D01BDF calculates an approximation to the integral of a function over a finite interval
$\left[a,b\right]$:
It is non-adaptive and as such is recommended for the integration of ‘smooth’ functions. These
exclude integrands with singularities, derivative singularities or high peaks on
$\left[a,b\right]$, or which oscillate too strongly on
$\left[a,b\right]$.
D01BDF is based on the QUADPACK routine QNG (see
Piessens et al. (1983)). It is a non-adaptive routine which uses as its basic rules, the Gauss
$10$-point and
$21$-point formulae. If the accuracy criterion is not met, formulae using
$43$ and
$87$ points are used successively, stopping whenever the accuracy criterion is satisfied.
This routine is designed for smooth integrands only.
There are no specific errors detected by D01BDF. However, if
ABSERR is greater than
this indicates that the routine has probably failed to achieve the requested accuracy within
$87$ function evaluations.
D01BDF attempts to compute an approximation,
RESULT, such that:
where
and
EPSABS and
EPSREL are user-specified absolute and relative error tolerances. There can be no guarantee that this is achieved, and you are advised to subdivide the interval if you have any doubts about the accuracy obtained. Note that
ABSERR contains an estimated bound on
$\left|I-{\mathbf{RESULT}}\right|$.
The time taken by D01BDF depends on the integrand and the accuracy required.
None.