# NAG Library Function Documentnag_airy_ai (s17agc)

## 1  Purpose

nag_airy_ai (s17agc) returns a value for the Airy function $\mathrm{Ai}\left(x\right)$.

## 2  Specification

#include <nag.h>
#include <nags.h>

double  nag_airy_ai (double x,  NagError *fail)

## 3  Description

nag_airy_ai (s17agc) evaluates an approximation to the Airy function, $\mathrm{Ai}\left(x\right)$. It is based on a number of Chebyshev expansions.
For large negative arguments, it is impossible to calculate the phase of the oscillatory function with any precision and so the function must fail. This occurs if $x<-{\left(3/2\epsilon \right)}^{2/3}$, 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 function must fail.

## 4  References

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

## 5  Arguments

1:     x – doubleInput
On entry: the argument $x$ of the function.
2:     fail – NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_REAL_ARG_GT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\le 〈\mathit{\text{value}}〉$.
x is too large and positive. The function returns zero.
NE_REAL_ARG_LT
On entry, x must not be less than $〈\mathit{\text{value}}〉$: ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
x is too large and negative. The function returns zero.

## 7  Accuracy

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\simeq \left|x{\mathrm{Ai}}^{\prime }\left(x\right)\right|\delta $, $\epsilon \simeq \left|x{\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)\right|\delta $.
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 $\text{amplitude}\left(E/\delta \right)\sim {\left|x\right|}^{5/4}/\sqrt{\pi }$. However the phase error will be growing roughly like $2\sqrt{{\left|x\right|}^{3}}/3$ and hence all accuracy will be lost for large negative arguments due to the difficulty in calculating sin and cos to any accuracy if $2\sqrt{{\left|x\right|}^{3}}/3>1/\delta $.
For large positive arguments, the relative error amplification is considerable, $\epsilon /\delta \sim \sqrt{{x}^{3}}$.
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.

None.

## 9  Example

The following program 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 (s17agce.c)

### 9.2  Program Data

Program Data (s17agce.d)

### 9.3  Program Results

Program Results (s17agce.r)
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