d01an Method
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d01an calculates an approximation to the sine or the cosine transform of a function g over a,b:
I=∫abgxsinω xdx  or  I=∫abgxcosω xdx
(for a user-specified value of ω).

Syntax

C#
public static void d01an(
	D01..::.D01AN_G g,
	double a,
	double b,
	double omega,
	int key,
	double epsabs,
	double epsrel,
	out double result,
	out double abserr,
	double[] w,
	out int subintvls,
	out int ifail
)
Visual Basic (Declaration)
Public Shared Sub d01an ( _
	g As D01..::.D01AN_G, _
	a As Double, _
	b As Double, _
	omega 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 d01an(
	D01..::.D01AN_G^ g, 
	double a, 
	double b, 
	double omega, 
	int key, 
	double epsabs, 
	double epsrel, 
	[OutAttribute] double% result, 
	[OutAttribute] double% abserr, 
	array<double>^ w, 
	[OutAttribute] int% subintvls, 
	[OutAttribute] int% ifail
)
F#
static member d01an : 
        g:D01..::.D01AN_G * 
        a:float * 
        b:float * 
        omega: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..::.D01AN_G
g must return the value of the function g at a given point x.

A delegate of type D01AN_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. It is not necessary that a<b.
omega
Type: System..::.Double
On entry: the parameter ω in the weight function of the transform.
key
Type: System..::.Int32
On entry: indicates which integral is to be computed.
key=1
wx=cosωx.
key=2
wx=sinωx.
Constraint: key=1 or 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]
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

d01an is based on the QUADPACK routine QFOUR (see Piessens et al. (1983)). It is an adaptive method, designed to integrate a function of the form gxwx, where wx is either sinωx or cosωx. If a sub-interval has length
L=b-a2-l
then the integration over this sub-interval is performed by means of a modified Clenshaw–Curtis procedure (see Piessens and Branders (1975)) if Lω>4 and l≤20. In this case a Chebyshev series approximation of degree 24 is used to approximate gx, while an error estimate is computed from this approximation together with that obtained using Chebyshev series of degree 12. If the above conditions do not hold then Gauss 7-point and Kronrod 15-point rules are used. The algorithm, described in Piessens et al. (1983), incorporates a global acceptance criterion (as defined in 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

Error Indicators and Warnings

Note: d01an 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 behaviour of gx 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, key≠1 or 2.
ifail=-8000
Negative dimension for array value
ifail=-6000
Invalid Parameters value

Accuracy

d01an cannot guarantee, but in practice usually achieves, the following accuracy:
I-result≤tol,
where
tol=maxepsabs,epsrel×I ,
and epsabs and epsrel are user-specified absolute and relative tolerances. Moreover, it returns the quantity abserr which in normal circumstances, satisfies
I-result≤abserr≤tol.

Further Comments

The time taken by d01an 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 d01an along with the integral contributions and error estimates over these 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, ∫aibigxwxdx≃ri and result=∑i=1nri unless d01an 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