TEST_NINT
Multidimensional Integration
Test Functions.
TEST_NINT,
a C++ code which
defines a set of 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";
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:
CLENSHAW_CURTIS_RULE,
a C++ library which
sets a Clenshaw Curtis quadrature grid in multiple dimensions.
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.
STROUD,
a C++ library which
defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and Ndimensions.
test_nint_test
TESTPACK,
a C++ library which
defines a set of integrands used to test multidimensional quadrature.
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Source Code:
Last revised on 23 April 2020.