d01aj Method
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d01aj is a general purpose integrator which calculates an approximation to the integral of a function fx over a finite interval a,b:
I= ∫ab fx dx .

Syntax

C#
public static void d01aj(
	D01..::.D01AJ_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 (Declaration)
Public Shared Sub d01aj ( _
	f As D01..::.D01AJ_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 d01aj(
	D01..::.D01AJ_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 d01aj : 
        f:D01..::.D01AJ_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..::.D01AJ_F
f must return the value of the integrand f at a given point.

A delegate of type D01AJ_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]
Note: lw must satisfy the constraint: lw≥4
On exit: details of the computation, as described in [Further Comments].
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 (see [Error Indicators and Warnings]).

Description

d01aj is based on the QUADPACK routine QAGS (see Piessens et al. (1983)). It is an adaptive method, 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 local error estimation is described in Piessens et al. (1983).
The method is suitable as a general purpose integrator, and can be used when the integrand has singularities, especially when these are of algebraic or logarithmic type.
d01aj requires you to supply a function to evaluate the integrand at a single point.
The method (d01at 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 (d01at not in this release) may be more efficient for some problem types and some machine architectures.

References

Error Indicators and Warnings

Note: d01aj 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 (lw, iw, liw, iuser, ruser) 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 the integrator 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
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=-8000
Negative dimension for array value
ifail=-6000
Invalid Parameters value

Accuracy

d01aj cannot guarantee, but in practice usually achieves, the following accuracy:
I-result ≤ tol ,
where
tol = max epsabs , epsrel × 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 ≤ abserr ≤ tol .

Further Comments

The time taken by d01aj depends on the integrand and the accuracy required.
If ifail≠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 d01aj along with the integral contributions and error estimates over the sub-intervals.
Specifically, for i=1,2,…,n, let ri denote the approximation to the value of the integral over the sub-interval ai,bi  in the partition of a,b  and ei  be the corresponding absolute error estimate. Then, ∫ ai bi fx dx ≃ ri  and result = ∑ i=1 n ri , unless d01aj 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 ai, bi, ei and ri are stored consecutively in the array w, that is:
  • ai=w[i-1],
  • bi=w[n+i-1],
  • ei=w[2n+i-1] and
  • ri=w[3n+i-1].

Example

See Also