a FORTRAN90 code 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:
integral on [0,1]x[0,1] of f(x,y) = 1 / ( 1 - x * y );
singular at [1,1].
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];
integral on [-1,1]x[-1,1] of f(x,y) = 1 / sqrt ( 2 - x - y );
singular at [1,1];
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.
integral on [0,1]x[0,1] of f(x,y) = sqrt ( x * y );
singular along the lines y = 0 and x = 0.
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.
integral on [0,1]x[0,1] of f(x,y) = sqrt ( abs ( x - y ) );
nondifferentiable along y = x.
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:
So once you have the calling sequences for these routines, you
can easily evaluate the function, or integrate it between the
appropriate limits, or compare your estimate of the integral
to the exact value.
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.
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
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:
a FORTRAN90 code which
defines some test integration problems over 1D finite intervals.
Practical Numerical Integration,
Last revised on 03 September 2020.