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

g (x) w (x)

, where the weight function

w (x) = \frac{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 \leq c \leq 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 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:

p – Pointer: On entry/exit: the pointer $comm \to p$ 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 nag_1d_quad_wt_cauchy_1 (d01sqc). If your code inadvertently does return any NaNs or infinities, nag_1d_quad_wt_cauchy_1 (d01sqc) is likely to produce unexpected results.

2: $a$ – doubleInput

On entry: the lower limit of integration,

a

3: $b$ – doubleInput

On entry: the upper limit of integration,

b

. It is not necessary that

a < b

4: $c$ – doubleInput

On entry: the argument

c

in the weight function.

Constraint:

c \neq a ​ or ​ b

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:

max_num_subint \geq 1

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:

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 $comm \to p$ , 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 $comm \to p$ 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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_2_REAL_ARG_EQ: On entry, $c = 〈value〉$ while $a = 〈value〉$ . These arguments must satisfy $c \neq a$ .

On entry, $c = 〈value〉$ while $b = 〈value〉$ . These arguments must satisfy $c \neq b$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, max_num_subint must not be less than 1: $max_num_subint = 〈value〉$ .
NE_QUAD_BAD_SUBDIV: Extremely bad integrand behaviour occurs around the sub-interval $(〈value〉, 〈value〉)$ .
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: $max_num_subint = 〈value〉$ .

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.
NE_QUAD_ROUNDOFF_TOL: 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

nag_1d_quad_wt_cauchy_1 (d01sqc) cannot guarantee, but in practice usually achieves, the following accuracy:

|I - result| \leq tol

where

tol = \max \{|epsabs|, |epsrel| \times |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| \leq abserr \leq tol .

8

Parallelism and Performance

nag_1d_quad_wt_cauchy_1 (d01sqc) is not threaded in any implementation.

9

Further Comments

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

i = 1, 2, \dots, 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,

\int_{a_{i}}^{b_{i}} g (x) w (x) d x ≃ r_{i}

and

result = \sum_{i = 1}^{n} r_{i}

The value of

n

is returned in

qp \to num_subint

, and the values

a_{i}

b_{i}

r_{i}

and

e_{i}

are stored in the structure qp as

$a_{i} = qp \to sub_int_beg_pts [i - 1]$ ,
$b_{i} = qp \to sub_int_end_pts [i - 1]$ ,
$r_{i} = qp \to sub_int_result [i - 1]$ and
$e_{i} = qp \to sub_int_error [i - 1]$ .

10

Example

This example computes

\int_{- 1}^{1} \frac{d x}{(x^{2} + {0.01}^{2}) (x - \frac{1}{2})} .

10.1

Program Text

Program Text (d01sqce.c)

10.2

Program Data

None.

10.3

Program Results

Program Results (d01sqce.r)

nag_1d_quad_wt_cauchy_1 (d01sqc) (PDF version)

d01 Chapter Contents

d01 Chapter Introduction

NAG Library Manual

NAG Library Function Document

nag_1d_quad_wt_cauchy_1 (d01sqc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification