# NAG Library Routine Document

## 1Purpose

s17ahf returns a value of the Airy function, $\mathrm{Bi}\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s17ahf ( x,
 Real (Kind=nag_wp) :: s17ahf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include nagmk26.h
 double s17ahf_ ( const double *x, Integer *ifail)

## 3Description

s17ahf evaluates an approximation to the Airy function $\mathrm{Bi}\left(x\right)$. It is based on a number of Chebyshev expansions.
For $x<-5$,
 $Bix=atcos⁡z+btsin⁡z-x1/4,$
where $z=\frac{\pi }{4}+\frac{2}{3}\sqrt{-{x}^{3}}$ and $a\left(t\right)$ and $b\left(t\right)$ are expansions in the variable $t=-2{\left(\frac{5}{x}\right)}^{3}-1$.
For $-5\le x\le 0$,
 $Bix=3ft+xgt,$
where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^{3}-1$.
For $0,
 $Bix=e11x/8yt,$
where $y$ is an expansion in $t=4x/9-1$.
For $4.5\le x<9$,
 $Bix=e5x/2vt,$
where $v$ is an expansion in $t=4x/9-3$.
For $x\ge 9$,
 $Bix=ezutx1/4,$
where $z=\frac{2}{3}\sqrt{{x}^{3}}$ and $u$ is an expansion in $t=2\left(\frac{18}{z}\right)-1$.
For , the result is set directly to $\mathrm{Bi}\left(0\right)$. This both saves time and avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate the phase of the oscillating function with any accuracy so the routine must fail. This occurs if $x<-{\left(\frac{3}{2\epsilon }\right)}^{2/3}$, where $\epsilon$ is the machine precision.
For large positive arguments, there is a danger of causing overflow since Bi grows in an essentially exponential manner, so the routine must fail.

## 4References

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

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
x is too large and positive. On soft failure, the routine returns zero. (see the Users' Note for your implementation for details)
${\mathbf{ifail}}=2$
x is too large and negative. On soft failure, the routine returns zero. See also the Users' Note for your implementation.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

For negative arguments the function is oscillatory and hence absolute error is the appropriate measure. In the positive region the function is essentially exponential-like and here relative error is appropriate. The absolute error, $E$, and the relative error, $\epsilon$, are related in principle to the relative error in the argument, $\delta$, by
 $E≃ x Bi′x δ,ε≃ x Bi′x Bix δ.$
In practice, approximate equality is the best that can be expected. When $\delta$, $\epsilon$ or $E$ is of the order of the machine precision, the errors in the result will be somewhat larger.
For small $x$, errors are strongly damped and hence will be bounded essentially by the machine precision.
For moderate to large negative $x$, the error behaviour is clearly oscillatory but the amplitude of the error grows like amplitude $\left(\frac{E}{\delta }\right)\sim \frac{{\left|x\right|}^{5/4}}{\sqrt{\pi }}$.
However the phase error will be growing roughly as $\frac{2}{3}\sqrt{{\left|x\right|}^{3}}$ and hence all accuracy will be lost for large negative arguments. This is due to the impossibility of calculating sin and cos to any accuracy if $\frac{2}{3}\sqrt{{\left|x\right|}^{3}}>\frac{1}{\delta }$.
For large positive arguments, the relative error amplification is considerable:
 $εδ∼x3.$
This means a loss of roughly two decimal places accuracy for arguments in the region of $20$. However very large arguments are not possible due to the danger of causing overflow and errors are therefore limited in practice.

## 8Parallelism and Performance

s17ahf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s17ahfe.f90)

### 10.2Program Data

Program Data (s17ahfe.d)

### 10.3Program Results

Program Results (s17ahfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017