Header Integrate.h

Detailed Description

Integrates a function.

Synopsis

#include <lal/Integrate.h>

This header covers the routines for integrating a function.

Description

The routine LALSRombergIntegrate() performs the integral specified by the structure input and the result is returned as result. Any additional parameters (other than the integration variable \(x\)) can be passed as params. The routine LALSRombergIntegrate() does not use params but just passes it to the integrand. The routine LALDRombergIntegrate() is the same but for double precision.

Operating Instructions

The following program performs the integral \(\int_0^2F(x)dx\) where \(F(x)=x^4\log(x+\sqrt{x^2+1})\).

#include <math.h>
#include <lal/LALStdlib.h>
#include <lal/Integrate.h>
 
static void F( LALStatus *s, REAL4 *y, REAL4 x, void *p )
{
REAL4 x2 = x*x;
REAL4 x4 = x2*x2;
INITSTATUS(s);
ASSERT( !p, s, 1, "Non-null pointer" );
y = x4 * log( x + sqrt( x2 + 1 ) );
RETURN( s );
}
 
int main ()
{
const REAL4       epsilon = 1e-6;
const long double expect  = 8.153364119811650205L;
static LALStatus  status;
SIntegrateIn      intinp;
REAL4             result;
 
intinp.function = F;
intinp.xmin     = 0;
intinp.xmax     = 2;
intinp.type     = ClosedInterval;
 
LALSRombergIntegrate( &status, &result, &intinp, NULL );
if ( fabs( result - expect ) > epsilon * fabs( expect ) )
{
// integration did not achieve desired accuracy --- exit failure
return 1;
}
 
return 0;
}

Algorithm

This is an implementation of the Romberg integrating function qromb in Numerical Recipes [22] .

Prototypes
REAL8	XLALREAL8RombergIntegrate (REAL8(*f)(REAL8 x, void *params), void *params, REAL8 xmin, REAL8 xmax, IntegralType type)

Enumerations
enum	IntegralType {   ClosedInterval , OpenInterval , SingularLowerLimit , SingularUpperLimit ,   InfiniteDomainPow , InfiniteDomainExp }
	Type of integral. More...

Files
file	IntegrateTest.c
	Tests the routines in Header Integrate.h by performing a suite of numerical integrations and checking the accuracy of the results.

Function Documentation

XLALREAL8RombergIntegrate()

REAL8 XLALREAL8RombergIntegrate	(	REAL8(*)(REAL8 x, void *params)	f,
		void *	params,
		REAL8	xmin,
		REAL8	xmax,
		IntegralType	type
	)

See also

See Header Integrate.h for documentation

Definition at line 233 of file Integrate.c.

Enumeration Type Documentation

IntegralType

enum IntegralType

Type of integral.

The types of integration are the following:

I. ClosedInterval indicates that the integral should be computed on equal-spaced domain intervals including the boundary.

II. OpenInterval indicates that the integral should be computed on intervals of the domain not including the boundary.

III. SingularLowerLimit indicates that the integral should be evaluated on an open interval with a transformation so that a inverse-square-root singularity at the lower limit can be integrated.

IV. SingularUpperLimit is the same as above but for a singularity at the upper limit.

V. InfiniteDomainPow indicates that the integral should be evaluated over an semi-infinite domain—appropriate when both limits have the same sign (though one is very large) and when the integrand vanishes faster than \(x^{-1}\) at infinity.

VI. InfiniteDomainExp indicates that the integral should be evaluated over an infinite domain starting at xmin and going to infinity (xmax is ignored)—the integrand should vanish exponentially for large \(x\).

Enumerator
ClosedInterval	evaluate integral on a closed interval
OpenInterval	evaluate integral on an open interval
SingularLowerLimit	integrate an inv sqrt singularity at lower limit
SingularUpperLimit	integrate an inv sqrt singularity at upper limit
InfiniteDomainPow	integrate infinite domain with power-law falloff
InfiniteDomainExp	integrate infinite domain with exponential falloff

Definition at line 134 of file Integrate.h.