naginterfaces.library.quad.dim1_inf_general(f, bound, inf, epsabs, epsrel, maxsub, data=None)[source]

dim1_inf_general calculates an approximation to the integral of a function over an infinite or semi-infinite interval :

For full information please refer to the NAG Library document for d01rm

https://www.nag.com/numeric/nl/nagdoc_29/flhtml/d01/d01rmf.html

Parameters
fcallable (fv, iflag) = f(x, iflag, data=None)

must return the values of the integrand at a set of points.

Parameters
xfloat, ndarray, shape

The abscissae, , for , at which function values are required.

iflagint

.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
fvfloat, array-like, shape

must contain the values of the integrand . for all .

iflagint

Set to force an immediate exit with = -1.

boundfloat

The finite limit of the integration range (if present). is not used if the interval is doubly infinite.

infint

Indicates the kind of integration range.

The range is .

The range is .

The range is .

epsabsfloat

, the absolute accuracy required. If is negative, . See Accuracy.

epsrelfloat

, the relative accuracy required. If is negative, . See Accuracy.

maxsubint

, the upper bound on the total number of subdivisions dim1_inf_general may use to generate new segments. If , only the initial segment will be evaluated.

Suggested value: a value in the range to is adequate for most problems.

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 .

rinfofloat, ndarray, shape

iinfoint, ndarray, shape

Raises
NagValueError
(errno )

On entry, .

Constraint: , or .

(errno )

On entry, .

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

Exit from with .

(errno )

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

(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_inf_general is based on the QUADPACK routine QAGI (see Piessens et al. (1983)). The entire infinite integration range is first transformed to using one of the identities:

where represents a finite integration limit. An adaptive procedure, based on the Gauss -point and Kronrod -point rules, is then employed on the transformed integral. 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).

dim1_inf_general requires you to supply a function to evaluate the integrand at an array of points.

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