naginterfaces.library.quad.dim1_​fin_​wsing

naginterfaces.library.quad.dim1_fin_wsing(g, a, b, alfa, beta, key, epsabs, epsrel, lw=800, liw=None, data=None)

dim1_fin_wsing is an adaptive integrator which calculates an approximation to the integral of a function $g\left(x\right)w\left(x\right)$ over a finite interval $[a,b]$:

$$I={\int }_a^{b}g\left(x\right)w\left(x\right)dx$$

where the weight function $w$ has end point singularities of algebraico-logarithmic type.

For full information please refer to the NAG Library document for d01ap

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/d01/d01apf.html

Parameters

gcallable retval = g(x, data=None)

$g$ must return the value of the function $g$ at a given point $x$.

Parameters

xfloat

The point at which the function $g$ must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of $g\left(x\right)$ evaluated at $x$.

afloat

$a$, the lower limit of integration.

bfloat

$b$, the upper limit of integration.

alfafloat

The argument $\alpha$ in the weight function.

betafloat

The argument $\beta$ in the weight function.

keyint

Indicates which weight function is to be used.

$key=1$

$w\left(x\right)={(x-a)}^{\alpha }{(b-x)}^{\beta }$.

$key=2$

$w\left(x\right)={(x-a)}^{\alpha }{(b-x)}^{\beta }ln(x-a)$.

$key=3$

$w\left(x\right)={(x-a)}^{\alpha }{(b-x)}^{\beta }ln(b-x)$.

$key=4$

$w\left(x\right)={(x-a)}^{\alpha }{(b-x)}^{\beta }ln(x-a)ln(b-x)$.

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 $4$)

On entry, $key=⟨value⟩$.

Constraint: $key\le 4$.

(errno $4$)

On entry, $key=⟨value⟩$.

Constraint: $key\ge 1$.

(errno $4$)

On entry, $beta=⟨value⟩$.

Constraint: $beta>-1.0$.

(errno $4$)

On entry, $alfa=⟨value⟩$.

Constraint: $alfa>-1.0$.

(errno $4$)

On entry, $b=⟨value⟩$ and $a=⟨value⟩$.

Constraint: $b>a$.

(errno $5$)

On entry, $liw=⟨value⟩$.

Constraint: $liw\ge 2$.

(errno $5$)

On entry, $lw=⟨value⟩$.

Constraint: $lw\ge 8$.

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

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

dim1_fin_wsing is based on the QUADPACK routine QAWSE (see Piessens et al. (1983)) and integrates a function of the form $g\left(x\right)w\left(x\right)$, where the weight function $w\left(x\right)$ may have algebraico-logarithmic singularities at the end points $a$ and/or $b$. The strategy is a modification of that in dim1_fin_osc_fn(). We start by bisecting the original interval and applying modified Clenshaw–Curtis integration of orders $12$ and $24$ to both halves. Clenshaw–Curtis integration is then used on all sub-intervals which have $a$ or $b$ as one of their end points (see Piessens et al. (1974)). On the other sub-intervals Gauss–Kronrod ($7$–$15$ point) integration is carried out.

A ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation control is described in Piessens et al. (1983).

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

Piessens, R, Mertens, I and Branders, M, 1974, Integration of functions having end-point singularities, Angew. Inf. (16), 65–68