inthp3

Purpose

Integrates a user-defined function over the \([a,+∞)\) interval that is oscillatory.

Format

y = inthp3(&f, pds, ctl, l_lim)
Parameters:
  • &f (scalar) – pointer to the procedure containing the function to be integrated.
  • pds (scalar) –

    pointer to instance of a DS structure. The members of the DS are:

    pds->dataMatrix NxK matrix.
    pds->dataArray NxKxL… array.
    pds->vnames string array.
    pds->dsname string.
    pds->type scalar.

    The contents, if any, are set by the user and are passed by inthp1() to the user-provided function without modification.

  • ctl (struct) –

    instance of an inthpControl structure with members

    ctl.maxEvaluations scalar, maximum number of function evaluations, default = 1e5;
    ctl.p scalar, termination parameter
    0:heuristic termination, default.
    1:deterministic termination with infinity norm.
    2,…:deterministic termination with p-th norm.
    ctl.d scalar termination parameter
    1:if heuristic termination
    0 < ctl.d < π/2:
     if deterministic termination
    ctl.eps scalar, relative error bound. Default = 1e-6.

    A default ctl can be generated by calling inthpControlCreate().

  • l_lim (1xN vector) – lower limits of integration.
Returns:

y (Nx1 vector) – the estimated integrals of \(f(x)\) evaluated over the interval \([a,+∞)\).

Examples

/*
** Function to be integrated
** note that ds pointer is
** first input (*pd0)
*/
proc fct(struct DS *pd0, x);
   local m, a;

   m = pd0->dataMatrix[1];
   a = pd0->dataMatrix[2];

   retp( exp(-a*x)*cos(m*x));
endp;

// Define DS structure
struct DS d0;

// Define DS structure pointer
struct DS *pd0;

/*
** Declare instance of inthpControl
** structure
*/
struct inthpControl c0;
c0 = inthpControlCreate;

// Set termination parameter
c0.p = 2;

// Set termination parameter
c0.d = pi/3;

// Set pointer to d0
pd0 = &d0;

// Set d0.dataMatrix
m = 2;
a = 1;
d0.dataMatrix = m | a;

// Set lower
l_lim = 0;

// Integrate function fct
r = inthp3(&fct, pd0, c0, l_lim);

// Print results
format /ld 16,10;
print r;
print a/(a*a + m*m);

produces the following output:

0.2000000000
0.2000000000

References

  1. “Optimal Quadratures in H_p Spaces” by K. Sikorski and F. Stenger, ACM Transactions on Mathematical Software, 10(2):140-151, June 1984.

Remarks

This procedure is designed especially for oscillatory functions.

The user-provided function must have the following format

f(struct DS *pds, x)

where

pds scalar, pointer to an instance of a DS structure.
x scalar, value at which integral will be evaluated.

If ctl.d can be specified (see Sikorski and Stenger, 1984), deterministic termination can be specified and accuracy guaranteed. if not, the heuristic method can be used and the value of ctl.d is disregarded.

The pointer to the instance of the data structure, pds, is passed untouched to the user-provided procedure computing the function to be integrated. Any information needed by that function can be put into that data structure.

Source

inthp.src