It is non-adaptive and as such is recommended for the integration of ‘smooth’ functions. These exclude integrands with singularities, derivative singularities or high peaks on

[a, b]

, or which oscillate too strongly on

[a, b]

2 Specification

#include <nag.h>

void

d01bdc (

double

(*f)(double x, Nag_Comm *comm),

double a, double b, double epsabs, double epsrel, double *result, double *abserr, Nag_Comm *comm)

The function may be called by the names: d01bdc, nag_quad_dim1_fin_smooth or nag_quad_1d_fin_smooth.

3 Description

d01bdc is based on the QUADPACK routine QNG (see Piessens et al. (1983)). It is a non-adaptive function which uses as its basic rules, the Gauss

10

-point and

21

-point formulae. If the accuracy criterion is not met, formulae using

43

and

87

points are used successively, stopping whenever the accuracy criterion is satisfied.

This function is designed for smooth integrands only.

4 References

Patterson T N L (1968) The Optimum addition of points to quadrature formulae Math. Comput. 22 847–856

Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

5 Arguments

1: $f$ – function, supplied by the user External Function

f must return the value of the integrand

f

at a given point.

The specification of f is:

double

f (double x, Nag_Comm *comm)

1: $x$ – double Input

On entry: the point at which the integrand

f

must be evaluated.

2: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to f.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling d01bdc you may allocate memory and initialize these pointers with various quantities for use by f when called from d01bdc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01bdc. If your code inadvertently does return any NaNs or infinities, d01bdc is likely to produce unexpected results.

2: $a$ – double Input

On entry:

a

, the lower limit of integration.

3: $b$ – double Input

On entry:

b

, the upper limit of integration. It is not necessary that

a < b

4: $epsabs$ – double Input

On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.

5: $epsrel$ – double Input

On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.

6: $result$ – double * Output

On exit: the approximation to the integral

I

7: $abserr$ – double * Output

On exit: an estimate of the modulus of the absolute error, which should be an upper bound for

|I - result|

8: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

There are no specific errors detected by d01bdc. However, if abserr is greater than

\max \{epsabs, epsrel \times |result|\}

this indicates that the function has probably failed to achieve the requested accuracy within

87

function evaluations.

7 Accuracy

d01bdc attempts to compute an approximation, result, such that:

|I - result| \leq tol,

where

tol = \max \{|epsabs|, |epsrel| \times |I|\},

and epsabs and epsrel are user-specified absolute and relative error tolerances. There can be no guarantee that this is achieved, and you are advised to subdivide the interval if you have any doubts about the accuracy obtained. Note that abserr contains an estimated bound on

|I - result|

8 Parallelism and Performance

d01bdc is not threaded in any implementation.

9 Further Comments

The time taken by d01bdc depends on the integrand and the accuracy required.

10 Example

This example computes

\int_{0}^{1} x^{2} \sin (10 π x) d x .

Interfaces: FL CL AD

NAG CL Interface Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction

d01bd: FL CL AD

NAG CL Interfaced01bdc (dim1_​fin_​smooth)

▸▿ 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 CL Interface
d01bdc (dim1_fin_smooth)