Gaussian quadrature (CERNLIB) More...

#include <inte_gauss_cern.h>

Inheritance diagram for o2scl::inte_gauss_cern< func_t, fp_t, weights_t >:

Detailed Description

template<class func_t = funct, class fp_t = double, class weights_t = inte_gauss_coeffs_double>
class o2scl::inte_gauss_cern< func_t, fp_t, weights_t >

For any interval $ (a,b) $ , we define $ g_8(a,b) $ and $g_{16}(a,b)$ to be the 8- and 16-point Gaussian quadrature approximations to

$I=\int_a^b f(x)~dx$

and define

$r(a,b)=\frac{|g_{16}(a,b)-g_{8}(a,b)|}{1+g_{16}(a,b)}$

The function integ() returns $ G $ given by

$G=\sum_{i=1}^{k} g_{16}(x_{i-1},x_i)$

where $ x_0=a $ and $ x_k=b $ and the subdivision points $ x_i $ are given by

$x_i=x_{i-1}+\lambda(B-x_{i-1})$

where $\lambda$ is the first number in the sequence $1,\frac{1}{2},\frac{1}{4},...$ for which

$r(x_{i-1},x_i)<\mathrm{eps}.$

If, at any stage, the ratio

$q=\left| \frac{x_i-x_{i-1}}{b-a} \right|$

is so small so that $ 1+0.005 q $ is indistinguishable from unity, then the accuracy is required is not reachable and the error handler is called.

Unless there is severe cancellation, inte::tol_rel may be considered as specifying a bound on the relative error of the integral in the case that $ |I|>1 $ and an absolute error if $ |I|<1 $ . More precisely, if $ k $ is the number of subintervals from above, and if

$I_{abs} = \int_a^b |f(x)|~dx$

then

$\frac{|G-I|}{I_{abs}+k}<\mathrm{tol_rel}$

will nearly always be true when no error is returned. For functions with no singualarities in the interval, the accuracy will usually be higher than this.

If the desired accuracy is not achieved, the integration functions will call the error handler and return the best guess, unless inte::err_nonconv is false, in which case the error handler is not called.

This function is based on the CERNLIB routines GAUSS and DGAUSS which are documented at http://wwwasdoc.web.cern.ch/wwwasdoc/shortwrupsdir/d103/top.html

Note: Currently O2scl supports only types double and long double for the floating point type fp_t .

Idea for Future:: Allow user to change cst?

Definition at line 293 of file inte_gauss_cern.h.

Public Member Functions
virtual int	integ_err (func_t &func, fp_t a, fp_t b, fp_t &res, fp_t &err)
	Integrate function `func` from `a` to `b`.

Public Member Functions inherited from o2scl::inte< funct, double >
virtual double	integ (funct &func, double a, double b)
	Integrate function `func` from `a` to `b`.

virtual int	integ_err (funct &func, double a, double b, double &res, double &err)=0
	Integrate function `func` from `a` to `b` and place the result in `res` and the error in `err`.

double	get_error ()
	Return the numerically estimated error in the result from the last call to integ() More...

virtual const char *	type ()
	Return string denoting type ("inte")

Public Attributes
const fp_t *	w

const fp_t *	x

weights_t	wgts

Public Attributes inherited from o2scl::inte< funct, double >
int	verbose
	Verbosity.

size_t	last_iter
	The most recent number of iterations taken.

double	tol_rel
	The maximum relative uncertainty in the value of the integral (default $10^{-8}$ )

double	tol_abs
	The maximum absolute uncertainty in the value of the integral (default $10^{-8}$ )

bool	err_nonconv
	If true, call the error handler if the routine does not converge or reach the desired tolerance (default true) More...

Additional Inherited Members
Protected Attributes inherited from o2scl::inte< funct, double >
double	interror
	The uncertainty for the last integration computation.

The documentation for this class was generated from the following file:

/home/awsteiner/wcs/o2scl/src/inte/inte_gauss_cern.h

Detailed Description

template<class func_t = funct, class fp_t = double, class weights_t = inte_gauss_coeffs_double> class o2scl::inte_gauss_cern< func_t, fp_t, weights_t >

Public Member Functions

Public Attributes

Additional Inherited Members

template<class func_t = funct, class fp_t = double, class weights_t = inte_gauss_coeffs_double>
class o2scl::inte_gauss_cern< func_t, fp_t, weights_t >