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NAG C Library Manual

# NAG Library Function Documentnag_fresnel_s (s20acc)

## 1  Purpose

nag_fresnel_s (s20acc) returns a value for the Fresnel Integral $S\left(x\right)$.

## 2  Specification

 #include #include
 double nag_fresnel_s (double x)

## 3  Description

nag_fresnel_s (s20acc) evaluates an approximation to the Fresnel Integral
 $S x = ∫ 0 x sin π 2 t 2 dt .$
The function is based on Chebyshev expansions.

## 4  References

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

## 5  Arguments

1:     xdoubleInput
On entry: the argument $x$ of the function.

None.

## 7  Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is somewhat larger than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by $\epsilon \simeq \left|x\mathrm{sin}\left(\pi {x}^{2}/2\right)/S\left(x\right)\right|\delta$.
However, if $\delta$ is of the same order as the machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.
For small $x$, $\epsilon \simeq 3\delta$ 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 $x$, $\left|\epsilon \right|\simeq \left|2x\mathrm{sin}\left(\pi {x}^{2}/2\right)\right|\left|\delta \right|$ 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 $1/{x}^{2}$ is of the order of the machine precision); in this region the relative error in the result is essentially bounded by $2/\pi x$.
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.

None.

## 9  Example

The following program reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

### 9.1  Program Text

Program Text (s20acce.c)

### 9.2  Program Data

Program Data (s20acce.d)

### 9.3  Program Results

Program Results (s20acce.r)