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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_fresnel_s (s20ac)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_fresnel_s (s20ac) returns a value for the Fresnel integral Sx, via the function name.


[result, ifail] = s20ac(x)
[result, ifail] = nag_specfun_fresnel_s(x)


nag_specfun_fresnel_s (s20ac) evaluates an approximation to the Fresnel integral
Note:  Sx=-S-x, so the approximation need only consider x0.0.
The function is based on three Chebyshev expansions:
For 0<x3,
Sx=x3r=0arTrt,   with ​ t=2 x3 4-1.  
For x>3,
Sx=12-fxxcosπ2x2-gxx3sinπ2x2 ,  
where fx=r=0brTrt,
and gx=r=0crTrt,
with t=2 3x 4-1.
For small x, Sx π6x3. 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, fx 1π  and gx 1π2 . Therefore for moderately large x, when 1π2x3  is negligible compared with 12 , the second term in the approximation for x>3 may be dropped. For very large x, when 1πx  becomes negligible, Sx12 . However there will be considerable difficulties in calculating cos π2x2 accurately before this final limiting value can be used. Since cos π2x2 is periodic, its value is essentially determined by the fractional part of x2. If x2=N+θ where N is an integer and 0θ<1, then cos π2x2 depends on θ and on N modulo 4. By exploiting this fact, it is possible to retain significance in the calculation of cos π2x2 either all the way to the very large x limit, or at least until the integer part of x2  is equal to the maximum integer allowed on the machine.


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


Compulsory Input Parameters

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

Optional Input Parameters


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_s (s20ac). The argument ifail has been included for consistency with other functions in this chapter.


Let δ and ε be the relative errors in the argument and result respectively.
If 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 x sin π2 x2 Sx δ.  
Figure 1 shows the behaviour of the error amplification factor x sin π2 x2 Sx .
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, ε3δ 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 2x sin π2 x2 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 1x2  is of the order of the machine precision); in this region the relative error in the result is essentially bounded by 2πx .
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
Figure 1

Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
function s20ac_example

fprintf('s20ac 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] = s20ac(x(j));

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


function s20ac_plot
  x = [-10:0.02:10];
  for j = 1:numel(x)
    [S(j), ifail] = s20ac(x(j));

  fig1 = figure;
  title('Fresnel Integral S(x)');

s20ac example results

      x          S(x)
   0.000e+00   0.000e+00
   5.000e-01   6.473e-02
   1.000e+00   4.383e-01
   2.000e+00   3.434e-01
   4.000e+00   4.205e-01
   5.000e+00   4.992e-01
   6.000e+00   4.470e-01
   8.000e+00   4.602e-01
   1.000e+01   4.682e-01
  -1.000e+00  -4.383e-01
   1.000e+03   4.997e-01

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Chapter Contents
Chapter Introduction
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