naginterfaces.library.quad.dim1_​inf_​general

naginterfaces.library.quad.dim1_inf_general(f, bound, inf, epsabs, epsrel, maxsub, data=None)

dim1_inf_general calculates an approximation to the integral of a function $f\left(x\right)$ over an infinite or semi-infinite interval $[a,b]$:

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

For full information please refer to the NAG Library document for d01rm

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/d01/d01rmf.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.

boundfloat

The finite limit of the integration range (if present). $bound$ is not used if the interval is doubly infinite.

infint

Indicates the kind of integration range.

$inf=1$

The range is $[bound,+\infty )$.

$inf=-1$

The range is $(-\infty ,bound]$.

$inf=2$

The range is $(-\infty ,+\infty )$.

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_inf_general may use to generate new segments. If ${max}_{\text{sdiv}}=1$, only the initial segment 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 $(4\times maxsub)$

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

iinfoint, ndarray, shape $(max(maxsub,4))$

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

Raises

NagValueError

(errno $31$)

On entry, $inf=⟨value⟩$.

Constraint: $inf=-1$, $1$ or $2$.

(errno $61$)

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_inf_general is based on the QUADPACK routine QAGI (see Piessens et al. (1983)). The entire infinite integration range is first transformed to $[0,1]$ using one of the identities:

$${\int }_{-\infty }^{a}f\left(x\right)dx={\int }_0^{1}f(a-\frac{1-t}{t})\frac{1}{t^2}dt$$

$${\int }_a^{\infty }f\left(x\right)dx={\int }_0^{1}f(a+\frac{1-t}{t})\frac{1}{t^2}dt$$

$${\int }_{-\infty }^{\infty }f\left(x\right)dx={\int }_0^{\infty }(f\left(x\right)+f(-x))dx={\int }_0^1[f\left(\frac{1-t}{t}\right)+f\left(\frac{-1+t}{t}\right)]\frac{1}{t^2}dt\text{,}$$

where $a$ represents a finite integration limit. An adaptive procedure, based on the Gauss $7$-point and Kronrod $15$-point rules, is then employed on the transformed integral. 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).

dim1_inf_general 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