test_nint

test_nint, a C++ code which defines a set of test problems for the approximate computation of integrals over multi-dimensional 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 m-dimensional 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:

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";

An Important Quote:

"When good results are obtained in integrating a high-dimensional 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 71-102.

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++ code which sets a Clenshaw Curtis quadrature grid in multiple dimensions.

NINTLIB, a C++ code which numerically estimates integrals in multiple dimensions.

PRODUCT_RULE, a C++ code which creates a multidimensional quadrature rule as a product of one dimensional rules.

STROUD, a C++ code which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TESTPACK, a C++ code which defines a set of integrands used to test multidimensional quadrature.

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