naginterfaces.library.quad.dim1_​fin_​brkpts

naginterfaces.library.quad.dim1_fin_brkpts(f, a, b, points, epsabs, epsrel, maxsub, data=None)

dim1_fin_brkpts is a general purpose integrator which calculates an approximation to the integral of a function $f\left(x\right)$ over a finite interval $[a,b]$:

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

where the integrand may have local singular behaviour at a finite number of points within the integration interval.

For full information please refer to the NAG Library document for d01rl

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d01/d01rlf.html

Parameters

fcallable (fv, iflag) = f(x, iflag, data=None)

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

Parameters

xfloat, ndarray, shape $\left(\text{nx}\right)$

The abscissae, $x_i$, for $\text{i}=1,2,\dots ,\text{nx}$, at which function values are required.

iflagint

$iflag=0$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

fvfloat, array-like, shape $\left(\text{nx}\right)$

$fv$ must contain the values of the integrand $f$. $fv[i-1]=f\left(x_i\right)$ for all $i=1,2,\dots ,\text{nx}$.

iflagint

Set $iflag<0$ to force an immediate exit with $errno$ = -1.

afloat

$a$, the lower limit of integration.

bfloat

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

Note: if $a=b$, the function will immediately return with $result=0.0$, $abserr=0.0$, $rinfo=0.0$ and $iinfo=0$.

pointsfloat, array-like, shape $\left(\text{npts}\right)$

The user-specified break-points.

epsabsfloat

${ϵ}_a$, the absolute accuracy required. If $epsabs$ is negative, ${ϵ}_a=\left|epsabs\right|$. See Accuracy.

epsrelfloat

${ϵ}_r$, the relative accuracy required. If $epsrel$ is negative, ${ϵ}_r=\left|epsrel\right|$. See Accuracy.

maxsubint

${max}_{\text{sdiv}}$, the upper bound on the total number of subdivisions dim1_fin_brkpts may use to generate new segments. This does not include the initial segments of the interval generated from the supplied break-points. If ${max}_{\text{sdiv}}=1$, only the initial segments will be evaluated.

Suggested value: a value in the range $200$ to $500$ is adequate for most problems.

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 $|I-result|$.

rinfofloat, ndarray, shape $(4\times max(maxsub,\text{npts})+\text{npts}+6)$

Details of the computation. See Further Comments for more information.

iinfoint, ndarray, shape $(2\times max(maxsub,\text{npts})+\text{npts}+4)$

Details of the computation. See Further Comments for more information.

Raises

NagValueError

(errno $41$)

On entry, $\text{npts}=⟨value⟩$.

Constraint: $\text{npts}\ge 0$.

(errno $51$)

On entry, break-points are specified outside $(a,b)$: $a=⟨value⟩$ and $b=⟨value⟩$.

(errno $81$)

On entry, $maxsub=⟨value⟩$.

Constraint: $maxsub\ge 1$.

Warns

NagAlgorithmicWarning

(errno $-1$)

Exit from $f$ with $iflag<0$.

(errno $1$)

The maximum number of subdivisions ($maxsub$) has been reached: $maxsub=⟨value⟩$.

(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

dim1_fin_brkpts is based on the QUADPACK routine QAGP (see Piessens et al. (1983)). It is very similar to dim1_fin_general(), but allows you to supply ‘break-points’, points at which the integrand is known to be difficult. It employs an adaptive algorithm, 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 user-supplied ‘break-points’ always occur as the end points of some sub-interval during the adaptive process. The local error estimation is described in Piessens et al. (1983).

dim1_fin_brkpts requires you to supply a function to evaluate the integrand at an array of points.

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