Integrates a user-defined function over an infinite interval.


y = inthp1(&f, pds, ctl)
  • &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().


y (scalar) – estimated integral of \(f(x)\) evaluated over the interval \((-∞,+∞)\).


** Function to be integrated
** note that ds pointer is
** first input (*pds)
proc fct(struct DS *pds, x);
    local var;
    var = pds->dataMatrix;
    retp( exp( -(x*x) / (2*var) ));

// Define DS structure
struct DS d0;

// Define DS structure pointer
struct DS *pds;

// Define variance
variance = 3;

// Set pointer to d0
pds = &d0;

// Set d0 data matrix
d0.dataMatrix = variance;

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

// Integrate function fct
r = inthp1(&fct, pds, c0);

// Print results
format /ld 16,10;
print r;
print sqrt(2*pi*variance);

results in the following output:



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


The user-provided function must have the following format

f(struct DS *pds, x)


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 clt.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.