nag_1d_quad_wt_alglog_1 (d01spc) (PDF version)
d01 Chapter Contents
d01 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_1d_quad_wt_alglog_1 (d01spc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_1d_quad_wt_alglog_1 (d01spc) is an adaptive integrator which calculates an approximation to the integral of a function g x w x  over a finite interval a,b :
I = a b g x w x dx  
where the weight function w  has end-point singularities of algebraico-logarithmic type.

2  Specification

#include <nag.h>
#include <nagd01.h>
void  nag_1d_quad_wt_alglog_1 (
double (*g)(double x, Nag_User *comm),
double a, double b, double alfa, double beta, Nag_QuadWeight 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_alglog_1 (d01spc) is based upon the QUADPACK routine QAWSE (Piessens et al. (1983)) and integrates a function of the form g x w x , where the weight function w x  may have algebraico-logarithmic singularities at the end-points a  and/or b . The strategy is a modification of that in nag_1d_quad_osc_1 (d01skc). We start by bisecting the original interval and applying modified Clenshaw–Curtis integration of orders 12 and 24 to both halves. Clenshaw–Curtis integration is then used on all sub-intervals which have a  or b  as one of their end-points (Piessens et al. (1974)). On the other sub-intervals Gauss–Kronrod (7–15 point) integration is carried out.
A ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation control 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, Mertens I and Branders M (1974) Integration of functions having end-point singularities Angew. Inf. 16 65–68

5  Arguments

1:     g function, supplied by the userExternal 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 doubleInput
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.)
2:     a doubleInput
On entry: the lower limit of integration, a .
3:     b doubleInput
On entry: the upper limit of integration, b .
Constraint: b>a .
4:     alfa doubleInput
On entry: the argument α  in the weight function.
Constraint: alfa > -1.0 .
5:     beta doubleInput
On entry: the argument β  in the weight function.
Constraint: beta > -1.0 .
6:     wt_func Nag_QuadWeightInput
On entry: indicates which weight function is to be used:
  • if wt_func=Nag_Alg, w x = x-a α b-x β ;
  • if wt_func=Nag_Alg_loga, w x = x-a α b-x β ln x-a ;
  • if wt_func=Nag_Alg_logb, w x = x-a α b-x β ln b-x ;
  • if wt_func=Nag_Alg_loga_logb, w x = x-a α b-x β ln x-a ln b-x .
Constraint: wt_func=Nag_Alg, Nag_Alg_loga, Nag_Alg_logb or Nag_Alg_loga_logb.
7:     epsabs doubleInput
On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.
8:     epsrel doubleInput
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.
9:     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: max_num_subint2 .
10:   result double *Output
On exit: the approximation to the integral I .
11:   abserr double *Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for I - result .
12:   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 nag_1d_quad_wt_alglog_1 (d01spc).
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_BAD_PARAM, NE_REAL_ARG_LE, NE_2_REAL_ARG_LE 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_alglog_1 (d01spc) 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.
13:   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 *.
14:   fail NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, b=value  while a=value . These arguments must satisfy b>a .
Dynamic memory allocation failed.
On entry, argument wt_func had an illegal value.
On entry, max_num_subint must not be less than 2: 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. If the position of a discontinuity or a singularity of algebraico-logarithmic type within the interval can be determined, the interval must be split up at this point and the integrator called 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.
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.
On entry, alfa=value.
Constraint: alfa-1.0.
On entry, beta=value.
Constraint: beta-1.0.

7  Accuracy

nag_1d_quad_wt_alglog_1 (d01spc) cannot guarantee, but in practice usually achieves, the following accuracy:
I - result tol  
where 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

Not applicable.

9  Further Comments

The time taken by nag_1d_quad_wt_alglog_1 (d01spc) 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, NE_REAL_ARG_LE, NE_2_REAL_ARG_LE 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_alglog_1 (d01spc) along with the integral contributions and error estimates over these sub-intervals.
Specifically, 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
0 1 lnx cos10πx dx  
0 1 sin10x x 1-x dx .  

10.1  Program Text

Program Text (d01spce.c)

10.2  Program Data


10.3  Program Results

Program Results (d01spce.r)

nag_1d_quad_wt_alglog_1 (d01spc) (PDF version)
d01 Chapter Contents
d01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015