# NAG Library Routine Document

## 1Purpose

s17ajf returns a value of the derivative of the Airy function $\mathrm{Ai}\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s17ajf ( x,
 Real (Kind=nag_wp) :: s17ajf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nagmk26.h>
 double s17ajf_ (const double *x, Integer *ifail)

## 3Description

s17ajf evaluates an approximation to the derivative of the Airy function $\mathrm{Ai}\left(x\right)$. It is based on a number of Chebyshev expansions.
For $x<-5$,
 $Ai′x=-x4 atcos⁡z+btζsin⁡z ,$
where $z=\frac{\pi }{4}+\zeta$, $\zeta =\frac{2}{3}\sqrt{-{x}^{3}}$ and $a\left(t\right)$ and $b\left(t\right)$ are expansions in variable $t=-2{\left(\frac{5}{x}\right)}^{3}-1$.
For $-5\le x\le 0$,
 $Ai′x=x2ft-gt,$
where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^{3}-1$.
For $0,
 $Ai′x=e-11x/8yt,$
where $y\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-1$.
For $4.5\le x<9$,
 $Ai′x=e-5x/2vt,$
where $v\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-3$.
For $x\ge 9$,
 $Ai′x = x4 e-z ut ,$
where $z=\frac{2}{3}\sqrt{{x}^{3}}$ and $u\left(t\right)$ is an expansion in $t=2\left(\frac{18}{z}\right)-1$.
For $\left|x\right|<\text{}$ the square of the machine precision, the result is set directly to ${\mathrm{Ai}}^{\prime }\left(0\right)$. This both saves time and avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy and so the routine must fail. This occurs for $x<-{\left(\frac{\sqrt{\pi }}{\epsilon }\right)}^{4/7}$, where $\epsilon$ is the machine precision.
For large positive arguments, where ${\mathrm{Ai}}^{\prime }$ decays in an essentially exponential manner, there is a danger of underflow so the routine must fail.

## 4References

NIST Digital Library of Mathematical Functions

## 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 $0$, . 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  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  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:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
x is too large and positive. The function returns zero.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
x is too large and negative. The function returns zero.
${\mathbf{ifail}}=-99$
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 in character and here relative error is needed. The absolute error, $E$, and the relative error, $\epsilon$, are related in principle to the relative error in the argument, $\delta$, by
 $E≃ x2 Aix δ ε≃ x2 Aix Ai′x δ.$
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 $x$, positive or negative, errors are strongly attenuated by the function and hence will be roughly bounded by the machine precision.
For moderate to large negative $x$, the error, like the function, is oscillatory; however, the amplitude of the error grows like
 $x7/4π.$
Therefore it becomes impossible to calculate the function with any accuracy if ${\left|x\right|}^{7/4}>\frac{\sqrt{\pi }}{\delta }$.
For large positive $x$, the relative error amplification is considerable:
 $εδ≃x3.$
However, very large arguments are not possible due to the danger of underflow. Thus in practice error amplification is limited.

## 8Parallelism and Performance

s17ajf 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 (s17ajfe.f90)

### 10.2Program Data

Program Data (s17ajfe.d)

### 10.3Program Results

Program Results (s17ajfe.r)