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NAG Toolbox: nag_quad_1d_inf_wtrig (d01as)

Purpose

nag_quad_1d_inf_wtrig (d01as) calculates an approximation to the sine or the cosine transform of a function gg over [a,)[a,):
I = g(x)sin(ωx)dx  or  I = g(x)cos(ωx)dx
a a
I = a g(x) sin(ωx) dx   or   I = a g(x) cos(ωx) dx
(for a user-specified value of ωω).

Syntax

[result, abserr, lst, erlst, rslst, ierlst, iw, ifail] = d01as(g, a, omega, key, epsabs, 'limlst', limlst, 'lw', lw, 'liw', liw)
[result, abserr, lst, erlst, rslst, ierlst, iw, ifail] = nag_quad_1d_inf_wtrig(g, a, omega, key, epsabs, 'limlst', limlst, 'lw', lw, 'liw', liw)

Description

nag_quad_1d_inf_wtrig (d01as) is based on the QUADPACK routine QAWFE (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) over a semi-infinite interval, where w(x)w(x) is either sin(ωx)sin(ωx) or cos(ωx)cos(ω x).
Over successive intervals
Ck = [a + (k1)c,a + kc] ,   k = 1,2,,lst
Ck = [a+(k-1)c,a+kc] ,   k=1,2,,lst
integration is performed by the same algorithm as is used by nag_quad_1d_fin_wtrig (d01an). The intervals CkCk are of constant length
c = {2[|ω|] + 1} π / |ω| ,   ω0 ,
c = { 2[|ω|] +1 } π/|ω| ,   ω0 ,
where [|ω|][|ω|] represents the largest integer less than or equal to |ω||ω|. Since cc equals an odd number of half periods, the integral contributions over succeeding intervals will alternate in sign when the function gg is positive and monotonically decreasing over [a,)[a,). The algorithm, described in Piessens et al. (1983), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the εε-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described by Piessens et al. (1983).
If ω = 0ω=0 and key = 1key=1, the function uses the same algorithm as nag_quad_1d_inf (d01am) (with epsrel = 0.0epsrel=0.0).
In contrast to the other functions in Chapter D01, nag_quad_1d_inf_wtrig (d01as) works only with an absolute error tolerance (epsabs). Over the interval CkCk it attempts to satisfy the absolute accuracy requirement
EPSAk = Uk × epsabs ,
EPSAk = Uk×epsabs ,
where Uk = (1p)pk1Uk=(1-p)pk-1, for k = 1,2,k=1,2, and p = 0.9p=0.9.
However, when difficulties occur during the integration over the kkth sub-interval CkCk such that the error flag ierlst(k)ierlstk is nonzero, the accuracy requirement over subsequent intervals is relaxed. See Piessens et al. (1983) for more details.

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
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:     omega – double scalar
The parameter ωω in the weight function of the transform.
4:     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.
5:     epsabs – double scalar
The absolute accuracy required. If epsabs is negative, the absolute value is used. See Section [Accuracy].

Optional Input Parameters

1:     limlst – int64int32nag_int scalar
An upper bound on the number of intervals CkCk needed for the integration.
Default: 5050
Constraint: limlst3limlst3.
2:     lw – int64int32nag_int scalar
The dimension of the array w as declared in the (sub)program from which nag_quad_1d_inf_wtrig (d01as) is called. The value of lw (together with that of liw) imposes a bound on the number of sub-intervals into which each interval CkCk 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: a value in the range 800800 to 20002000 is adequate for most problems.
Constraint: lw4lw4.
3:     liw – int64int32nag_int scalar
The dimension of the array iw as declared in the (sub)program from which nag_quad_1d_inf_wtrig (d01as) is called. The number of sub-intervals into which each interval CkCk may be divided cannot exceed liw / 2liw/2.
Default: lw / 2lw/2 
Constraint: liw2liw2.

Input Parameters Omitted from the MATLAB Interface

w

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:     lst – int64int32nag_int scalar
The number of intervals CkCk actually used for the integration.
4:     erlst(limlst) – double array
erlst(k)erlstk contains the error estimate corresponding to the integral contribution over the interval CkCk, for k = 1,2,,lstk=1,2,,lst.
5:     rslst(limlst) – double array
rslst(k)rslstk contains the integral contribution over the interval CkCk, for k = 1,2,,lstk=1,2,,lst.
6:     ierlst(limlst) – int64int32nag_int array
ierlst(k)ierlstk contains the error flag corresponding to rslst(k)rslstk, for k = 1,2,,lstk=1,2,,lst. See Section [Error Indicators and Warnings].
7:     iw(liw) – int64int32nag_int array
iw(1)iw1 contains the maximum number of sub-intervals actually used for integrating over any of the intervals CkCk. The rest of the array is used as workspace.
8:     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_inf_wtrig (d01as) 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 nag_quad_1d_inf_wtrig (d01as) on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by epsabs 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 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 = 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.
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 nag_quad_1d_inf_wtrig (d01as) on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by epsabs or increasing the amount of workspace.
Please note that divergence can occur with any nonzero value of ifail.
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,
orlimlst < 3limlst<3.
W ifail = 7ifail=7
Bad integration behaviour occurs within one or more of the intervals CkCk. Location and type of the difficulty involved can be determined from the vector ierlst.
W ifail = 8ifail=8
Maximum number of intervals CkCk ( = limlst=limlst) allowed has been achieved. Increase the value of limlst to allow more cycles.
W ifail = 9ifail=9
The extrapolation table constructed for convergence acceleration of the series formed by the integral contribution over the intervals CkCk, does not converge to the required accuracy.
  ifail = 10ifail=10
On entry,lw < 4lw<4,
orliw < 2liw<2.
In the cases ifail = 7ifail=7, 88 or 99, additional information about the cause of the error can be obtained from the array ierlst, as follows:
ierlst(k) = 1ierlstk=1
The maximum number of subdivisions = min (lw / 4,liw / 2)subdivisions=min(lw/4,liw/2) has been achieved on the kkth interval.
ierlst(k) = 2ierlstk=2
Occurrence of round-off error is detected and prevents the tolerance imposed on the kkth interval from being achieved.
ierlst(k) = 3ierlstk=3
Extremely bad integrand behaviour occurs at some points of the kkth interval.
ierlst(k) = 4ierlstk=4
The integration procedure over the kkth interval does not converge (to within the required accuracy) due to round-off in the extrapolation procedure invoked on this interval. It is assumed that the result on this interval is the best which can be obtained.
ierlst(k) = 5ierlstk=5
The integral over the kkth interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of ierlst(k)ierlstk.

Accuracy

nag_quad_1d_inf_wtrig (d01as) cannot guarantee, but in practice usually achieves, the following accuracy:
|Iresult||epsabs|,
|I-result||epsabs|,
where epsabs is the user-specified absolute error tolerance. Moreover, it returns the quantity abserr, which, in normal circumstances, satisfies
|Iresult|abserr|epsabs|.
|I-result|abserr|epsabs|.

Further Comments

None.

Example

function nag_quad_1d_inf_wtrig_example
a = 0;
omega = 1.570796326794897;
key = int64(1);
epsabs = 0.001;
[result, abserr, lst, erlst, rslst, ierlst, iw, ifail] = ...
    nag_quad_1d_inf_wtrig(@g, a, omega, key, epsabs);
 result, abserr, lst, ifail

function [result] = g(x)
  if (x > 0)
    result = 1/sqrt(x);
  else
    result = 0;
  end
 

result =

    1.0000


abserr =

   5.9234e-04


lst =

                    6


ifail =

                    0


function d01as_example
a = 0;
omega = 1.570796326794897;
key = int64(1);
epsabs = 0.001;
[result, abserr, lst, erlst, rslst, ierlst, iw, ifail] = ...
    d01as(@g, a, omega, key, epsabs);
 result, abserr, lst, ifail

function [result] = g(x)
  if (x > 0)
    result = 1/sqrt(x);
  else
    result = 0;
  end
 

result =

    1.0000


abserr =

   5.9234e-04


lst =

                    6


ifail =

                    0



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