d01ap is an adaptive integrator which calculates an approximation to the integral of a function

g (x) w (x)

over a finite interval

[a, b]

I = \int_{a}^{b} g (x) w (x) d x

where the weight function

w

has end point singularities of algebraico-logarithmic type.

Syntax

C#
public static void d01ap( D01..::..D01AP_G g, double a, double b, double alfa, double beta, int key, double epsabs, double epsrel, out double result, out double abserr, double[] w, out int subintvls, out int ifail )

public static void d01ap(
	D01..::..D01AP_G g,
	double a,
	double b,
	double alfa,
	double beta,
	int key,
	double epsabs,
	double epsrel,
	out double result,
	out double abserr,
	double[] w,
	out int subintvls,
	out int ifail
)

Visual Basic
Public Shared Sub d01ap ( _ g As D01..::..D01AP_G, _ a As Double, _ b As Double, _ alfa As Double, _ beta As Double, _ key As Integer, _ 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 d01ap ( _
	g As D01..::..D01AP_G, _
	a As Double, _
	b As Double, _
	alfa As Double, _
	beta As Double, _
	key As Integer, _
	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 d01ap( D01..::..D01AP_G^ g, double a, double b, double alfa, double beta, int key, double epsabs, double epsrel, [OutAttribute] double% result, [OutAttribute] double% abserr, array<double>^ w, [OutAttribute] int% subintvls, [OutAttribute] int% ifail )

Visual C++

public:
static void d01ap(
	D01..::..D01AP_G^ g, 
	double a, 
	double b, 
	double alfa, 
	double beta, 
	int key, 
	double epsabs, 
	double epsrel, 
	[OutAttribute] double% result, 
	[OutAttribute] double% abserr, 
	array<double>^ w, 
	[OutAttribute] int% subintvls, 
	[OutAttribute] int% ifail
)

F#
static member d01ap : g : D01..::..D01AP_G * a : float * b : float * alfa : float * beta : float * key : int * epsabs : float * epsrel : float * result : float byref * abserr : float byref * w : float[] * subintvls : int byref * ifail : int byref -> unit

static member d01ap : 
        g : D01..::..D01AP_G * 
        a : float * 
        b : float * 
        alfa : float * 
        beta : float * 
        key : int * 
        epsabs : float * 
        epsrel : float * 
        result : float byref * 
        abserr : float byref * 
        w : float[] * 
        subintvls : int byref * 
        ifail : int byref -> unit

Parameters

g: Type: NagLibrary..::..D01..::..D01AP_G
g must return the value of the function $g$ at a given point x.
A delegate of type D01AP_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.

Constraint: $b > a$ .

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

Constraint: $alfa > - 1.0$ .

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

Constraint: $beta > - 1.0$ .

key

Type: System..::..Int32

On entry: indicates which weight function is to be used.

$key = 1$: $w (x) = {(x - a)}^{α} {(b - x)}^{β}$ .
$key = 2$: $w (x) = {(x - a)}^{α} {(b - x)}^{β} \ln (x - a)$ .
$key = 3$: $w (x) = {(x - a)}^{α} {(b - x)}^{β} \ln (b - x)$ .
$key = 4$: $w (x) = {(x - a)}^{α} {(b - x)}^{β} \ln (x - a) \ln (b - x)$ .

Constraint:

key = 1

2

3

4

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

d01ap is based on the QUADPACK routine QAWSE (see Piessens et al. (1983)) and integrates a function of the form

g (x) w (x)

, where the weight function

w (x)

may have algebraico-logarithmic singularities at the end points

a

and/or

b

. The strategy is a modification of that in d01ak. 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

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

Error Indicators and Warnings

Note: d01ap 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 discontinuity or a singularity of algebraico-logarithmic type within the interval can be determined, the interval must be split up at this point and the integrator called on the subranges. 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,	$b \leq a$ ,
or	$alfa \leq - 1.0$ ,
or	$beta \leq - 1.0$ ,
or	$key \neq 1$ , $2$ , $3$ or $4$ .

$ifail = 5$

On entry,	$lw < 8$ ,
or	$liw < 2$ .

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

Accuracy

d01ap 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 d01ap 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 d01ap 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) 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

\int_{0}^{1} \ln x \cos (10 π x) d x and \int_{0}^{1} \frac{\sin (10 x)}{\sqrt{x (1 - x)}} d x .

Example program (C#): d01ape.cs

Example program results: d01ape.r