TEST_INT_2D
Quadrature Tests for 2D Finite Intervals

TEST_INT_2D is a FORTRAN90 program which evaluates test integrands of a 2D argument.

The test integrands would normally be used to testing 2D quadrature software. It is possible to invoke a particular function by number, or to try out all available functions, as demonstrated in the sample calling program.

The current set of problems is:

1. integral on [0,1]x[0,1] of f(x,y) = 1 / ( 1 - x * y ); singular at [1,1].
2. integral on [-1,1]x[-1,1] of f(x,y) = 1 / sqrt ( 1 - x * x * y * y ); singular at [1,1], [1,-1], [-1,1], [-1,-1];
3. integral on [-1,1]x[-1,1] of f(x,y) = 1 / sqrt ( 2 - x - y ); singular at [1,1];
4. integral on [-1,1]x[-1,1] of f(x,y) = 1 / sqrt ( 3 - x - 2 * y ); singular along the line y = ( 3 - x ) / 2.
5. integral on [0,1]x[0,1] of f(x,y) = sqrt ( x * y ); singular along the lines y = 0 and x = 0.
6. integral on [-1,1]x[-1,1] of f(x,y) = abs ( x * x + y * y - 1/4 ); nondifferentiable along x*x+y*y=1/4.
7. integral on [0,1]x[0,1] of f(x,y) = sqrt ( abs ( x - y ) ); nondifferentiable along y = x.
8. integral on [0,5]x[0,5] of f(x,y) = exp ( - ( (x-4)^2 + (y-1)^2 ) ), highly localized near (4,1).

The library includes not just the integrand, but also the interval of integration, and the exact value of the integral. Thus, for each integrand function, three subroutines are supplied. For instance, for function #5, we have the routines:

• P05_FUN evaluates the integrand for problem 5.
• P05_LIM returns the integration limits for problem 5.
• P05_EXACT returns the exact integral for problem 5.

Moreover, since the same interface is used for each function, if you wish to work with problem 2 instead, you simply change the "05" to "02" in your routine calls.

If you wish to call all of the functions, then you simply use the generic interface, which again has three subroutines, but which requires you to specify the problem number as an extra input argument:

• P00_FUN evaluates the integrand for any problem.
• P00_LIM returns the integration limits for any problem.
• P00_EXACT returns the exact integral for any problem.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

TEST_INT_2D is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

TEST_INT, a FORTRAN90 library which defines some test integration problems over 1D finite intervals.

TEST_INT_HERMITE, a FORTRAN90 library which defines some test integration problems over infinite intervals.

TEST_INT_LAGUERRE, a FORTRAN90 library which defines test integrands for integration over [ALPHA,+oo).

Reference:

1. Gwynne Evans,
   Practical Numerical Integration,
   Wiley, 1993,
   ISBN: 047193898X,
   LC: QA299.3E93.

Source Code:

• test_int_2d.f90, the source code.

Examples and Tests:

• test_int_2d_test.f90, a sample calling program.
• test_int_2d_test.txt, the output file.

List of Routines:

• LEGENDRE_SET sets abscissas and weights for Gauss-Legendre quadrature.
• P00_EXACT returns the exact integral for any problem.
• P00_FUN evaluates the integrand for any problem.
• P00_LIM returns the integration limits for any problem.
• P00_PROBLEM_NUM returns the number of test integration problems.
• P01_EXACT returns the exact integral for problem 1.
• P01_FUN evaluates the integrand for problem 1.
• P01_LIM returns the integration limits for problem 1.
• P02_EXACT returns the exact integral for problem 2.
• P02_FUN evaluates the integrand for problem 2.
• P02_LIM returns the integration limits for problem 2.
• P03_EXACT returns the exact integral for problem 3.
• P03_FUN evaluates the integrand for problem 3.
• P03_LIM returns the integration limits for problem 3.
• P04_EXACT returns the exact integral for problem 4.
• P04_FUN evaluates the integrand for problem 4.
• P04_LIM returns the integration limits for problem 4.
• P05_EXACT returns the exact integral for problem 5.
• P05_FUN evaluates the integrand for problem 5.
• P05_LIM returns the integration limits for problem 5.
• P06_EXACT returns the exact integral for problem 6.
• P06_FUN evaluates the integrand for problem 6.
• P06_LIM returns the integration limits for problem 6.
• P07_EXACT returns the exact integral for problem 7.
• P07_FUN evaluates the integrand for problem 7.
• P07_LIM returns the integration limits for problem 7.
• P08_EXACT returns the exact integral for problem 8.
• P08_FUN evaluates the integrand for problem 8.
• P08_LIM returns the integration limits for problem 8.
• R8_CSEVL evaluates a Chebyshev series.
• R8_ERF evaluates the error function of an R8 argument.
• R8_ERFC evaluates the co-error function of an R8 argument.
• R8_INITS initializes a Chebyshev series.
• R8_MACH returns real ( kind = 8 ) real machine-dependent constants.
• R8MAT_UNIFORM_01 returns a unit pseudorandom R8MAT.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.