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NAG Toolbox

NAG Toolbox: nag_specfun_fresnel_s_vector (s20aq)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_fresnel_s_vector (s20aq) returns an array of values for the Fresnel integral Sx.

Syntax

[f, ifail] = s20aq(x, 'n', n)
[f, ifail] = nag_specfun_fresnel_s_vector(x, 'n', n)

Description

nag_specfun_fresnel_s_vector (s20aq) evaluates an approximation to the Fresnel integral
Sxi=0xisinπ2t2dt  
for an array of arguments xi, for i=1,2,,n.
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.

References

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

Parameters

Compulsory Input Parameters

1:     xn – double array
The argument xi of the function, for i=1,2,,n.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array x.
n, the number of points.
Constraint: n0.

Output Parameters

1:     fn – double array
Sxi, the function values.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ifail=1
Constraint: n0.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

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

None.

Example

This example reads values of x from a file, evaluates the function at each value of xi and prints the results.
function s20aq_example


fprintf('s20aq example results\n\n');

x = [0; 0.5; 1; 2; 4; 5; 6; 8; 10; -1; 1000];

[f, ifail] = s20aq(x);

fprintf('     x           S(x)\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e\n', x(i), f(i));
end


s20aq 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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