public static double s20ad(
	double x
)

Public Shared Function s20ad ( _
	x As Double _
) As Double

public:
static double s20ad(
	double x
)

static member s20ad : 
        x:float -> float 

s20ad returns a value for the Fresnel Integral Cx.

s20ad evaluates an approximation to the Fresnel Integral Note:  Cx=-C-x, so the approximation need only consider x≥0.0.

The method is based on three Chebyshev expansions:

For 0<x≤3, For x>3, where fx=∑'r=0brTrt,

and gx=∑'r=0crTrt,

with t=2 3x 4-1.

For small x, Cx≃x. This approximation is used when x is sufficiently small for the result to be correct to machine precision.

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, Cx≃ 12 . However there will be considerable difficulties in calculating sin π2x2 accurately before this final limiting value can be used. Since sin π2x2 is periodic, its value is essentially determined by the fractional part of x2. If x2=N+θ, where N is an integer and 0≤θ<1, then sin π2x2 depends on θ and on N modulo 4. By exploiting this fact, it is possible to retain some significance in the calculation of sin 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.

There are no failure exits from s20ad.

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 cos π2 x2 Cx δ.

Figure 1 shows the behaviour of the error amplification factor x cos π2 x2 Cx .

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, 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.