s20aq: FL CL CPP AD

NAG CL Interface
s20aqc (fresnel_s_vector)

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s20aq: 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

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1 Purpose

s20aqc returns an array of values for the Fresnel integral

S (x)

2 Specification

#include <nag.h>

void	s20aqc (Integer n, const double x[], double f[], NagError *fail)

The function may be called by the names: s20aqc, nag_specfun_fresnel_s_vector or nag_fresnel_s_vector.

3 Description

s20aqc evaluates an approximation to the Fresnel integral

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

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

S (x) = - S (- x)

, so the approximation need only consider

x \geq 0.0

The function 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: $n$ – Integer Input: On entry: $n$ , the number of points.

Constraint: $n \geq 0$ .
2: $x [n]$ – const double Input: On entry: the argument $x_{i}$ of the function, for $i = 1, 2, \dots, n$ .
3: $f [n]$ – double Output: On exit: $S (x_{i})$ , the function values.
4: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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

s20aqc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

s20aq: FL CL CPP AD

NAG CL Interfaces20aqc (fresnel_​s_​vector)

▸▿ 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
s20aqc (fresnel_s_vector)