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NAG Library Manual

# NAG Library Routine DocumentS13ACF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

S13ACF returns the value of the cosine integral
 $Cix=γ+ln⁡x+∫0xcos⁡u-1udu, x>0$
via the routine name where $\gamma$ denotes Euler's constant.

## 2  Specification

 FUNCTION S13ACF ( X, IFAIL)
 REAL (KIND=nag_wp) S13ACF
 INTEGER IFAIL REAL (KIND=nag_wp) X

## 3  Description

S13ACF calculates an approximate value for $\mathrm{Ci}\left(x\right)$.
For $0 it is based on the Chebyshev expansion
 $Cix=ln⁡x+∑r=0′arTrt,t=2 x16 2-1.$
For $16 where the value of ${x}_{\mathrm{hi}}$ is given in the Users' Note for your implementation,
 $Cix=fxsin⁡xx-gxcos⁡xx2$
where $f\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{f}_{r}{T}_{r}\left(t\right)$ and $g\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{g}_{r}{T}_{r}\left(t\right)$, $t=2{\left(\frac{16}{x}\right)}^{2}-1$.
For $x\ge {x}_{\mathrm{hi}}$, $\mathrm{Ci}\left(x\right)=0$ to within the accuracy possible (see Section 7).

## 4  References

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

## 5  Parameters

1:     X – REAL (KIND=nag_wp)Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{X}}>0.0$.
2:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
The routine has been called with an argument less than or equal to zero for which the function is not defined. The result returned is zero.

## 7  Accuracy

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
 $E≃ δ cos⁡x and ​ ε≃ δ cos⁡x Cix .$
That is accuracy will be limited by machine precision near the origin and near the zeros of $\mathrm{cos}x$, 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.
Figure 1
For large values of $x$, $\mathrm{Ci}\left(x\right)\sim \frac{\mathrm{sin}x}{x}$ therefore $\epsilon \sim \delta x\mathrm{cot}x$ 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.

## 9  Example

This example 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 (s13acfe.f90)

### 9.2  Program Data

Program Data (s13acfe.d)

### 9.3  Program Results

Program Results (s13acfe.r)