s20ac: FL CL CPP AD

NAG FL Interface
s20acf (fresnel_s)

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NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s20ac: FL CL CPP AD

▸▿ 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 <nag.h>

double

s20acf_ (const double *x, Integer *ifail)

The routine may be called by the names s20acf or nagf_specfun_fresnel_s.

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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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.

s20ac: FL CL CPP AD

NAG FL Interfaces20acf (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

NAG FL Interface
s20acf (fresnel_s)