_region.m,
returns the region type for problem 10.
-
p10_title.m,
prints a title for problem 10.
-
p11_default.m,
sets default values for problem 11.
-
p11_exact.m,
sets the exact integral for problem 11.
-
p11_f.m,
evaluates the integrand for problem 11.
-
p11_i4.m,
sets or gets I4 parameters for problem 11.
-
p11_lim.m,
returns the integration limits for problem 11.
-
p11_name.m,
returns a name for problem 11.
-
p11_region.m,
returns the region type for problem 11.
-
p11_title.m,
prints a title for problem 11.
-
p12_default.m,
sets default values for problem 12.
-
p12_exact.m,
sets the exact integral for problem 12.
-
p12_f.m,
evaluates the integrand for problem 12.
-
p12_i4.m,
sets or gets I4 parameters for problem 12.
-
p12_lim.m,
returns the integration limits for problem 12.
-
p12_name.m,
returns a name for problem 12.
-
-
p12_region.m,
returns the region type for problem 12.
-
p12_title.m,
prints a title for problem 12.
-
p13_default.m,
sets default values for problem 13.
-
p13_exact.m,
sets the exact integral for problem 13.
-
p13_f.m,
evaluates the integrand for problem 13.
-
p13_i4.m,
sets or gets I4 parameters for problem 13.
-
p13_lim.m,
returns the integration limits for problem 13.
-
p13_name.m,
returns a name for problem 13.
-
p13_region.m,
returns the region type for problem 13.
-
p13_title.m,
prints a title for problem 13.
-
p14_default.m,
sets default values for problem 14.
-
p14_exact.m,
sets the exact integral for problem 14.
-
p14_f.m,
evaluates the integrand for problem 14.
-
p14_i4.m,
sets or gets I4 parameters for problem 14.
-
p14_lim.m,
returns the integration limits for problem 14.
-
p14_name.m,
returns a name for problem 14.
-
p14_region.m,
returns the region type for problem 14.
-
p14_title.m,
prints a title for problem 14.
-
p15_default.m,
sets default values for problem 15.
-
p15_exact.m,
sets the exact integral for problem 15.
-
p15_f.m,
evaluates the integrand for problem 15.
-
p15_i4.m,
sets or gets I4 parameters for problem 15.
-
p15_lim.m,
returns the integration limits for problem 151.
-
p15_name.m,
returns a name for problem 15.
-
p15_region.m,
returns the region type for problem 15.
-
p15_title.m,
prints a title for problem 15.
-
p16_default.m,
sets default values for problem 16.
-
p16_exact.m,
sets the exact integral for problem 16.
-
p16_f.m,
evaluates the integrand for problem 16.
-
p16_i4.m,
sets or gets I4 parameters for problem 16.
-
p16_lim.m,
returns the integration limits for problem 16.
-
p16_name.m,
returns a name for problem 16.
-
p16_r8vec.m,
sets or gets R8VEC parameters for problem 16.
-
p16_region.m,
returns the region type for problem 16.
-
p16_title.m,
prints a title for problem 16.
-
p17_default.m,
sets default values for problem 17.
-
p17_exact.m,
sets the exact integral for problem 17.
-
p17_f.m,
evaluates the integrand for problem 17.
-
p17_i4.m,
sets or gets I4 parameters for problem 17.
-
p17_lim.m,
returns the integration limits for problem 17.
-
p17_name.m,
returns a name for problem 17.
-
p17_r8vec.m,
sets or gets R8VEC parameters for problem 17.
-
p17_region.m,
returns the region type for problem 17.
-
p17_title.m,
prints a title for problem 17.
-
p18_default.m,
sets default values for problem 18.
-
p18_exact.m,
sets the exact integral for problem 18.
-
p18_f.m,
evaluates the integrand for problem 18.
-
p18_i4.m,
sets or gets I4 parameters for problem 18.
-
p18_lim.m,
returns the integration limits for problem 18.
-
p18_name.m,
returns a name for problem 18.
-
p18_r8.m,
sets or gets R8 parameters for problem 18.
-
p18_r8vec.m,
sets or gets R8VEC parameters for problem 18.
-
p18_region.m,
returns the region type for problem 18.
-
p18_title.m,
prints a title for problem 18.
-
p19_default.m,
sets default values for problem 19.
-
p19_exact.m,
sets the exact integral for problem 19.
-
p19_f.m,
evaluates the integrand for problem 19.
-
p19_i4.m,
sets or gets I4 parameters for problem 19.
-
p19_lim.m,
returns the integration limits for problem 19.
-
p19_name.m,
returns a name for problem 19.
-
p19_r8vec.m,
sets or gets R8VEC parameters for problem 19.
-
p19_region.m,
returns the region type for problem 19.
-
p19_title.m,
prints a title for problem 19.
-
p20_default.m,
sets default values for problem 20.
-
p20_exact.m,
sets the exact integral for problem 20.
-
p20_f.m,
evaluates the integrand for problem 20.
-
p20_i4.m,
sets or gets I4 parameters for problem 20.
-
p20_lim.m,
returns the integration limits for problem 20.
-
p20_name.m,
returns a name for problem 20.
-
p20_r8.m,
sets or gets R8 parameters for problem 20.
-
p20_region.m,
returns the region type for problem 20.
-
p20_title.m,
prints a title for problem 20.
-
p21_default.m,
sets default values for problem 21.
-
p21_exact.m,
sets the exact integral for problem 21.
-
p21_f.m,
evaluates the integrand for problem 21.
-
p21_i4.m,
sets or gets I4 parameters for problem 21.
-
p21_i4vec.m,
sets or gets I4VEC parameters for problem 21.
-
p21_lim.m,
returns the integration limits for problem 21.
-
p21_name.m,
returns a name for problem 21.
-
p21_r8.m,
sets or gets R8 parameters for problem 21.
-
p21_region.m,
returns the region type for problem 21.
-
p21_title.m,
prints a title for problem 21.
-
p22_default.m,
sets default values for problem 22.
-
p22_exact.m,
sets the exact integral for problem 22.
-
p22_f.m,
evaluates the integrand for problem 22.
-
p22_i4.m,
sets or gets I4 parameters for problem 22.
-
p22_i4vec.m,
sets or gets I4VEC parameters for problem 22.
-
p22_lim.m,
returns the integration limits for problem 22.
-
p22_name.m,
returns a name for problem 22.
-
p22_r8.m,
sets or gets R8 parameters for problem 22.
-
p22_region.m,
returns the region type for problem 22.
-
p22_title.m,
prints a title for problem 22.
-
p23_default.m,
sets default values for problem 23.
-
p23_exact.m,
sets the exact integral for problem 23.
-
p23_f.m,
evaluates the integrand for problem 23.
-
p23_i4.m,
sets or gets I4 parameters for problem 23.
-
p23_i4vec.m,
sets or gets I4VEC parameters for problem 23.
-
p23_lim.m,
returns the integration limits for problem 23.
-
p23_name.m,
returns a name for problem 23.
-
p23_r8.m,
sets or gets R8 parameters for problem 23.
-
p23_region.m,
returns the region type for problem 23.
-
p23_title.m,
prints a title for problem 23.
-
p24_default.m,
sets default values for problem 24.
-
p24_exact.m,
sets the exact integral for problem 24.
-
p24_f.m,
evaluates the integrand for problem 24.
-
p24_i4.m,
sets or gets I4 parameters for problem 24.
-
p24_lim.m,
returns the integration limits for problem 24.
-
p24_name.m,
returns a name for problem 24.
-
p24_r8vec.m,
sets or gets R8VEC parameters for problem 24.
-
p24_region.m,
returns the region type for problem 24.
-
p24_title.m,
prints a title for problem 24.
-
p25_default.m,
sets default values for problem 25.
-
p25_exact.m,
sets the exact integral for problem 25.
-
p25_f.m,
evaluates the integrand for problem 25.
-
p25_i4.m,
sets or gets I4 parameters for problem 25.
-
p25_lim.m,
returns the integration limits for problem 25.
-
p25_name.m,
returns a name for problem 25.
-
p25_r8.m,
sets or gets R8 parameters for problem 25.
-
p25_region.m,
returns the region type for problem 25.
-
p25_title.m,
prints a title for problem 25.
-
p26_default.m,
sets default values for problem 26.
-
p26_exact.m,
sets the exact integral for problem 26.
-
p26_f.m,
evaluates the integrand for problem 26.
-
p26_i4.m,
sets or gets I4 parameters for problem 26.
-
p26_lim.m,
returns the integration limits for problem 26.
-
p26_name.m,
returns a name for problem 26.
-
p26_r8vec.m,
sets or gets R8VEC parameters for problem 26.
-
p26_region.m,
returns the region type for problem 26.
-
p26_title.m,
prints a title for problem 26.
-
p27_default.m,
sets default values for problem 27.
-
p27_exact.m,
sets the exact integral for problem 27.
-
p27_f.m,
evaluates the integrand for problem 27.
-
p27_i4.m,
sets or gets I4 parameters for problem 27.
-
p27_lim.m,
returns the integration limits for problem 27.
-
p27_name.m,
returns a name for problem 27.
-
p27_r8.m,
sets or gets R8 parameters for problem 27.
-
p27_r8vec.m,
sets or gets R8VEC parameters for problem 27.
-
p27_region.m,
returns the region type for problem 27.
-
p27_title.m,
prints a title for problem 27.
-
p28_default.m,
sets default values for problem 28.
-
p28_exact.m,
sets the exact integral for problem 28.
-
p28_f.m,
evaluates the integrand for problem 28.
-
p28_i4.m,
sets or gets I4 parameters for problem 28.
-
p28_lim.m,
returns the integration limits for problem 28.
-
p28_name.m,
returns a name for problem 28.
-
p28_r8vec.m,
sets or gets R8VEC parameters for problem 281.
-
p28_region.m,
returns the region type for problem 28.
-
p28_title.m,
prints a title for problem 28.
-
p29_default.m,
sets default values for problem 29.
-
p29_exact.m,
sets the exact integral for problem 29.
-
p29_f.m,
evaluates the integrand for problem 29.
-
p29_i4.m,
sets or gets I4 parameters for problem 29.
-
p29_lim.m,
returns the integration limits for problem 29.
-
p29_name.m,
returns a name for problem 29.
-
p29_r8.m,
sets or gets R8 parameters for problem 29.
-
p29_r8vec.m,
sets or gets R8VEC parameters for problem 29.
-
p29_region.m,
returns the region type for problem 29.
-
p29_title.m,
prints a title for problem 29.
-
p30_default.m,
sets default values for problem 30.
-
p30_exact.m,
sets the exact integral for problem 30.
-
p30_f.m,
evaluates the integrand for problem 30.
-
p30_i4.m,
sets or gets I4 parameters for problem 30.
-
p30_lim.m,
returns the integration limits for problem 30.
-
p30_name.m,
returns a name for problem 30.
-
p30_r8vec.m,
sets or gets R8VEC parameters for problem 30.
-
p30_region.m,
returns the region type for problem 30.
-
p30_title.m,
prints a title for problem 30.
-
p31_default.m,
sets default values for problem 31.
-
p31_exact.m,
sets the exact integral for problem 31.
-
p31_f.m,
evaluates the integrand for problem 31.
-
p31_i4.m,
sets or gets I4 parameters for problem 31.
-
p31_lim.m,
returns the integration limits for problem 31.
-
p31_name.m,
returns a name for problem 31.
-
p31_r8vec.m,
sets or gets R8VEC parameters for problem 31.
-
p31_region.m,
returns the region type for problem 31.
-
p31_title.m,
prints a title for problem 31.
-
p32_default.m,
sets default values for problem 32.
-
p32_exact.m,
sets the exact integral for problem 32.
-
p32_f.m,
evaluates the integrand for problem 32.
-
p32_i4.m,
sets or gets I4 parameters for problem 32.
-
p32_lim.m,
returns the integration limits for problem 32.
-
p32_name.m,
returns a name for problem 32.
-
p32_r8vec.m,
sets or gets R8VEC parameters for problem 32.
-
p32_region.m,
returns the region type for problem 32.
-
p32_title.m,
prints a title for problem 32.
-
r8_choose.m,
computes the combinatorial coefficient C(N,K).
-
r8_huge.m,
returns a "huge" R8.
-
r8_sign.m,
returns the sign of an R8.
-
r8_tiny.m,
returns a "tiny" R8.
-
r8mat_data_read.m,
reads the data from an R8MAT file.
-
r8mat_header_read.m,
reads the header from an R8MAT file.
-
r8vec_print.m,
prints an R8VEC.
-
s_len_trim.m,
returns the length of a string to the last nonblank.
-
s_word_count.m,
returns the number of words in a string.
-
simplex_unit_volume_nd.m,
computes the volume of a unit simplex in ND.
-
sphere_unit_area_nd.m,
computes the area of a unit sphere in ND.
-
sphere_unit_volume_nd.m,
computes the volume of a unit sphere in ND.
-
sphere_volume_nd.m,
computes the volume of a sphere in ND.
-
timestamp.m,
prints the current YMDHMS date as a time stamp.
-
tuple_next.m,
computes the next element of a tuple space.
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.
You can go up one level to
the MATLAB source codes.
Last revised on 11 November 2009.