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>

#include <nagd01.h>

void

nag_quad_1d_fin_smooth (

double

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

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

3

Description

nag_quad_1d_fin_smooth (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 userExternal 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: $x$ – doubleInput

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 nag_quad_1d_fin_smooth (d01bdc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_quad_1d_fin_smooth (d01bdc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_quad_1d_fin_smooth (d01bdc). If your code inadvertently does return any NaNs or infinities, nag_quad_1d_fin_smooth (d01bdc) is likely to produce unexpected results.

2: $a$ – doubleInput

On entry:

a

, the lower limit of integration.

3: $b$ – doubleInput

On entry:

b

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

a < b

4: $epsabs$ – doubleInput

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

5: $epsrel$ – doubleInput

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 3.3.1.1 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

There are no specific errors detected by nag_quad_1d_fin_smooth (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

nag_quad_1d_fin_smooth (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

nag_quad_1d_fin_smooth (d01bdc) is not threaded in any implementation.

9

Further Comments

The time taken by nag_quad_1d_fin_smooth (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 .

NAG Library Function Document

nag_quad_1d_fin_smooth (d01bdc)

▸▿ 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