test_nint
test_nint,
a FORTRAN90 code which
defines test problems for the approximate computation of integrals
over multidimensional regions.
Routines are available to evaluate the integrand, return the exact
value of the integral, report the name of the problem,
report the integration limits, get, set or modify a base point.
The integrands is assigned an index. The integrands can be invoked
by index. Most integrands may be defined for any value of the
spatial dimension, which we denote here by m. Most integrands
are defined on the unit mdimensional hypercube. Some
integrands include one or more parameters. These generally have
default values, which can be altered by the user.
For each problem, a set of routines are available with a standard
interface, for manipulating and evaluating the problem. For a
problem with index "87", for instance, we might have the following
set of routines. The most important is P87_F which evaluates
the integrand. We probably also need P87_LIM to determine
the limits of integration, and P87_EXACT to get the exact
value of the integral (if known). A number of routines are available
to set, get, or randomize parameters associated with the problem.

P87_DEFAULT sets default values for problem 87.

P87_EXACT returns the exact integral for problem 87.

P87_F evaluates the integrand for problem 87.

P87_I4 sets or gets integer scalar parameters for problem 87.

P87_I4VEC sets or gets integer vector parameters for problem 87.

P87_LIM returns the integration limits for problem 87.

P87_NAME returns the name of problem 87.

P87_R8 sets or gets real scalar parameters for problem 87.

P87_R8VEC sets or gets real vector parameters for problem 87.

P87_REGION returns the name of the integration region
for problem 87.

P87_TITLE prints a title for problem 87.
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(i1,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 mdimensional 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 mdimensional unit sphere;

f(x) = c * product ( x(1:m)^e(1:m) ) in an mdimensional ball;

f(x) = c * product ( x(1:m)^e(1:m) ) in the unit mdimensional 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";

f(x) = sum ( x(1:n)^2 )
Ball R^2;
An Important Quote:
"When good results are obtained in integrating a highdimensional
function, we should conclude first of all that an especially tractable
integrand was tried and not that a generally successful method has
been found. A secondary conclusion is that we might have made a
very good choice in selecting an integration method to exploit whatever
features of f made it tractable."
Art Owen,
Latin Supercube Sampling for Very High Dimensional Simulation,
ACM Transactions on Modeling and Computer Simulations,
Volume 8, Number 1, January 1998, pages 71102.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
test_nint is available in
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
BALL_VOLUME_MONTE_CARLO,
a FORTRAN90 code which
applies a Monte Carlo method to estimate the volume of the unit 6D ball;
CUBPACK,
a FORTRAN90 code which
estimates the integral of a function over a collection of Ndimensional
hyperrectangles and simplices.
INTEGRAL_TEST,
a FORTRAN90 code which
tests the suitability of a set of N points for use in an equalweight quadrature rule over
the multidimensional unit hypercube.
NINT_EXACTNESS,
a FORTRAN90 code which
measures the polynomial exactness of a multidimensional quadrature rule.
NINTLIB,
a FORTRAN90 code which
estimates integrals over multidimensional regions.
PRODUCT_RULE,
a FORTRAN90 code which
constructs a product quadrature rule from 1D factor rules.
QUADRATURE_TEST,
a FORTRAN90 code which
reads files defining a quadrature rule, and
applies them to all the test integrals defined by TEST_NINT.
STROUD,
a FORTRAN90 code which
defines quadrature rules for a variety of multidimensional reqions.
TEST_INT_2D,
a FORTRAN90 code which
defines test integrands for 2D quadrature rules.
test_nint_test
TESTPACK,
a FORTRAN90 code 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 118121.

Paul Bratley, Bennett Fox, Harald Niederreiter,
Implementation and Tests of LowDiscrepancy Sequences,
ACM Transactions on Modeling and Computer Simulation,
Volume 2, Number 3, July 1992, pages 195213.

Roger Broucke,
Algorithm 446:
Ten Subroutines for the Manipulation of Chebyshev Series,
Communications of the ACM,
Volume 16, 1973, pages 254256.

William Cody, Kenneth Hillstrom,
Chebyshev Approximations for the Natural Logarithm of the
Gamma Function,
Mathematics of Computation,
Volume 21, Number 98, April 1967, pages 198203.

Richard Crandall,
Projects in Scientific Computing,
Springer, 2005,
ISBN: 0387950095,
LC: Q183.9.C733.

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Methods of Numerical Integration,
Second Edition,
Dover, 2007,
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LC: QA299.3.D28.

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How to Integrate a Polynomial Over a Sphere,
American Mathematical Monthly,
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Chebyshev Polynomials in Numerical Analysis,
Oxford Press, 1968,
LC: QA297.F65.

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Testing Multidimensional Integration Routines,
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QuasiMonte Carlo: halftoning in high dimensions?
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Source Code:
Last revised on 04 September 2020.