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

naginterfaces.library.quad.dim1_fin_bad_vec(f, a, b, epsabs, epsrel, lw=800, liw=None, data=None)

dim1_fin_bad_vec is a general purpose integrator 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_bad_vec will be removed in naginterfaces 31.3.0.0. Please use dim1_fin_general() instead. See also the Replacement Calls document.

For full information please refer to the NAG Library document for d01at

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/d01/d01atf.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.

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 $6$)

On entry, $liw=⟨value⟩$.

Constraint: $liw\ge 1$.

(errno $6$)

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

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

dim1_fin_bad_vec is based on the QUADPACK routine QAGS (see Piessens et al. (1983)). It is an adaptive function, using the Gauss $10$-point and Kronrod $21$-point rules. The algorithm, described in de Doncker (1978), 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 in Piessens et al. (1983).

The function is suitable as a general purpose integrator, and can be used when the integrand has singularities, especially when these are of algebraic or logarithmic type.

dim1_fin_bad_vec requires a function to evaluate the integrand at an array of different points and is, therefore, amenable to parallel execution. Otherwise the algorithm is identical to that used by dim1_fin_bad().

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 $e_m\left(S_n\right)$ transformation, Math. Tables Aids Comput. (10), 91–96