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NAG Toolbox: nag_quad_1d_fin_sing (d01al)

Purpose

nag_quad_1d_fin_sing (d01al) is a general purpose integrator which calculates an approximation to the integral of a function f(x)f(x) over a finite interval [a,b][a,b]:
b
I = f(x)dx
a
I= ab f(x) dx
where the integrand may have local singular behaviour at a finite number of points within the integration interval.

Syntax

[result, abserr, w, iw, ifail] = d01al(f, a, b, points, epsabs, epsrel, 'npts', npts, 'lw', lw, 'liw', liw)
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_sing(f, a, b, points, epsabs, epsrel, 'npts', npts, 'lw', lw, 'liw', liw)

Description

nag_quad_1d_fin_sing (d01al) is based on the QUADPACK routine QAGP (see Piessens et al. (1983)). It is very similar to nag_quad_1d_fin_bad (d01aj), but allows you to supply ‘break points’, points at which the integrand is known to be difficult. It employs an adaptive algorithm, using the Gauss 1010-point and Kronrod 2121-point rules. The algorithm, described in de Doncker (1978), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the εε-algorithm (see Wynn (1956)) to perform extrapolation. The user-supplied ‘break points’ always occur as the end points of some sub-interval during the adaptive process. 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 em(Sn)em(Sn) transformation Math. Tables Aids Comput. 10 91–96

Parameters

Compulsory Input Parameters

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

Input Parameters

1:     x – double scalar
The point at which the integrand ff 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:     points( : :) – double array
Note: the dimension of the array points must be at least max (1,npts)max(1,npts).
The user-specified break points.
Constraint: the break points must all lie within the interval of integration (but may be supplied in any order).
5:     epsabs – double scalar
The absolute accuracy required. If epsabs is negative, the absolute value is used. See Section [Accuracy].
6:     epsrel – double scalar
The relative accuracy required. If epsrel is negative, the absolute value is used. See Section [Accuracy].

Optional Input Parameters

1:     npts – int64int32nag_int scalar
Default: The dimension of the array points.
The number of user-supplied break points within the integration interval.
Constraint: npts0npts0 and npts < min ( (lw2 × npts4) / 4 , (liwnpts2) / 2 ) npts<min ( ( lw-2×npts-4 ) / 4 , ( liw-npts-2 ) / 2 ) .
2:     lw – int64int32nag_int scalar
The dimension of the array w as declared in the (sub)program from which nag_quad_1d_fin_sing (d01al) 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 (lw2 × npts4) / 4(lw-2×npts-4)/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: lw2 × npts + 8lw2×npts+8.
3:     liw – int64int32nag_int scalar
The dimension of the array iw as declared in the (sub)program from which nag_quad_1d_fin_sing (d01al) is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed (liwnpts2) / 2(liw-npts-2)/2.
Default: lw / 2lw/2 
Constraint: liwnpts + 4liwnpts+4.

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_sing (d01al) 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.) it should be supplied to the function as an element of the vector points. 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 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.
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
The input is invalid: break points are specified outside the integration range, npts > min ( (lw2 × npts4) / 4 , (liwnpts2) / 2 ) npts > min ( (lw-2×npts-4)/4 , (liw-npts-2)/2 )  or npts < 0npts<0. result and abserr are set to zero.
  ifail = 7ifail=7
On entry,lw < 2 × npts + 8lw<2×npts+8,
orliw < npts + 4liw<npts+4.

Accuracy

nag_quad_1d_fin_sing (d01al) 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 error 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_sing (d01al) 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_sing (d01al) 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 ei ei  be the corresponding absolute error estimate. Then, aibi f(x) dx ri ai bi f(x) dx ri  and result = i = 1n ri result = i=1 n ri  unless nag_quad_1d_fin_sing (d01al) 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_sing_example
a = 0;
b = 1;
points = [0.1428571428571428];
epsabs = 0;
epsrel = 0.001;
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_sing(@f, a, b, points, epsabs, epsrel);
 result, abserr, ifail

function [result] = f(x)
  a = abs(x-1.0/7.0);
  if (a ~= 0.0)
    result = a^(-0.5);
  else
    result = 0;
  end
 

result =

    2.6076


abserr =

   8.0824e-14


ifail =

                    0


function d01al_example
a = 0;
b = 1;
points = [0.1428571428571428];
epsabs = 0;
epsrel = 0.001;
[result, abserr, w, iw, ifail] = d01al(@f, a, b, points, epsabs, epsrel);
 result, abserr, ifail

function [result] = f(x)
  a = abs(x-1.0/7.0);
  if (a ~= 0.0)
    result = a^(-0.5);
  else
    result = 0;
  end
 

result =

    2.6076


abserr =

   8.0824e-14


ifail =

                    0



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