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NAG Toolbox: nag_quad_1d_fin_bad_vec (d01at)

Purpose

nag_quad_1d_fin_bad_vec (d01at) 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 .

Syntax

[result, abserr, w, iw, ifail] = d01at(f, a, b, epsabs, epsrel, 'lw', lw, 'liw', liw)
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_bad_vec(f, a, b, epsabs, epsrel, 'lw', lw, 'liw', liw)

Description

nag_quad_1d_fin_bad_vec (d01at) is based on the QUADPACK routine QAGS (see Piessens et al. (1983)). It is an adaptive function, 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 local error estimation is described in Piessens et al. (1983).
The function 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.
nag_quad_1d_fin_bad_vec (d01at) requires a function to evaluate the integrand at an array of different points and is therefore amenable to parallel execution (see Section [Parallelism and Performance]). Otherwise the algorithm is identical to that used by nag_quad_1d_fin_bad (d01aj).

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 values of the integrand ff at a set of points.
[fv] = f(x, n)

Input Parameters

1:     x(n) – double array
The points at which the integrand ff must be evaluated.
2:     n – int64int32nag_int scalar
The number of points at which the integrand is to be evaluated. The actual value of n is always 2121.

Output Parameters

1:     fv(n) – double array
fv(j)fvj must contain the value of ff at the point x(j)xj, for j = 1,2,,nj=1,2,,n.
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:     epsabs – double scalar
The absolute accuracy required. If epsabs is negative, the absolute value is used. See Section [Accuracy].
5:     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_bad_vec (d01at) 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_bad_vec (d01at) is called. The number of sub-intervals into which the interval of integration may be divided cannot exceed liw.
Default: lw / 4lw/4 
Constraint: liw1liw1.

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_bad_vec (d01at) 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 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
On entry,lw < 4lw<4,
orliw < 1liw<1.

Accuracy

nag_quad_1d_fin_bad_vec (d01at) 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| abserr tol .
|I-result| abserr tol .

Further Comments

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_bad_vec (d01at) along with the integral contributions and error estimates over the 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_bad_vec (d01at) 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_bad_vec_example
a = 0;
b = 6.283185307179586;
epsabs = 0;
epsrel = 0.0001;
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_bad_vec(@f, a, b, epsabs, epsrel);
 result, abserr, ifail

function [fv] = f(x,n)
  fv = zeros(n,1);
  for i = 1:double(n)
    fv(i) = x(i)*sin(30*x(i))/sqrt(1-x(i)^2/(4*pi^2));
  end
 

result =

   -2.5433


abserr =

   1.2752e-05


ifail =

                    0


function d01at_example
a = 0;
b = 6.283185307179586;
epsabs = 0;
epsrel = 0.0001;
[result, abserr, w, iw, ifail] = d01at(@f, a, b, epsabs, epsrel);
 result, abserr, ifail

function [fv] = f(x,n)
  fv = zeros(n,1);
  for i = 1:double(n)
    fv(i) = x(i)*sin(30*x(i))/sqrt(1-x(i)^2/(4*pi^2));
  end
 

result =

   -2.5433


abserr =

   1.2752e-05


ifail =

                    0



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