# NAG Library Function Documentnag_airy_bi_vector (s17avc)

## 1  Purpose

nag_airy_bi_vector (s17avc) returns an array of values of the Airy function, $\mathrm{Bi}\left(x\right)$.

## 2  Specification

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

void  nag_airy_bi_vector (Integer n, const double x[], double f[], Integer ivalid[],  NagError *fail)

## 3  Description

nag_airy_bi_vector (s17avc) evaluates an approximation to the Airy function $\mathrm{Bi}\left({x}_{i}\right)$ 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 $x<-5$,

$Bix=atcos⁡z+btsin⁡z-x1/4,$
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 $0<x<4.5$,

$Bix=e11x/8yt,$
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 function 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 function must fail.

## 4  References

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

## 5  Arguments

1:     n – IntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:     x[n] – const doubleInput
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:     f[n] – doubleOutput
On exit: $\mathrm{Bi}\left({x}_{i}\right)$, the function values.
4:     ivalid[n] – IntegerOutput
On exit: ${\mathbf{ivalid}}\left[\mathit{i}-1\right]$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
${x}_{i}$ is too large and positive. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains zero. The threshold value is the same as for NE_REAL_ARG_GT in nag_airy_bi (s17ahc), as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left[i-1\right]=2$
${x}_{i}$ is too large and negative. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains zero. The threshold value is the same as for NE_REAL_ARG_LT in nag_airy_bi (s17ahc), as defined in the Users' Note for your implementation.
5:     fail – NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NW_IVALID
On entry, at least one value of x was invalid.

## 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

$E≃ x Bi′x δ,ε≃ x Bi′x Bix δ.$
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.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (s17avce.c)

### 9.2  Program Data

Program Data (s17avce.d)

### 9.3  Program Results

Program Results (s17avce.r)
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