Object-oriented Scientific Computing Library: Version 0.910
Public Member Functions
cern_gauss< func_t > Class Template Reference

Gaussian quadrature (CERNLIB) More...

#include <cern_gauss.h>

Inheritance diagram for cern_gauss< func_t >:
inte< func_t >

Detailed Description

template<class func_t = funct>
class cern_gauss< func_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

Idea for Future:
Allow user to change cst?

Definition at line 93 of file cern_gauss.h.

Public Member Functions

virtual int integ_err (func_t &func, double a, double b, double &res, double &err)
 Integrate function func from a to b giving result res and error err.
virtual double integ (func_t &func, double a, double b)
 Integrate function func from a to b.

Protected Attributes

Integration constants (Set in the constructor)
double x [12]
double w [12]

The documentation for this class was generated from the following file:
 All Data Structures Namespaces Files Functions Variables Typedefs Enumerations Enumerator Friends Defines

Documentation generated with Doxygen. Provided under the GNU Free Documentation License (see License Information).

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