NAG Library Function Document
nag_1d_quad_wt_cauchy_1 (d01sqc)
1 Purpose
nag_1d_quad_wt_cauchy_1 (d01sqc) calculates an approximation to the Hilbert transform of a function
over
:
for user-specified values of
,
and
.
2 Specification
#include <nag.h> |
#include <nagd01.h> |
void |
nag_1d_quad_wt_cauchy_1 (
double |
(*g)(double x,
Nag_User *comm),
|
|
double a,
double b,
double c,
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_cauchy_1 (d01sqc) is based upon the QUADPACK routine QAWC (
Piessens et al. (1983)) and integrates a function of the form
, where the weight function
is that of the Hilbert transform. (If
the integral has to be interpreted in the sense of a Cauchy principal value.) It is an adaptive function which employs a ‘global’ acceptance criterion (as defined by
Malcolm and Simpson (1976)). Special care is taken to ensure that
is never the end-point of a sub-interval (
Piessens et al. (1976)). On each sub-interval
modified Clenshaw–Curtis integration of orders 12 and 24 is performed if
where
. Otherwise the Gauss 7-point and Kronrod 15-point rules are used. The local error estimation is described by
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, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag
Piessens R, van Roy–Branders M and Mertens I (1976) The automatic evaluation of Cauchy principal value integrals Angew. Inf. 18 31–35
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:
c – doubleInput
-
On entry: the argument in the weight function.
Constraint:
.
- 5:
epsabs – doubleInput
-
On entry: the absolute accuracy required. If
epsabs is negative, the absolute value is used. See
Section 7.
- 6:
epsrel – doubleInput
-
On entry: the relative accuracy required. If
epsrel is negative, the absolute value is used. See
Section 7.
- 7:
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:
.
- 8:
result – double *Output
-
On exit: the approximation to the integral .
- 9:
abserr – double *Output
-
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for .
- 10:
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_cauchy_1 (d01sqc).
- 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_2_REAL_ARG_EQ 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_cauchy_1 (d01sqc) 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.
- 11:
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 *.
- 12:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_2_REAL_ARG_EQ
-
On entry, while . These arguments must satisfy .
On entry, while . These arguments must satisfy .
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- 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. 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_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_cauchy_1 (d01sqc) 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 nag_1d_quad_wt_cauchy_1 (d01sqc) depends on the integrand and the accuracy required.
If the function fails with an error exit other than
NE_INT_ARG_LT,
NE_2_REAL_ARG_EQ 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_cauchy_1 (d01sqc) along with the integral contributions and error estimates over the sub-intervals.
Specifically, for , 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 .
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 (d01sqce.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (d01sqce.r)