naginterfaces.library.quad.dim1_​fin_​osc_​vec

naginterfaces.library.quad.dim1_fin_osc_vec(f, a, b, epsabs, epsrel, key=6, lw=800, liw=None, data=None)

dim1_fin_osc_vec is an adaptive integrator, especially suited to oscillating, nonsingular integrands, which calculates an approximation to the integral of a function $f\left(x\right)$ over a finite interval $[a,b]$:

$$I={\int }_a^{b}f\left(x\right)dx\text{.}$$

Deprecated since version 27.1.0.0: dim1_fin_osc_vec will be removed in naginterfaces 31.3.0.0. Please use dim1_fin_osc_fn() instead. See also the Replacement Calls document.

For full information please refer to the NAG Library document for d01au

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d01/d01auf.html

Parameters

fcallable fv = f(x, data=None)

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

Parameters

xfloat, ndarray, shape $\left(n\right)$

The points at which the integrand $f$ must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

fvfloat, array-like, shape $\left(n\right)$

$fv[\text{j}-1]$ must contain the value of $f$ at the point $x[\text{j}-1]$, for $\text{j}=1,2,\dots ,n$.

afloat

$a$, the lower limit of integration.

bfloat

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

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.

keyint, optional

Indicates which integration rule is to be used.

$key=1$

For the Gauss $7$-point and Kronrod $15$-point rule.

$key=2$

For the Gauss $10$-point and Kronrod $21$-point rule.

$key=3$

For the Gauss $15$-point and Kronrod $31$-point rule.

$key=4$

For the Gauss $20$-point and Kronrod $41$-point rule.

$key=5$

For the Gauss $25$-point and Kronrod $51$-point rule.

$key=6$

For the Gauss $30$-point and Kronrod $61$-point rule.

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: $lw/4$.

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

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 $4$)

On entry, $key=⟨value⟩$.

Constraint: $key\ge 1$ and $key\le 6$.

(errno $5$)

On entry, $liw=⟨value⟩$.

Constraint: $liw\ge 1$.

(errno $5$)

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 $(⟨value⟩,⟨value⟩)$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

dim1_fin_osc_vec is based on the QUADPACK routine QAG (see Piessens et al. (1983)). It is an adaptive function, offering a choice of six Gauss–Kronrod rules. A global acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation is described in Piessens et al. (1983).

Because dim1_fin_osc_vec is based on integration rules of high order, it is especially suitable for nonsingular oscillating integrands.

dim1_fin_osc_vec requires a function to evaluate the integrand at an array of different points and is, therefore, amenable to parallel execution. Otherwise this algorithm with $key=6$ is identical to that used by dim1_fin_osc().

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, 1973, An algorithm for automatic integration, Angew. Inf. (15), 399–401

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