Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_airy_bi_deriv (s17ak)

## Purpose

nag_specfun_airy_bi_deriv (s17ak) returns a value for the derivative of the Airy function $\mathrm{Bi}\left(x\right)$, via the function name.

## Syntax

[result, ifail] = s17ak(x)
[result, ifail] = nag_specfun_airy_bi_deriv(x)

## Description

nag_specfun_airy_bi_deriv (s17ak) calculates an approximate value for the derivative of the Airy function $\mathrm{Bi}\left(x\right)$. It is based on a number of Chebyshev expansions.
For $x<-5$,
 $Bi′x=-x4 -atsin⁡z+btζcos⁡z ,$
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 the variable $t=-2{\left(\frac{5}{x}\right)}^{3}-1$.
For $-5\le x\le 0$,
 $Bi′x=3x2ft+gt,$
where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^{3}-1$.
For $0,
 $Bi′x=e3x/2yt,$
where $y\left(t\right)$ is an expansion in $t=4x/9-1$.
For $4.5\le x<9$,
 $Bi′x=e21x/8ut,$
where $u\left(t\right)$ is an expansion in $t=4x/9-3$.
For $x\ge 9$,
 $Bi′x=x4ezvt,$
where $z=\frac{2}{3}\sqrt{{x}^{3}}$ and $v\left(t\right)$ 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 ${\mathrm{Bi}}^{\prime }\left(0\right)$. This saves time and avoids possible underflows in calculation.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy so the function 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 ${\mathrm{Bi}}^{\prime }$ grows in an essentially exponential manner, there is a danger of overflow so the function must fail.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}$ – double scalar
The argument $x$ of the function.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
x is too large and positive. On soft failure, the function returns zero.
${\mathbf{ifail}}=2$
x is too large and negative. On soft failure the function returns zero.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

For negative arguments the function is oscillatory and hence absolute error is appropriate. In the positive region the function has essentially exponential behaviour and hence 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
 $E≃ x2 Bix δ ε≃ x2 Bix Bi′x δ.$
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 effectively be bounded by the machine precision.
For moderate to large negative $x$, the error is, like the function, oscillatory. However, the amplitude of the absolute error grows like $\frac{{\left|x\right|}^{7/4}}{\sqrt{\pi }}$. 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: $\frac{\epsilon }{\delta }\sim \sqrt{{x}^{3}}$. However, very large arguments are not possible due to the danger of overflow. Thus in practice the actual amplification that occurs is limited.

None.

## Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.
```function s17ak_example

fprintf('s17ak example results\n\n');

x = [-10    -1    0    1    5    10   20];
n = size(x,2);
result = x;

for j=1:n
[result(j), ifail] = s17ak(x(j));
end

disp('      x         Bi''(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s17ak_plot;

function s17ak_plot
x = [-10:0.1:3];
for j = 1:numel(x)
[Bid(j), ifail] = s17ak(x(j));
end

fig1 = figure;
plot(x,Bid,'-r');
xlabel('x');
ylabel('Bi''(x)');
title('Derivative of Airy Function Bi(x)');
axis([-10 4 -2 10]);

```
```s17ak example results

x         Bi'(x)
-1.000e+01   1.194e-01
-1.000e+00   5.924e-01
0.000e+00   4.483e-01
1.000e+00   9.324e-01
5.000e+00   1.436e+03
1.000e+01   1.429e+09
2.000e+01   9.382e+25
```