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Chapter Contents

Chapter Introduction

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NAG Toolbox: nag_specfun_fresnel_c (s20ad)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_fresnel_c (s20ad) returns a value for the Fresnel integral

C (x)

, via the function name.

Syntax

[result, ifail] = s20ad(x)

[result, ifail] = nag_specfun_fresnel_c(x)

Description

nag_specfun_fresnel_c (s20ad) evaluates an approximation to the Fresnel integral

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

Note:

C (x) = - C (- x)

, so the approximation need only consider

x \geq 0.0

The function is based on three Chebyshev expansions:

For

0 < x \leq 3

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

For

x > 3

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

C (x) ≃ x

. This approximation is used when

x

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

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,

C (x) ≃ \frac{1}{2}

. However there will be considerable difficulties in calculating

\sin (\frac{π}{2} x^{2})

accurately before this final limiting value can be used. Since

\sin (\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

\sin (\frac{π}{2} x^{2})

depends on

θ

and on

N

modulo

4

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

\sin (\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.

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Parameters

Compulsory Input Parameters

1: $x$ – double scalar: The argument $x$ of the function.

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

There are no failure exits from nag_specfun_fresnel_c (s20ad). The argument ifail has been included for consistency with other functions in this chapter.

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 \cos (\frac{π}{2} x^{2})}{C (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x \cos (\frac{π}{2} x^{2})}{C (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 ≃ δ

and there is no amplification of relative error.

For moderately large values of

x

|ε| ≃ |2 x \cos (\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

Further Comments

None.

Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

Open in the MATLAB editor: s20ad_example

function s20ad_example


fprintf('s20ad example results\n\n');

x = [0   0.5   1   2   4   5   6   8   10   -1   1000];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s20ad(x(j));
end

disp('      x          C(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s20ad_plot;



function s20ad_plot
  x = [-10:0.02:10];
  for j = 1:numel(x)
    [C(j), ifail] = s20ad(x(j));
  end

  fig1 = figure;
  plot(x,C,'-r');
  xlabel('x');
  ylabel('C(x)');
  title('Fresnel Integral C(x)');

s20ad example results

      x          C(x)
   0.000e+00   0.000e+00
   5.000e-01   4.923e-01
   1.000e+00   7.799e-01
   2.000e+00   4.883e-01
   4.000e+00   4.984e-01
   5.000e+00   5.636e-01
   6.000e+00   4.995e-01
   8.000e+00   4.998e-01
   1.000e+01   4.999e-01
  -1.000e+00  -7.799e-01
   1.000e+03   5.000e-01

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Chapter Contents

Chapter Introduction

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