dim1_fin_bad is a general purpose integrator which calculates an approximation to the integral of a function over a finite interval :

For full information please refer to the NAG Library document for d01aj

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/d01/d01ajf.html

Parameters
fcallable retval = f(x, data=None)

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

Parameters
xfloat

The point at which the integrand must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of the integrand at .

afloat

, the lower limit of integration.

bfloat

, the upper limit of integration. It is not necessary that .

epsabsfloat

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

epsrelfloat

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

lwint, optional

The value of (together with that of ) imposes a bound on the number of sub-intervals into which the interval of integration may be divided by the function. The number of sub-intervals cannot exceed . The more difficult the integrand, the larger should be.

liwNone or int, optional

Note: if this argument is None then a default value will be used, determined as follows: .

The number of sub-intervals into which the interval of integration may be divided cannot exceed .

dataarbitrary, optional

User-communication data for callback functions.

Returns
resultfloat

The approximation to the integral .

abserrfloat

An estimate of the modulus of the absolute error, which should be an upper bound for .

wfloat, ndarray, shape

iwint, ndarray, shape

contains the actual number of sub-intervals used. The rest of the array is used as workspace.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

The maximum number of subdivisions () has been reached: , and .

(errno )

Round-off error prevents the requested tolerance from being achieved: and .

(errno )

Extremely bad integrand behaviour occurs around the sub-interval .

(errno )

Round-off error is detected in the extrapolation table.

(errno )

The integral is probably divergent or slowly convergent.

Notes

dim1_fin_bad is based on the QUADPACK routine QAGS (see Piessens et al. (1983)). It is an adaptive function, using the Gauss -point and Kronrod -point rules. The algorithm, described in de Doncker (1978), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the -algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described in Piessens et al. (1983).

The function is suitable as a general purpose integrator, and can be used when the integrand has singularities, especially when these are of algebraic or logarithmic type.

dim1_fin_bad requires you to supply a function to evaluate the integrand at a single point.

The function dim1_fin_bad_vec() uses an identical algorithm but requires you to supply a function to evaluate the integrand at an array of points. Therefore, dim1_fin_bad_vec() may be more efficient for some problem types and some machine architectures.

References

de Doncker, E, 1978, An adaptive extrapolation algorithm for automatic integration, ACM SIGNUM Newsl. (13(2)), 12–18

Malcolm, M A and Simpson, R B, 1976, Local versus global strategies for adaptive quadrature, ACM Trans. Math. Software (1), 129–146

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

Wynn, P, 1956, On a device for computing the transformation, Math. Tables Aids Comput. (10), 91–96