NAG Library Routine Document

1Purpose

s17auf returns an array of values for the Airy function, $\mathrm{Ai}\left(x\right)$.

2Specification

Fortran Interface
 Subroutine s17auf ( n, x, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(n) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include nagmk26.h
 void s17auf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)

3Description

s17auf evaluates an approximation to the Airy function, $\mathrm{Ai}\left({x}_{i}\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. It is based on a number of Chebyshev expansions:
For $x<-5$,
 $Aix=atsin⁡z-btcos⁡z-x1/4$
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 $-5\le x\le 0$,
 $Aix=ft-xgt,$
where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^{3}-1\text{.}$
For $0,
 $Aix=e-3x/2yt,$
where $y$ is an expansion in $t=4x/9-1$.
For $4.5\le x<9$,
 $Aix=e-5x/2ut,$
where $u$ is an expansion in $t=4x/9-3$.
For $x\ge 9$,
 $Aix=e-zvtx1/4,$
where $z=\frac{2}{3}\sqrt{{x}^{3}}$ and $v$ is an expansion in $t=2\left(\frac{18}{z}\right)-1$.
For , the result is set directly to $\mathrm{Ai}\left(0\right)$. This both saves time and guards against underflow in intermediate calculations.
For large negative arguments, it becomes impossible to calculate the phase of the oscillatory function with any precision and 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, where $\mathrm{Ai}$ decays in an essentially exponential manner, there is a danger of underflow so the routine must fail.

4References

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

5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:     $\mathbf{f}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: $\mathrm{Ai}\left({x}_{i}\right)$, the function values.
4:     $\mathbf{ivalid}\left({\mathbf{n}}\right)$ – Integer arrayOutput
On exit: ${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${x}_{i}$ is too large and positive. ${\mathbf{f}}\left(\mathit{i}\right)$ contains zero. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in s17agf, as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left(i\right)=2$
${x}_{i}$ is too large and negative. ${\mathbf{f}}\left(\mathit{i}\right)$ contains zero. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{2}}$ in s17agf, as defined in the Users' Note for your implementation.
5:     $\mathbf{ifail}$ – 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:
${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
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
 $E≃ x Ai′x δ, ε≃ x Ai′x Aix δ.$
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 by the function and hence will be bounded by the machine precision.
For moderate negative $x$, the error behaviour is oscillatory but the amplitude of the error grows like
 $amplitude Eδ ∼x5/4π.$
However the phase error will be growing roughly like $\frac{2}{3}\sqrt{{\left|x\right|}^{3}}$ and hence all accuracy will be lost for large negative arguments 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:
 $ε δ ∼x3.$
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 setting underflow and so the errors are limited in practice.

8Parallelism and Performance

s17auf is not threaded in any implementation.

None.

10Example

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

10.1Program Text

Program Text (s17aufe.f90)

10.2Program Data

Program Data (s17aufe.d)

10.3Program Results

Program Results (s17aufe.r)

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