NAG CL Interface
d01sqc (dim1_​quad_​wt_​cauchy_​1)

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1 Purpose

d01sqc calculates an approximation to the Hilbert transform of a function g (x) over [a,b] :
I = a b g (x) x-c dx  
for user-specified values of a , b and c .

2 Specification

#include <nag.h>
void  d01sqc (
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)
The function may be called by the names: d01sqc, nag_quad_dim1_quad_wt_cauchy_1 or nag_1d_quad_wt_cauchy_1.

3 Description

d01sqc is based upon the QUADPACK routine QAWC (Piessens et al. (1983)) and integrates a function of the form g (x) w (x) , where the weight function
w (x) = 1 x-c  
is that of the Hilbert transform. (If a < c < b 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 c is never the end-point of a sub-interval (Piessens et al. (1976)). On each sub-interval ( c 1 , c 2 ) modified Clenshaw–Curtis integration of orders 12 and 24 is performed if c 1 - d c c 2 + d where d = ( c 2 - c 1 ) / 20 . 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 user External Function
g must return the value of the function g at a given point.
The specification of g is:
double  g (double x, Nag_User *comm)
1: x double Input
On entry: the point at which the function g must be evaluated.
2: comm Nag_User *
Pointer to a structure of type Nag_User with the following member:
On entry/exit: the pointer commp 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.)
Note: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01sqc. If your code inadvertently does return any NaNs or infinities, d01sqc is likely to produce unexpected results.
2: a double Input
On entry: the lower limit of integration, a .
3: b double Input
On entry: the upper limit of integration, b . It is not necessary that a<b .
4: c double Input
On entry: the argument c in the weight function.
Constraint: ca ​ or ​ b .
5: epsabs double Input
On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.
6: epsrel double Input
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.
7: max_num_subint Integer Input
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: max_num_subint1 .
8: result double * Output
On exit: the approximation to the integral I .
9: abserr double * Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for |I-result| .
10: qp Nag_QuadProgress *
Pointer to structure of type Nag_QuadProgress with the following members:
On exit: the actual number of sub-intervals used.
On exit: the number of function evaluations performed by d01sqc.
sub_int_beg_ptsdouble *Output
sub_int_end_ptsdouble *Output
sub_int_resultdouble *Output
sub_int_errordouble *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 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:
On entry/exit: the pointer commp, 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 commp 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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

On entry, c=value while a=value . These arguments must satisfy ca .
On entry, c=value while b=value . These arguments must satisfy cb .
Dynamic memory allocation failed.
On entry, max_num_subint must not be less than 1: max_num_subint=value .
Extremely bad integrand behaviour occurs around the sub-interval (value,value) .
The same advice applies as in the case of NE_QUAD_MAX_SUBDIV.
The maximum number of subdivisions has been reached: max_num_subint=value .
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.
Round-off error prevents the requested tolerance from being achieved: epsabs=value , epsrel=value .
The error may be underestimated. Consider relaxing the accuracy requirements specified by epsabs and epsrel.

7 Accuracy

d01sqc cannot guarantee, but in practice usually achieves, the following accuracy:
|I-result| tol  
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
|I-result| abserr tol .  

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
d01sqc is not threaded in any implementation.

9 Further Comments

The time taken by 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 d01sqc along with the integral contributions and error estimates over the sub-intervals.
Specifically, for i = 1 , 2 , , n , let r i denote the approximation to the value of the integral over the sub-interval [ a i , b i ] in the partition of [a,b] and e i be the corresponding absolute error estimate.
Then, a i b i g (x) w (x) dx r i and result = i=1 n r i .
The value of n is returned in qpnum_subint, and the values a i , b i , r i and e i are stored in the structure qp as

10 Example

This example computes
−1 1 dx ( x 2 + 0.01 2 ) (x- 1 2 ) .  

10.1 Program Text

Program Text (d01sqce.c)

10.2 Program Data


10.3 Program Results

Program Results (d01sqce.r)