naginterfaces.library.quad.dim1_​fin_​smooth

naginterfaces.library.quad.dim1_fin_smooth(f, a, b, epsabs, epsrel, data=None)

dim1_fin_smooth calculates an approximation to the integral of a function over a finite interval $[a,b]$:

$$I={\int }_a^{b}f\left(x\right)dx\text{.}$$

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 $[a,b]$, or which oscillate too strongly on $[a,b]$.

For full information please refer to the NAG Library document for d01bd

https://www.nag.com/numeric/nl/nagdoc_30/flhtml/d01/d01bdf.html

Parameters

fcallable retval = f(x, data=None)

$f$ must return the value of the integrand $f$ at a given point.

Parameters

xfloat

The point at which the integrand $f$ must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of $f\left(x\right)$ evaluated at $x$.

afloat

$a$, the lower limit of integration.

bfloat

$b$, the upper limit of integration. It is not necessary that $a<b$.

epsabsfloat

The absolute accuracy required. If $epsabs$ is negative, the absolute value is used. See Accuracy.

epsrelfloat

The relative accuracy required. If $epsrel$ is negative, the absolute value is used. See Accuracy.

dataarbitrary, optional

User-communication data for callback functions.

Returns

resultfloat

The approximation to the integral $I$.

abserrfloat

An estimate of the modulus of the absolute error, which should be an upper bound for $|I-result|$.

Notes

dim1_fin_smooth is based on the QUADPACK routine QNG (see Piessens et al. (1983)). It is a non-adaptive function 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 function is designed for smooth integrands only.

References

Patterson, T N L, 1968, The Optimum addition of points to quadrature formulae, Math. Comput. (22), 847–856

Piessens, R, de Doncker–Kapenga, E, Überhuber, C and Kahaner, D, 1983, QUADPACK, A Subroutine Package for Automatic Integration, Springer–Verlag

See also

naginterfaces.library.examples.quad.dim1_fin_smooth_ex.main()