naginterfaces.library.quad.dim1_​inf_​wtrig

naginterfaces.library.quad.dim1_inf_wtrig(g, a, omega, key, epsabs, limlst=50, lw=800, liw=None, data=None)

dim1_inf_wtrig calculates an approximation to the sine or the cosine transform of a function $g$ over $[a,\infty )$:

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

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

For full information please refer to the NAG Library document for d01as

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d01/d01asf.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.

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.

limlstint, optional

An upper bound on the number of intervals $C_k$ needed for the integration.

lwint, optional

The value of $lw$ (together with that of $liw$) imposes a bound on the number of sub-intervals into which each interval $C_k$ 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 each interval $C_k$ 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 $|I-result|$.

lstint

The number of intervals $C_k$ actually used for the integration.

erlstfloat, ndarray, shape $\left(limlst\right)$

$erlst[\text{k}-1]$ contains the error estimate corresponding to the integral contribution over the interval $C_{\text{k}}$, for $\text{k}=1,2,\dots ,lst$.

rslstfloat, ndarray, shape $\left(limlst\right)$

$rslst[\text{k}-1]$ contains the integral contribution over the interval $C_{\text{k}}$, for $\text{k}=1,2,\dots ,lst$.

ierlstint, ndarray, shape $\left(limlst\right)$

$ierlst[\text{k}-1]$ contains the error flag corresponding to $rslst[\text{k}-1]$, for $\text{k}=1,2,\dots ,lst$.

In the cases $errno$ = 7, 8 or 9, additional information about the cause of the error can be obtained from the array $ierlst$, as follows:

$ierlst[k-1]=1$

The maximum number of $\text{subdivisions}=min(lw/4,liw/2)$ has been achieved on the $k$th interval.

$ierlst[k-1]=2$

Occurrence of round-off error is detected and prevents the tolerance imposed on the $k$th interval from being achieved.

$ierlst[k-1]=3$

Extremely bad integrand behaviour occurs at some points of the $k$th interval.

$ierlst[k-1]=4$

The integration procedure over the $k$th interval does not converge (to within the required accuracy) due to round-off in the extrapolation procedure invoked on this interval. It is assumed that the result on this interval is the best which can be obtained.

$ierlst[k-1]=5$

The integral over the $k$th interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of $ierlst[k-1]$.

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

$iw\left[0\right]$ contains the maximum number of sub-intervals actually used for integrating over any of the intervals $C_k$. The rest of the array is used as workspace.

Raises

NagValueError

(errno $6$)

On entry, $limlst=⟨value⟩$.

Constraint: $limlst\ge 3$.

(errno $6$)

On entry, $key=⟨value⟩$.

Constraint: $key\le 2$.

(errno $6$)

On entry, $key=⟨value⟩$.

Constraint: $key\ge 1$.

(errno $10$)

On entry, $liw=⟨value⟩$.

Constraint: $liw\ge 2$.

(errno $10$)

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: $epsabs=⟨value⟩$.

(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.

(errno $7$)

Bad integration behaviour occurs within interval $\text{K}$: $\text{K}=⟨value⟩$: $(⟨value⟩,⟨value⟩)$. Inspect $ierlst$ for more details.

(errno $8$)

The maximum number of intervals ($limlst$) has been reached: $limlst=⟨value⟩$. Inspect $ierlst$ for more details.

(errno $9$)

Extrapolation does not converge to the requested accuracy. Inspect $ierlst$ for more details.

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_inf_wtrig is based on the QUADPACK routine QAWFE (see Piessens et al. (1983)). It is an adaptive function, designed to integrate a function of the form $g\left(x\right)w\left(x\right)$ over a semi-infinite interval, where $w\left(x\right)$ is either $\sin \left(\omega x\right)$ or $\cos \left(\omega x\right)$.

Over successive intervals

$$C_k=[a+(k-1)c,a+kc]\text{,}k=1,2,\dots ,lst$$

integration is performed by the same algorithm as is used by dim1_fin_wtrig(). The intervals $C_k$ are of constant length

$$c=\{2\left[\left|\omega \right|\right]+1\}\pi /\left|\omega \right|\text{,}\omega \ne 0\text{,}$$

where $\left[\left|\omega \right|\right]$ represents the largest integer less than or equal to $\left|\omega \right|$. Since $c$ equals an odd number of half periods, the integral contributions over succeeding intervals will alternate in sign when the function $g$ is positive and monotonically decreasing over $[a,\infty )$. The algorithm, described in Piessens et al. (1983), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the $ϵ$-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described by Piessens et al. (1983).

If $\omega =0$ and $key=1$, the function uses the same algorithm as dim1_inf_general() (with $\text{epsrel}=0.0$).

In contrast to the other functions in submodule quad, dim1_inf_wtrig works only with an absolute error tolerance ($epsabs$). Over the interval $C_k$ it attempts to satisfy the absolute accuracy requirement

$${\text{EPSA}}_k=U_k\times epsabs\text{,}$$

where $U_{\text{k}}=(1-p)p^{\text{k}-1}$, for $\text{k}=1,2,\dots ,$ and $p=0.9$.

However, when difficulties occur during the integration over the $k$th sub-interval $C_k$ such that the error flag $ierlst[k-1]$ is nonzero, the accuracy requirement over subsequent intervals is relaxed. See Piessens et al. (1983) for more details.

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, 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