NAG Library Function Document
nag_1d_quad_wt_trig_1 (d01snc)
1 Purpose
nag_1d_quad_wt_trig_1 (d01snc) calculates an approximation to the sine or the cosine transform of a function
over
:
(for a user-specified value of
).
2 Specification
#include <nag.h> |
#include <nagd01.h> |
void |
nag_1d_quad_wt_trig_1 (
double |
(*g)(double x,
Nag_User *comm),
|
|
double a,
double b,
double omega,
Nag_TrigTransform wt_func,
double epsabs,
double epsrel,
Integer max_num_subint,
double *result,
double *abserr,
Nag_QuadProgress *qp,
Nag_User *comm,
NagError *fail) |
|
3 Description
nag_1d_quad_wt_trig_1 (d01snc) is based upon the QUADPACK routine QFOUR (
Piessens et al. (1983)). It is an adaptive function, designed to integrate a function of the form
, where
is either
or
. If a sub-interval has length
then the integration over this sub-interval is performed by means of a modified Clenshaw–Curtis procedure (
Piessens and Branders (1975)) if
and
. In this case a Chebyshev series approximation of degree 24 is used to approximate
, while an error estimate is computed from this approximation together with that obtained using Chebyshev series of degree 12. If the above conditions do not hold then Gauss 7-point and Kronrod 15-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
-algorithm (
Wynn (1956)) to perform extrapolation. The local error estimation is described in
Piessens et al. (1983).
4 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 transformation Math. Tables Aids Comput. 10 91–96
5 Arguments
- 1:
g – function, supplied by the userExternal Function
-
g must return the value of the function
at a given point.
The specification of
g is:
double |
g (double x,
Nag_User *comm)
|
|
- 1:
x – doubleInput
-
On entry: the point at which the function must be evaluated.
- 2:
comm – Nag_User *
-
Pointer to a structure of type Nag_User with the following member:
- p – Pointer
-
On entry/exit: the pointer
should be cast to the required type, e.g.,
struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument
comm below.)
- 2:
a – doubleInput
-
On entry: the lower limit of integration, .
- 3:
b – doubleInput
-
On entry: the upper limit of integration, . It is not necessary that .
- 4:
omega – doubleInput
-
On entry: the argument in the weight function of the transform.
- 5:
wt_func – Nag_TrigTransformInput
On entry: indicates which integral is to be computed:
- if , ;
- if , .
Constraint:
or .
- 6:
epsabs – doubleInput
-
On entry: the absolute accuracy required. If
epsabs is negative, the absolute value is used. See
Section 7.
- 7:
epsrel – doubleInput
-
On entry: the relative accuracy required. If
epsrel is negative, the absolute value is used. See
Section 7.
- 8:
max_num_subint – IntegerInput
-
On entry: the upper bound on the number of sub-intervals into which the interval of integration may be divided by the function. The more difficult the integrand, the larger
max_num_subint should be.
Constraint:
.
- 9:
result – double *Output
-
On exit: the approximation to the integral .
- 10:
abserr – double *Output
-
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for .
- 11:
qp – Nag_QuadProgress *
-
Pointer to structure of type Nag_QuadProgress with the following members:
- num_subint – IntegerOutput
-
On exit: the actual number of sub-intervals used.
- fun_count – IntegerOutput
-
On exit: the number of function evaluations performed by nag_1d_quad_wt_trig_1 (d01snc).
- sub_int_beg_pts – double *Output
- sub_int_end_pts – double *Output
- sub_int_result – double *Output
- sub_int_error – double *Output
-
On exit: these pointers are allocated memory internally with
max_num_subint elements. If an error exit other than
NE_INT_ARG_LT,
NE_BAD_PARAM or
NE_ALLOC_FAIL occurs, these arrays will contain information which may be useful. For details, see
Section 9.
Before a subsequent call to nag_1d_quad_wt_trig_1 (d01snc) is made, or when the information contained in these arrays is no longer useful, you should free the storage allocated by these pointers using the NAG macro NAG_FREE.
- 12:
comm – Nag_User *
-
Pointer to a structure of type Nag_User with the following member:
- p – Pointer
-
On entry/exit: the pointer
, of type Pointer, allows you to communicate information to and from
g(). An object of the required type should be declared, e.g., a structure, and its address assigned to the pointer
by means of a cast to Pointer in the calling program, e.g.,
comm.p = (Pointer)&s. The type Pointer is
void *.
- 13:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument
wt_func had an illegal value.
- NE_INT_ARG_LT
-
On entry,
max_num_subint must not be less than 1:
.
- NE_QUAD_BAD_SUBDIV
-
Extremely bad integrand behaviour occurs around the sub-interval
.
The same advice applies as in the case of
NE_QUAD_MAX_SUBDIV.
- NE_QUAD_MAX_SUBDIV
-
The maximum number of subdivisions has been reached: .
The maximum number of subdivisions 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 sub-intervals. 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 value of
max_num_subint.
- NE_QUAD_NO_CONV
-
The integral is probably divergent or slowly convergent.
Please note that divergence can also occur with any error exit other than
NE_INT_ARG_LT,
NE_BAD_PARAM or
NE_ALLOC_FAIL.
-
Round-off error is detected during extrapolation.
The requested tolerance cannot be achieved, because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best that can be obtained.
The same advice applies as in the case of
NE_QUAD_MAX_SUBDIV.
- NE_QUAD_ROUNDOFF_TOL
-
Round-off error prevents the requested tolerance from being achieved:
,
.
The error may be underestimated. Consider relaxing the accuracy requirements specified by
epsabs and
epsrel.
7 Accuracy
nag_1d_quad_wt_trig_1 (d01snc) cannot guarantee, but in practice usually achieves, the following accuracy:
where
and
epsabs and
epsrel are user-specified absolute and relative error tolerances. Moreover it returns the quantity
abserr which, in normal circumstances, satisfies
8 Parallelism and Performance
Not applicable.
The time taken by tnag_1d_quad_wt_trig_1 (d01snc) depends on the integrand and the accuracy required.
If the function fails with an error exit other than
NE_INT_ARG_LT,
NE_BAD_PARAM or
NE_ALLOC_FAIL, then you may wish to examine the contents of the structure
qp. These contain the end-points of the sub-intervals used by nag_1d_quad_wt_trig_1 (d01snc) along with the integral contributions and error estimates over the sub-intervals.
Specifically, , let denote the approximation to the value of the integral over the sub-interval in the partition of and be the corresponding absolute error estimate.
Then,
and
unless the function 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
is returned in
, and the values
,
,
and
are stored in the structure
qp as
- ,
- ,
- and
- .
10 Example
10.1 Program Text
Program Text (d01snce.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (d01snce.r)