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NAG Toolbox: nag_quad_1d_fin_wtrig (d01an)

Purpose

nag_quad_1d_fin_wtrig (d01an) calculates an approximation to the sine or the cosine transform of a function gg over [a,b][a,b]:
b b
I = g(x)sin(ωx)dx  or  I = g(x)cos(ωx)dx
a a
I=abg(x)sin(ω x)dx  or  I=abg(x)cos(ω x)dx
(for a user-specified value of ωω).

Syntax

[result, abserr, w, iw, ifail] = d01an(g, a, b, omega, key, epsabs, epsrel, 'lw', lw, 'liw', liw)
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_wtrig(g, a, b, omega, key, epsabs, epsrel, 'lw', lw, 'liw', liw)

Description

nag_quad_1d_fin_wtrig (d01an) is based on the QUADPACK routine QFOUR (see Piessens et al. (1983)). It is an adaptive function, designed to integrate a function of the form g(x)w(x)g(x)w(x), where w(x)w(x) is either sin(ωx)sin(ωx) or cos(ωx)cos(ωx). If a sub-interval has length
L = |ba|2l
L=|b-a|2-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 > 4Lω>4 and l20.l20. In this case a Chebyshev series approximation of degree 2424 is used to approximate g(x)g(x), while an error estimate is computed from this approximation together with that obtained using Chebyshev series of degree 1212. If the above conditions do not hold then Gauss 77-point and Kronrod 1515-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
εε-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation 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 and Branders M (1975) Algorithm 002: computation of oscillating integrals J. Comput. Appl. Math. 1 153–164
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 em(Sn)em(Sn) transformation Math. Tables Aids Comput. 10 91–96

Parameters

Compulsory Input Parameters

1:     g – function handle or string containing name of m-file
g must return the value of the function gg at a given point x.
[result] = g(x)

Input Parameters

1:     x – double scalar
The point at which the function gg must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
2:     a – double scalar
aa, the lower limit of integration.
3:     b – double scalar
bb, the upper limit of integration. It is not necessary that a < ba<b.
4:     omega – double scalar
The parameter ωω in the weight function of the transform.
5:     key – int64int32nag_int scalar
Indicates which integral is to be computed.
key = 1key=1
w(x) = cos(ωx)w(x)=cos(ωx).
key = 2key=2
w(x) = sin(ωx)w(x)=sin(ωx).
Constraint: key = 1key=1 or 22.
6:     epsabs – double scalar
The absolute accuracy required. If epsabs is negative, the absolute value is used. See Section [Accuracy].
7:     epsrel – double scalar
The relative accuracy required. If epsrel is negative, the absolute value is used. See Section [Accuracy].

Optional Input Parameters

1:     lw – int64int32nag_int scalar
The dimension of the array w as declared in the (sub)program from which nag_quad_1d_fin_wtrig (d01an) is called. The value of lw (together with that of liw) imposes a bound on the number of sub-intervals into which the interval of integration may be divided by the function. The number of sub-intervals cannot exceed lw / 4lw/4. The more difficult the integrand, the larger lw should be.
Default: 800800 
Constraint: lw4lw4.
2:     liw – int64int32nag_int scalar
The dimension of the array iw as declared in the (sub)program from which nag_quad_1d_fin_wtrig (d01an) is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed liw / 2liw/2.
Default: lw / 2lw/2 
Constraint: liw2liw2.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     result – double scalar
The approximation to the integral II.
2:     abserr – double scalar
An estimate of the modulus of the absolute error, which should be an upper bound for |Iresult||I-result|.
3:     w(lw) – double array
Details of the computation see Section [Further Comments] for more information.
4:     iw(liw) – int64int32nag_int array
iw(1)iw1 contains the actual number of sub-intervals used. The rest of the array is used as workspace.
5:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Note: nag_quad_1d_fin_wtrig (d01an) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail = 1ifail=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.
W ifail = 2ifail=2
Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.
W ifail = 3ifail=3
Extremely bad local behaviour of g(x)g(x) causes a very strong subdivision around one (or more) points of the interval. The same advice applies as in the case of ifail = 1ifail=1.
W ifail = 4ifail=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 = 1ifail=1.
W ifail = 5ifail=5
The integral is probably divergent, or slowly convergent. Please note that divergence can occur with any nonzero value of ifail.
  ifail = 6ifail=6
On entry, key1key1 or 22.
  ifail = 7ifail=7
On entry,lw < 4lw<4,
orliw < 2liw<2.

Accuracy

nag_quad_1d_fin_wtrig (d01an) cannot guarantee, but in practice usually achieves, the following accuracy:
|Iresult|tol,
|I-result|tol,
where
tol = max {|epsabs|,|epsrel| × |I|} ,
tol=max{|epsabs|,|epsrel|×|I|} ,
and epsabs and epsrel are user-specified absolute and relative tolerances. Moreover, it returns the quantity abserr which in normal circumstances, satisfies
|Iresult|abserrtol.
|I-result|abserrtol.

Further Comments

The time taken by nag_quad_1d_fin_wtrig (d01an) depends on the integrand and the accuracy required.
If ifail0ifail0 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 nag_quad_1d_fin_wtrig (d01an) along with the integral contributions and error estimates over these sub-intervals.
Specifically, for i = 1,2,,ni=1,2,,n, let riri denote the approximation to the value of the integral over the sub-interval [ai,bi][ai,bi] in the partition of [a,b][a,b] and eiei be the corresponding absolute error estimate. Then, aibig(x)w(x)dxriaibig(x)w(x)dxri and result = i = 1nriresult=i=1nri unless nag_quad_1d_fin_wtrig (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 nn is returned in iw(1)iw1, and the values aiai, bibi, eiei and riri are stored consecutively in the array w, that is:

Example

function nag_quad_1d_fin_wtrig_example
a = 0;
b = 1;
omega = 31.41592653589793;
key = int64(2);
epsabs = 0;
epsrel = 0.0001;
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_wtrig(@g, a, b, omega, key, epsabs, epsrel);
 result, abserr, ifail

function [result] = g(x)
  if (x > 0)
    result = log(x);
  else
    result = 0;
  end
 

result =

   -0.1281


abserr =

   3.5796e-06


ifail =

                    0


function d01an_example
a = 0;
b = 1;
omega = 31.41592653589793;
key = int64(2);
epsabs = 0;
epsrel = 0.0001;
[result, abserr, w, iw, ifail] = d01an(@g, a, b, omega, key, epsabs, epsrel);
 result, abserr, ifail

function [result] = g(x)
  if (x > 0)
    result = log(x);
  else
    result = 0;
  end
 

result =

   -0.1281


abserr =

   3.5796e-06


ifail =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
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