naginterfaces.library.quad.dim1_fin_general¶

naginterfaces.library.quad.dim1_fin_general(f, a, b, epsabs, epsrel, maxsub, data=None)[source]¶

dim1_fin_general is a general purpose integrator which calculates an approximation to the integral of a function $f (x)$ over a finite interval $[a, b]$ :

I = \int_{a}^{b} f (x) d x .

For full information please refer to the NAG Library document for d01rj

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d01/d01rjf.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 $(nx)$: The abscissae, $x_{i}$ , for $i = 1, 2, \dots, nx$ , at which function values are required.
iflagint: $i f l a g = 0$ .
dataarbitrary, optional, modifiable in place: User-communication data for callback functions.

Returns

fvfloat, array-like, shape $(nx)$: $f v$ must contain the values of the integrand $f$ . $f v [i - 1] = f (x_{i})$ for all $i = 1, 2, \dots, nx$ .
iflagint: Set $i f l a g < 0$ to force an immediate exit with $e r r n o$ = -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 $r e s u l t = 0.0$ , $a b s e r r = 0.0$ , $r i n f o = 0.0$ and $i i n f o = 0$ .

epsabsfloat

$ϵ_{a}$ , the absolute accuracy required. If $e p s a b s$ is negative, $ϵ_{a} = | e p s a b s |$ . See Accuracy.

epsrelfloat

$ϵ_{r}$ , the relative accuracy required. If $e p s r e l$ is negative, $ϵ_{r} = | e p s r e l |$ . See Accuracy.

maxsubint

${m a x}_{sdiv}$ , the upper bound on the total number of subdivisions dim1_fin_general may use to generate new segments. If ${m a x}_{sdiv} = 1$ , 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 $| I - r e s u l t |$ .
rinfofloat, ndarray, shape $(4 \times m a x s u b)$: Details of the computation. See Further Comments for more information.
iinfoint, ndarray, shape $(max (m a x s u b, 4))$: Details of the computation. See Further Comments for more information.

Raises

NagValueError

(errno $61$ )

On entry, $m a x s u b = ⟨ v a l u e ⟩$ .

Constraint: $m a x s u b \geq 1$ .

Warns

NagAlgorithmicWarning

(errno $- 1$ ): Exit from $f$ with $i f l a g < 0$ .
(errno $1$ ): The maximum number of subdivisions ( $m a x s u b$ ) has been reached: $m a x s u b = ⟨ v a l u e ⟩$ .
(errno $2$ ): Round-off error prevents the requested tolerance from being achieved: $e p s a b s = ⟨ v a l u e ⟩$ and $e p s r e l = ⟨ v a l u e ⟩$ .
(errno $3$ ): Extremely bad integrand behaviour occurs around the sub-interval $(⟨ v a l u e ⟩, ⟨ v a l u e ⟩)$ .
(errno $4$ ): Round-off error is detected in the extrapolation table.
(errno $5$ ): The integral is probably divergent or slowly convergent.

Notes

dim1_fin_general 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_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 $e_{m} (S_{n})$ transformation, Math. Tables Aids Comput. (10), 91–96

NAG and Python

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naginterfaces.library.quad.dim1_fin_general¶

naginterfaces.library.quad.dim1_​fin_​general¶

naginterfaces.library.quad.dim1_fin_general¶