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NAG Toolbox: nag_quad_1d_fin_osc_vec (d01au)

Purpose

nag_quad_1d_fin_osc_vec (d01au) is an adaptive integrator, especially suited to oscillating, nonsingular integrands, 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] = d01au(f, a, b, epsabs, epsrel, 'key', key, 'lw', lw, 'liw', liw)
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_osc_vec(f, a, b, epsabs, epsrel, 'key', key, 'lw', lw, 'liw', liw)

Description

nag_quad_1d_fin_osc_vec (d01au) is based on the QUADPACK routine QAG (see Piessens et al. (1983)). It is an adaptive function, offering a choice of six Gauss–Kronrod rules. A global acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation is described in Piessens et al. (1983).
Because nag_quad_1d_fin_osc_vec (d01au) is based on integration rules of high order, it is especially suitable for nonsingular oscillating integrands.
nag_quad_1d_fin_osc_vec (d01au) 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 this algorithm with key = 6key=6 is identical to that used by nag_quad_1d_fin_osc (d01ak).

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 (1973) An algorithm for automatic integration Angew. Inf. 15 399–401
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

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 equal to the number of points in the Kronrod rule (see specification of key).

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:     key – int64int32nag_int scalar
Indicates which integration rule is to be used.
key = 1key=1
For the Gauss 77-point and Kronrod 1515-point rule.
key = 2key=2
For the Gauss 1010-point and Kronrod 2121-point rule.
key = 3key=3
For the Gauss 1515-point and Kronrod 3131-point rule.
key = 4key=4
For the Gauss 2020-point and Kronrod 4141-point rule.
key = 5key=5
For the Gauss 2525-point and Kronrod 5151-point rule.
key = 6key=6
For the Gauss 3030-point and Kronrod 6161-point rule.
Default: 66
Constraint: key = 1key=1, 22, 33, 44, 55 or 66.
2:     lw – int64int32nag_int scalar
The dimension of the array w as declared in the (sub)program from which nag_quad_1d_fin_osc_vec (d01au) 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.
3:     liw – int64int32nag_int scalar
The dimension of the array iw as declared in the (sub)program from which nag_quad_1d_fin_osc_vec (d01au) 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_osc_vec (d01au) 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 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.
  ifail = 4ifail=4
On entry, key1key1, 22, 33, 44, 55 or 66.
  ifail = 5ifail=5
On entry,lw < 4lw<4,
orliw < 1liw<1.

Accuracy

nag_quad_1d_fin_osc_vec (d01au) 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

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_osc_vec (d01au) 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 . 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_osc_vec_example
a = 0;
b = 6.283185307179586;
epsabs = 0;
epsrel = 0.001;
[result, abserr, w, iw, ifail] = nag_quad_1d_fin_osc_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))*cos(x(i));
  end
 

result =

   -0.2097


abserr =

   4.4659e-14


ifail =

                    0


function d01au_example
a = 0;
b = 6.283185307179586;
epsabs = 0;
epsrel = 0.001;
[result, abserr, w, iw, ifail] = d01au(@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))*cos(x(i));
  end
 

result =

   -0.2097


abserr =

   4.4659e-14


ifail =

                    0



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