s20ac returns a value for the Fresnel integral Sx.


public static double s20ac(
	double x
Visual Basic
Public Shared Function s20ac ( _
	x As Double _
) As Double
Visual C++
static double s20ac(
	double x
static member s20ac : 
        x : float -> float 


Type: System..::..Double
On entry: the argument x of the function.

Return Value

s20ac returns a value for the Fresnel integral Sx.


s20ac evaluates an approximation to the Fresnel integral
Note:  Sx=-S-x, so the approximation need only consider x0.0.
The method is based on three Chebyshev expansions:
For 0<x3,
Sx=x3r=0arTrt,   with ​t=2x34-1.
For x>3,
where fx=r=0brTrt,
and gx=r=0crTrt,
with t=23x4-1.
For small x, Sxπ6x3. 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, fx1π and gx1π2. 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 cosπ2x2 accurately before this final limiting value can be used. Since cosπ2x2 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 cosπ2x2 depends on θ and on N modulo 4. By exploiting this fact, it is possible to retain significance in the calculation of cosπ2x2 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

Error Indicators and Warnings

There are no failure exits from s20ac. The parameter _ifail has been included for consistency with other methods 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:
Figure 1 shows the behaviour of the error amplification factor xsinπ2x2Sx.
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,
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

Parallelism and Performance


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.

Example program (C#): s20ace.cs

Example program data: s20ace.d

Example program results: s20ace.r

See Also