test_nint

test_nint, a FORTRAN90 code which defines 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";
33. f(x) = sum ( x(1:n)^2 )
    Ball R^2;

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."

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT 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 N-dimensional hyperrectangles and simplices.

INTEGRAL_TEST, a FORTRAN90 code which tests the suitability of a set of N points for use in an equal-weight quadrature rule over the multi-dimensional 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:

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Source Code:

• test_nint.f90, the source code.
• test_nint.sh, compiles the source code.