NAG Library Function Document

nag_1d_quad_wt_alglog_1 (d01spc)

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

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:

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.)

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 $wt_func = Nag_Alg_loga$ , $w (x) = {(x - a)}^{α} {(b - x)}^{β} l n (x - a)$ ;
if $wt_func = Nag_Alg_logb$ , $w (x) = {(x - a)}^{α} {(b - x)}^{β} l n (b - x)$ ;
if $wt_func = Nag_Alg_loga_logb$ , $w (x) = {(x - a)}^{α} {(b - x)}^{β} l n (x - a) l n (b - x)$ .

Constraint:

wt_func = Nag_Alg

Nag_Alg_loga

Nag_Alg_logb

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_subint \geq 2

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:

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_alglog_1 (d01spc).

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, 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:

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 *.

14: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_REAL_ARG_LE: On entry, $b = 〈value〉$ while $a = 〈value〉$ . These arguments must satisfy $b > a$ .
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 2: $max_num_subint = 〈value〉$ .
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. 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.
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.
NE_REAL_ARG_LE: On entry, $alfa = 〈value〉$ .
Constraint: $alfa \leq - 1.0$ .

On entry, $beta = 〈value〉$ .
Constraint: $beta \leq - 1.0$ .

7 Accuracy

nag_1d_quad_wt_alglog_1 (d01spc) 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

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, \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_{0}^{1} \ln x \cos (10 π x) d x

and

\int_{0}^{1} \frac{\sin (10 x)}{\sqrt{x (1 - x)}} d x .

10.1 Program Text

Program Text (d01spce.c)

10.2 Program Data

None.

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

NAG Library Function Documentnag_1d_quad_wt_alglog_1 (d01spc)

▸▿ 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

NAG Library Function Document

nag_1d_quad_wt_alglog_1 (d01spc)