naginterfaces.library.quad.dim1_fin_wtrig(g, a, b, omega, key, epsabs, epsrel, lw=800, liw=None, data=None)[source]

dim1_fin_wtrig calculates an approximation to the sine or the cosine transform of a function over :

(for a user-specified value of ).

For full information please refer to the NAG Library document for d01an

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/d01/d01anf.html

Parameters
gcallable retval = g(x, data=None)

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

Parameters
xfloat

The point at which the function must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of evaluated at .

afloat

, the lower limit of integration.

bfloat

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

omegafloat

The argument in the weight function of the transform.

keyint

Indicates which integral is to be computed.

.

.

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: and .

(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_wtrig is based on the QUADPACK routine QFOUR (see Piessens et al. (1983)). It is an adaptive function, designed to integrate a function of the form , where is either or . If a sub-interval has length

then the integration over this sub-interval is performed by means of a modified Clenshaw–Curtis procedure (see Piessens and Branders (1975)) if and In this case a Chebyshev series approximation of degree is used to approximate , while an error estimate is computed from this approximation together with that obtained using Chebyshev series of degree . If the above conditions do not hold then Gauss -point and Kronrod -point rules are used. The algorithm, described in Piessens et al. (1983), incorporates a global acceptance criterion (as defined in 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).

References

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

Piessens, R and Branders, M, 1975, Algorithm 002: computation of oscillating integrals, J. Comput. Appl. Math. (1), 153–164

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