d01aq calculates an approximation to the Hilbert transform of a function

g (x)

over

[a, b]

I = \int_{a}^{b} \frac{g (x)}{x - c} d x

for user-specified values of

a

b

and

c

Syntax

C#
public static void d01aq( D01..::..D01AQ_G g, double a, double b, double c, double epsabs, double epsrel, out double result, out double abserr, double[] w, out int subintvls, out int ifail )

Visual Basic
Public Shared Sub d01aq ( _ g As D01..::..D01AQ_G, _ a As Double, _ b As Double, _ c As Double, _ epsabs As Double, _ epsrel As Double, _ <OutAttribute> ByRef result As Double, _ <OutAttribute> ByRef abserr As Double, _ w As Double(), _ <OutAttribute> ByRef subintvls As Integer, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub d01aq ( _
	g As D01..::..D01AQ_G, _
	a As Double, _
	b As Double, _
	c As Double, _
	epsabs As Double, _
	epsrel As Double, _
	<OutAttribute> ByRef result As Double, _
	<OutAttribute> ByRef abserr As Double, _
	w As Double(), _
	<OutAttribute> ByRef subintvls As Integer, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void d01aq( D01..::..D01AQ_G^ g, double a, double b, double c, double epsabs, double epsrel, [OutAttribute] double% result, [OutAttribute] double% abserr, array<double>^ w, [OutAttribute] int% subintvls, [OutAttribute] int% ifail )

Visual C++

public:
static void d01aq(
	D01..::..D01AQ_G^ g, 
	double a, 
	double b, 
	double c, 
	double epsabs, 
	double epsrel, 
	[OutAttribute] double% result, 
	[OutAttribute] double% abserr, 
	array<double>^ w, 
	[OutAttribute] int% subintvls, 
	[OutAttribute] int% ifail
)

F#
static member d01aq : g : D01..::..D01AQ_G * a : float * b : float * c : float * epsabs : float * epsrel : float * result : float byref * abserr : float byref * w : float[] * subintvls : int byref * ifail : int byref -> unit

static member d01aq : 
        g : D01..::..D01AQ_G * 
        a : float * 
        b : float * 
        c : float * 
        epsabs : float * 
        epsrel : float * 
        result : float byref * 
        abserr : float byref * 
        w : float[] * 
        subintvls : int byref * 
        ifail : int byref -> unit

Parameters

g: Type: NagLibrary..::..D01..::..D01AQ_G
g must return the value of the function $g$ at a given point x.
A delegate of type D01AQ_G.

a: Type: System..::..Double
On entry: $a$ , the lower limit of integration.

b: Type: System..::..Double
On entry: $b$ , the upper limit of integration. It is not necessary that $a < b$ .

c: Type: System..::..Double
On entry: the parameter $c$ in the weight function.

Constraint: $c$ must not equal a or b.

epsabs: Type: System..::..Double
On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See [Accuracy].

epsrel: Type: System..::..Double
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See [Accuracy].

result: Type: System..::..Double%
On exit: the approximation to the integral $I$ .

abserr: Type: System..::..Double%
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $|I - result|$ .

w: Type: array<System..::..Double>[]()[][]
An array of size [lw]
On exit: details of the computation see [Further Comments] for more information.

subintvls: Type: System..::..Int32%
On exit: subintvls contains the actual number of sub-intervals used.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

d01aq is based on the QUADPACK routine QAWC (see Piessens et al. (1983)) and integrates a function of the form

g (x) w (x)

, where the weight function

w (x) = \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 method 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 \leq c \leq 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

Error Indicators and Warnings

Note: d01aq may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface (IW) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 1$: The maximum number of subdivisions allowed with the given workspace has been reached without the accuracy requirements being achieved. Look at the integrand in order to determine the integration difficulties. If necessary, another integrator, which is designed for handling the type of difficulty involved, must be used. Alternatively, consider relaxing the accuracy requirements specified by epsabs and epsrel, or increasing the amount of workspace.

$ifail = 2$: Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.

$ifail = 3$: Extremely bad local behaviour of $g (x)$ causes a very strong subdivision around one (or more) points of the interval. The same advice applies as in the case of $ifail = 1$ .

$ifail = 4$

On entry,

c = a

c = b

$ifail = 5$

On entry,	$lw < 4$ ,
or	$liw < 1$ .

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

d01aq cannot guarantee, but in practice usually achieves, the following accuracy:

|I - result| \leq tol,

where

tol = \max \{|epsabs|, |epsrel| \times |I|\},

and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover, it returns the quantity abserr which, in normal circumstances satisfies:

|I - result| \leq abserr \leq tol .

Parallelism and Performance

None.

Further Comments

The time taken by d01aq depends on the integrand and the accuracy required.

ifail \neq 0

on exit, then you may wish to examine the contents of the array w, which contains the end points of the sub-intervals used by d01aq along with the integral contributions and error estimates over these sub-intervals.

Specifically, for

i = 1, 2, \dots, n

, let

r_{i}

denote the approximation to the value of the integral over the sub-interval [

a_{i}, b_{i}

] in the partition of

[a, b]

and

e_{i}

be the corresponding absolute error estimate. Then,

\int_{a_{i}}^{b_{i}} g (x) w (x) d x ≃ r_{i}

and

result = \sum_{i = 1}^{n} r_{i}

. The value of

n

is returned in

_iw [0]

, and the values

a_{i}

b_{i}

e_{i}

and

r_{i}

are stored consecutively in the array w, that is:

$a_{i} = w [i - 1]$ ,
$b_{i} = w [n + i - 1]$ ,
$e_{i} = w [2 n + i - 1]$ and
$r_{i} = w [3 n + i - 1]$ .

Example

This example computes the Cauchy principal value of

\int_{- 1}^{1} \frac{d x}{(x^{2} + {0.01}^{2}) (x - \frac{1}{2})} .

Example program (C#): d01aqe.cs

Example program results: d01aqe.r