d01am calculates an approximation to the integral of a function

f (x)

over an infinite or semi-infinite interval

[a, b]

I = \int_{a}^{b} f (x) d x .

Syntax

C#
public static void d01am( D01..::..D01AM_F f, double bound, int inf, double epsabs, double epsrel, out double result, out double abserr, double[] w, out int subintvls, out int ifail )

Visual Basic
Public Shared Sub d01am ( _ f As D01..::..D01AM_F, _ bound As Double, _ inf 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 d01am ( _
	f As D01..::..D01AM_F, _
	bound As Double, _
	inf 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 d01am( D01..::..D01AM_F^ f, double bound, int inf, double epsabs, double epsrel, [OutAttribute] double% result, [OutAttribute] double% abserr, array<double>^ w, [OutAttribute] int% subintvls, [OutAttribute] int% ifail )

Visual C++

public:
static void d01am(
	D01..::..D01AM_F^ f, 
	double bound, 
	int inf, 
	double epsabs, 
	double epsrel, 
	[OutAttribute] double% result, 
	[OutAttribute] double% abserr, 
	array<double>^ w, 
	[OutAttribute] int% subintvls, 
	[OutAttribute] int% ifail
)

F#
static member d01am : f : D01..::..D01AM_F * bound : float * inf : int * epsabs : float * epsrel : float * result : float byref * abserr : float byref * w : float[] * subintvls : int byref * ifail : int byref -> unit

static member d01am : 
        f : D01..::..D01AM_F * 
        bound : float * 
        inf : int * 
        epsabs : float * 
        epsrel : float * 
        result : float byref * 
        abserr : float byref * 
        w : float[] * 
        subintvls : int byref * 
        ifail : int byref -> unit

Parameters

f: Type: NagLibrary..::..D01..::..D01AM_F
f must return the value of the integrand $f$ at a given point.
A delegate of type D01AM_F.

bound: Type: System..::..Double
On entry: the finite limit of the integration range (if present). bound is not used if the interval is doubly infinite.

inf

Type: System..::..Int32

On entry: indicates the kind of integration range.

$\inf = 1$: The range is $[bound, + \infty)$ .
$\inf = -1$: The range is $(- \infty, bound]$ .
$\inf = 2$: The range is $(- \infty, + \infty)$ .

Constraint:

\inf = -1

1

2

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

d01am is based on the QUADPACK routine QAGI (see Piessens et al. (1983)). The entire infinite integration range is first transformed to

[0, 1]

using one of the identities:

\int_{- \infty}^{a} f (x) d x = \int_{0}^{1} f (a - \frac{1 - t}{t}) \frac{1}{t^{2}} d t

\int_{a}^{\infty} f (x) d x = \int_{0}^{1} f (a + \frac{1 - t}{t}) \frac{1}{t^{2}} d t

\int_{- \infty}^{\infty} f (x) d x = \int_{0}^{\infty} (f (x) + f (- x)) d x = \int_{0}^{1} ​ ​ [f (\frac{1 - t}{t}) + f (\frac{- 1 + t}{t})] \frac{1}{t^{2}} d t

where

a

represents a finite integration limit. An adaptive procedure, based on the Gauss

7

-point and Kronrod

15

-point rules, is then employed on the transformed integral. 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 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: d01am 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.) you will probably gain from splitting up the interval at this point and calling d01am on the infinite subrange and an appropriate integrator on the finite subrange. 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.

$ifail = 6$

On entry,	$lw < 4$ ,
or	$liw < 1$ ,
or	$\inf \neq - 1$ , $1$ or $2$ .

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

Accuracy

d01am 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 d01am 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 d01am 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 d01am 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
$r_{i} = w [3 n + i - 1]$ .

Note: this information applies to the integral transformed to

[0, 1]

as described in [Description], not to the original integral.

Example

This example computes

\int_{0}^{\infty} \frac{1}{(x + 1) \sqrt{x}} d x .

The exact answer is

π

Example program (C#): d01ame.cs

Example program results: d01ame.r