d01al 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

where the integrand may have local singular behaviour at a finite number of points within the integration interval.

Syntax

C#
public static void d01al( D01..::..D01AL_F f, double a, double b, int npts, double[] points, double epsabs, double epsrel, out double result, out double abserr, double[] w, out int subintvls, out int ifail )

public static void d01al(
	D01..::..D01AL_F f,
	double a,
	double b,
	int npts,
	double[] points,
	double epsabs,
	double epsrel,
	out double result,
	out double abserr,
	double[] w,
	out int subintvls,
	out int ifail
)

Visual Basic
Public Shared Sub d01al ( _ f As D01..::..D01AL_F, _ a As Double, _ b As Double, _ npts As Integer, _ points 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 d01al ( _
	f As D01..::..D01AL_F, _
	a As Double, _
	b As Double, _
	npts As Integer, _
	points 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 d01al( D01..::..D01AL_F^ f, double a, double b, int npts, array<double>^ points, double epsabs, double epsrel, [OutAttribute] double% result, [OutAttribute] double% abserr, array<double>^ w, [OutAttribute] int% subintvls, [OutAttribute] int% ifail )

Visual C++

public:
static void d01al(
	D01..::..D01AL_F^ f, 
	double a, 
	double b, 
	int npts, 
	array<double>^ points, 
	double epsabs, 
	double epsrel, 
	[OutAttribute] double% result, 
	[OutAttribute] double% abserr, 
	array<double>^ w, 
	[OutAttribute] int% subintvls, 
	[OutAttribute] int% ifail
)

F#
static member d01al : f : D01..::..D01AL_F * a : float * b : float * npts : int * points : float[] * epsabs : float * epsrel : float * result : float byref * abserr : float byref * w : float[] * subintvls : int byref * ifail : int byref -> unit

static member d01al : 
        f : D01..::..D01AL_F * 
        a : float * 
        b : float * 
        npts : int * 
        points : 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..::..D01AL_F
f must return the value of the integrand $f$ at a given point.
A delegate of type D01AL_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 $a < b$ .

npts: Type: System..::..Int32
On entry: the number of user-supplied break-points within the integration interval.

Constraint: $npts \geq 0$ and $npts < \min ((lw - 2 \times npts - 4) / 4, (liw - npts - 2) / 2)$ .

points: Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array points must be at least $\max (1, npts)$ .
On entry: the user-specified break-points.

Constraint: the break-points must all lie within the interval of integration (but may be supplied in any order).

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

d01al is based on the QUADPACK routine QAGP (see Piessens et al. (1983)). It is very similar to d01aj, but allows you to supply ‘break-points’, points at which the integrand is known to be difficult. It employs an adaptive algorithm, 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 user-supplied ‘break-points’ always occur as the end points of some sub-interval during the adaptive process. The local error estimation is described in Piessens et al. (1983).

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

Error Indicators and Warnings

Note: d01al 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 the position of a local difficulty within the interval can be determined (e.g., a singularity of the integrand or its derivative, a peak, a discontinuity, etc.) it should be supplied to the method as an element of the vector points. 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 $ifail = 1$ .

$ifail = 4$: The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best which can be obtained. The same advice applies as in the case of $ifail = 1$ .

$ifail = 5$: The integral is probably divergent, or slowly convergent. Please note that divergence can occur with any nonzero value of ifail.

$ifail = 6$: The input is invalid: break-points are specified outside the integration range, $npts > \min ((lw - 2 \times npts - 4) / 4, (liw - npts - 2) / 2)$ or $npts < 0$ . result and abserr are set to zero.

$ifail = 7$

On entry,	$lw < 2 \times npts + 8$ ,
or	$liw < npts + 4$ .

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

Accuracy

d01al 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 d01al 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 d01al 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}

unless d01al terminates while testing for divergence of the integral (see Section 3.4.3 of Piessens et al. (1983)). In this case, result (and abserr) are taken to be the values returned from the extrapolation process. 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}^{1} \frac{1}{\sqrt{|x - 1 / 7|}} d x .

A break-point is specified at

x = 1 / 7

, at which point the integrand is infinite. (For definiteness the function FST returns the value

0.0

at this point.)

Example program (C#): d01ale.cs

Example program results: d01ale.r