naginterfaces.library.quad.dim1_​fin_​wcauchy

naginterfaces.library.quad.dim1_fin_wcauchy(g, a, b, c, epsabs, epsrel, lw=800, liw=None, data=None)

dim1_fin_wcauchy calculates an approximation to the Hilbert transform of a function $g\left(x\right)$ over $[a,b]$:

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

for user-specified values of $a$, $b$ and $c$.

For full information please refer to the NAG Library document for d01aq

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d01/d01aqf.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. It is not necessary that $a<b$.

cfloat

The argument $c$ in the weight function.

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, $a=⟨value⟩$, $b=⟨value⟩$ and $c=⟨value⟩$.

Constraint: $c\ne a$ and $c\ne b$.

(errno $5$)

On entry, $liw=⟨value⟩$.

Constraint: $liw\ge 1$.

(errno $5$)

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 $(⟨value⟩,⟨value⟩)$.

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_wcauchy is based on the QUADPACK routine QAWC (see Piessens et al. (1983)) and integrates a function of the form $g\left(x\right)w\left(x\right)$, where the weight function

$$w\left(x\right)=\frac{1}{x-c}$$

is that of the Hilbert transform. (If $a<c<b$ the integral has to be interpreted in the sense of a Cauchy principal value.) It is an adaptive function which employs a ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)). Special care is taken to ensure that $c$ is never the end point of a sub-interval (see Piessens et al. (1976)). On each sub-interval $(c_1,c_2)$ modified Clenshaw–Curtis integration of orders $12$ and $24$ is performed if $c_1-d\le c\le c_2+d$ where $d=(c_2-c_1)/20$. Otherwise the Gauss $7$-point and Kronrod $15$-point rules are used. The local error estimation is described by 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, van Roy–Branders, M and Mertens, I, 1976, The automatic evaluation of Cauchy principal value integrals, Angew. Inf. (18), 31–35