naginterfaces.library.quad.dim1_​fin_​wtrig

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

dim1_fin_wtrig calculates an approximation to the sine or the cosine transform of a function $g$ over $[a,b]$:

$$I={\int }_a^{b}g\left(x\right)\sin \left(\omega x\right)dx\text{or}I={\int }_a^{b}g\left(x\right)\cos \left(\omega x\right)dx$$

(for a user-specified value of $\omega$).

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

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d01/d01anf.html

Parameters

gcallable retval = g(x, data=None)

$g$ must return the value of the function $g$ at a given point $x$.

Parameters

xfloat

The point at which the function $g$ must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

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

afloat

$a$, the lower limit of integration.

bfloat

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

omegafloat

The argument $\omega$ in the weight function of the transform.

keyint

Indicates which integral is to be computed.

$key=1$

$w\left(x\right)=\cos \left(\omega x\right)$.

$key=2$

$w\left(x\right)=\sin \left(\omega x\right)$.

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.

lwint, optional

The value of $lw$ (together with that of $liw$) 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 $lw/4$. The more difficult the integrand, the larger $lw$ should be.

liwNone or int, optional

Note: if this argument is None then a default value will be used, determined as follows: $lw/2$.

The number of sub-intervals into which the interval of integration may be divided cannot exceed $liw/2$.

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 $|I-result|$.

wfloat, ndarray, shape $\left(lw\right)$

Details of the computation see Further Comments for more information.

iwint, ndarray, shape $\left(liw\right)$

$iw\left[0\right]$ contains the actual number of sub-intervals used. The rest of the array is used as workspace.

Raises

NagValueError

(errno $6$)

On entry, $key=⟨value⟩$.

Constraint: $key\ge 1$ and $key\le 2$.

(errno $7$)

On entry, $liw=⟨value⟩$.

Constraint: $liw\ge 2$.

(errno $7$)

On entry, $lw=⟨value⟩$.

Constraint: $lw\ge 4$.

Warns

NagAlgorithmicWarning

(errno $1$)

The maximum number of subdivisions ($LIMIT$) has been reached: $LIMIT=⟨value⟩$, $lw=⟨value⟩$ and $liw=⟨value⟩$.

(errno $2$)

Round-off error prevents the requested tolerance from being achieved: $epsabs=⟨value⟩$ and $epsrel=⟨value⟩$.

(errno $3$)

Extremely bad integrand behaviour occurs around the sub-interval $(⟨value⟩,⟨value⟩)$.

(errno $4$)

Round-off error is detected in the extrapolation table.

(errno $5$)

The integral is probably divergent or slowly convergent.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

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 $g\left(x\right)w\left(x\right)$, where $w\left(x\right)$ is either $\sin \left(\omega x\right)$ or $\cos \left(\omega x\right)$. If a sub-interval has length

$$L=|b-a|2^{-l}$$

then the integration over this sub-interval is performed by means of a modified Clenshaw–Curtis procedure (see Piessens and Branders (1975)) if $L\omega >4$ and $l\le 20\text{.}$ In this case a Chebyshev series approximation of degree $24$ is used to approximate $g\left(x\right)$, while an error estimate is computed from this approximation together with that obtained using Chebyshev series of degree $12$. If the above conditions do not hold then Gauss $7$-point and Kronrod $15$-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 $ϵ$-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 $e_m\left(S_n\right)$ transformation, Math. Tables Aids Comput. (10), 91–96