NAG Library Routine Document

s20acf (fresnel_s)

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NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

S (specfun) Chapter Contents

S (specfun) Chapter Introduction

▸▿ 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

s20acf returns a value for the Fresnel integral

S (x)

, via the function name.

2

Specification

Fortran Interface

Function s20acf (

x, ifail)

Real (Kind=nag_wp)	::	s20acf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include <nagmk26.h>

double

s20acf_ (const double *x, Integer *ifail)

3

Description

s20acf evaluates an approximation to the Fresnel integral

S (x) = \int_{0}^{x} \sin (\frac{π}{2} t^{2}) d t .

Note:

S (x) = - S (- x)

, so the approximation need only consider

x \geq 0.0

The routine is based on three Chebyshev expansions:

For

0 < x \leq 3

S (x) = x^{3} \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t),   with ​ t = 2 {(\frac{x}{3})}^{4} - 1 .

For

x > 3

S (x) = \frac{1}{2} - \frac{f (x)}{x} \cos (\frac{π}{2} x^{2}) - \frac{g (x)}{x^{3}} \sin (\frac{π}{2} x^{2}),

where

f (x) = \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t)

and

g (x) = \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t)

with

t = 2 {(\frac{3}{x})}^{4} - 1

For small

x

S (x) ≃ \frac{π}{6} x^{3}

. This approximation is used when

x

is sufficiently small for the result to be correct to machine precision. For very small

x

, this approximation would underflow; the result is then set exactly to zero.

For large

x

f (x) ≃ \frac{1}{π}

and

g (x) ≃ \frac{1}{π^{2}}

. Therefore for moderately large

x

, when

\frac{1}{π^{2} x^{3}}

is negligible compared with

\frac{1}{2}

, the second term in the approximation for

x > 3

may be dropped. For very large

x

, when

\frac{1}{π x}

becomes negligible,

S (x) ≃ \frac{1}{2}

. However, there will be considerable difficulties in calculating

\cos (\frac{π}{2} x^{2})

accurately before this final limiting value can be used. Since

\cos (\frac{π}{2} x^{2})

is periodic, its value is essentially determined by the fractional part of

x^{2}

. If

x^{2} = N + θ

where

N

is an integer and

0 \leq θ < 1

, then

\cos (\frac{π}{2} x^{2})

depends on

θ

and on

N

modulo

4

. By exploiting this fact, it is possible to retain significance in the calculation of

\cos (\frac{π}{2} x^{2})

either all the way to the very large

x

limit, or at least until the integer part of

\frac{x}{2}

is equal to the maximum integer allowed on the machine.

4

References

NIST Digital Library of Mathematical Functions

5

Arguments

1: $x$ – Real (Kind=nag_wp)Input: On entry: the argument $x$ of the function.
2: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

There are no failure exits from s20acf. The argument ifail has been included for consistency with other routines in this chapter.

7

Accuracy

Let

δ

and

ε

be the relative errors in the argument and result respectively.

δ

is somewhat larger than the machine precision (i.e., if

δ

is due to data errors etc.), then

ε

and

δ

are approximately related by:

ε ≃ |\frac{x \sin (\frac{π}{2} x^{2})}{S (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x \sin (\frac{π}{2} x^{2})}{S (x)}|

However, if

δ

is of the same order as the machine precision, then rounding errors could make

ε

slightly larger than the above relation predicts.

For small

x

ε ≃ 3 δ

and hence there is only moderate amplification of relative error. Of course for very small

x

where the correct result would underflow and exact zero is returned, relative error-control is lost.

For moderately large values of

x

|ε| ≃ |2 x \sin (\frac{π}{2} x^{2})| |δ|

and the result will be subject to increasingly large amplification of errors. However, the above relation breaks down for large values of

x

(i.e., when

\frac{1}{x^{2}}

is of the order of the machine precision); in this region the relative error in the result is essentially bounded by

\frac{2}{π x}

Hence the effects of error amplification are limited and at worst the relative error loss should not exceed half the possible number of significant figures.

Figure 1

8

Parallelism and Performance

s20acf is not threaded in any implementation.

9

Further Comments

None.

10

Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Routine Document

s20acf (fresnel_s)

▸▿ 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