public static double s13ac(
	double x,
	out int ifail
)

Public Shared Function s13ac ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s13ac(
	double x, 
	[OutAttribute] int% ifail
)

static member s13ac : 
        x : float * 
        ifail : int byref -> float 

s13ac returns the value of the cosine integral where $\gamma$ denotes Euler's constant.

s13ac calculates an approximate value for $\mathrm{Ci}\left(x\right)$.

For $0<x\le 16$ it is based on the Chebyshev expansion For $16<x<x_{\mathrm{hi}}$ where the value of $x_{\mathrm{hi}}$ is given in the Users' Note for your implementation, where $f\left(x\right)={\displaystyle \underset{r=0}{{\sum }^{\prime }}}{f}_r{T}_r\left(t\right)$ and $g\left(x\right)={\displaystyle \underset{r=0}{{\sum }^{\prime }}}{g}_r{T}_r\left(t\right)$, $t=2{\left(\frac{16}{x}\right)}^2-1$.

For $x\ge x_{\mathrm{hi}}$, $\mathrm{Ci}\left(x\right)=0$ to within the accuracy possible (see [Accuracy]).

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

Errors or warnings detected by the method:

If $E$ and $\epsilon$ are the absolute and relative errors in the result and $\delta$ is the relative error in the argument then in principle these are related by That is accuracy will be limited by machine precision near the origin and near the zeros of $\cos x$, but near the zeros of $\mathrm{Ci}\left(x\right)$ only absolute accuracy can be maintained.

The behaviour of this amplification is shown in Figure 1.

For large values of $x$, $\mathrm{Ci}\left(x\right)\sim \frac{\sin x}{x}$ therefore $\epsilon \sim \delta x\cot x$ and since $\delta$ is limited by the finite precision of the machine it becomes impossible to return results which have any relative accuracy. That is, when $x\ge 1/\delta$ we have that $\left|\mathrm{Ci}\left(x\right)\right|\le 1/x\sim E$ and hence is not significantly different from zero.

Hence $x_{\mathrm{hi}}$ is chosen such that for values of $x\ge x_{\mathrm{hi}}$, $\mathrm{Ci}\left(x\right)$ in principle would have values less than the machine precision and so is essentially zero.

None.

None.

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