inthp4#

Purpose#

Integrates a user-defined function over the \([a, b]\) interval.

Format#

y = inthp4(&f, pds, ctl, lims)#

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().

lims (2×N vector) – upper and lower limits of integration, the ﬁrst row contains upper limits and the second row the lower.

Returns:

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

Examples#

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

   a = pd0->dataMatrix[1];
   b = pd0->dataMatrix[2];
   c = pd0->dataMatrix[3];

   retp( 1/sqrt(a*x*x + b*x + c));
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/2;

// Set d0.dataMatrix
a = -1;
b = -2;
c = 3;
d0.dataMatrix = a|b|c;

// Set pointer to d0
pd0 = &d0;

// Set limits of integration
lims = 1 | -1;

// Integrate function
r = inthp4(&fct, pd0, c0, lims);

// Print results
format /ld 16,10;
print r;
print pi/2;

produces the following output:

1.5707962283
1.5707963268

References#

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

Remarks#

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

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