d01ak is an adaptive integrator, especially suited to oscillating, nonsingular integrands, 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 .

Syntax

C#
public static void d01ak( D01..::..D01AK_F f, double a, double b, double epsabs, double epsrel, out double result, out double abserr, double[] w, out int subintvls, out int ifail )

Visual Basic
Public Shared Sub d01ak ( _ f As D01..::..D01AK_F, _ a As Double, _ b 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 d01ak ( _
	f As D01..::..D01AK_F, _
	a As Double, _
	b 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 d01ak( D01..::..D01AK_F^ f, double a, double b, double epsabs, double epsrel, [OutAttribute] double% result, [OutAttribute] double% abserr, array<double>^ w, [OutAttribute] int% subintvls, [OutAttribute] int% ifail )

Visual C++

public:
static void d01ak(
	D01..::..D01AK_F^ f, 
	double a, 
	double b, 
	double epsabs, 
	double epsrel, 
	[OutAttribute] double% result, 
	[OutAttribute] double% abserr, 
	array<double>^ w, 
	[OutAttribute] int% subintvls, 
	[OutAttribute] int% ifail
)

F#
static member d01ak : f : D01..::..D01AK_F * a : float * b : float * epsabs : float * epsrel : float * result : float byref * abserr : float byref * w : float[] * subintvls : int byref * ifail : int byref -> unit

static member d01ak : 
        f : D01..::..D01AK_F * 
        a : float * 
        b : float * 
        epsabs : float * 
        epsrel : float * 
        result : float byref * 
        abserr : float byref * 
        w : float[] * 
        subintvls : int byref * 
        ifail : int byref -> unit

Parameters

f: Type: NagLibrary..::..D01..::..D01AK_F
f must return the value of the integrand $f$ at a given point.
A delegate of type D01AK_F.

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 $a < b$ .

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

d01ak is based on the QUADPACK routine QAG (see Piessens et al. (1983)). It is an adaptive method, using the Gauss

30

-point and Kronrod

61

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

Because d01ak is based on integration rules of high order, it is especially suitable for nonsingular oscillating integrands.

d01ak requires you to supply a function to evaluate the integrand at a single point.

The method (D01AUF not in this release) uses an identical algorithm but requires you to supply a method to evaluate the integrand at an array of points. Therefore (D01AUF not in this release) will be more efficient if the evaluation can be performed in vector mode on a vector-processing machine.

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 (1973) An algorithm for automatic integration Angew. Inf. 15 399–401

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

Error Indicators and 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 integrand behaviour 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,	$lw < 4$ ,
or	$liw < 1$ .

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

Accuracy

d01ak 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 d01ak 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 d01ak 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}} f (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

\int_{0}^{2 π} x \sin (30 x) \cos x d x .

Example program (C#): d01ake.cs

Example program results: d01ake.r