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# NAG Toolbox: nag_specfun_airy_bi_deriv_vector (s17ax)

## Purpose

nag_specfun_airy_bi_deriv_vector (s17ax) returns an array of values for the derivative of the Airy function $\mathrm{Bi}\left(x\right)$.

## Syntax

[f, ivalid, ifail] = s17ax(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_airy_bi_deriv_vector(x, 'n', n)

## Description

nag_specfun_airy_bi_deriv_vector (s17ax) calculates an approximate value for the derivative of 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 $x<-5$,
 $Bi′x=-x4 -atsin⁡z+btζcos⁡z ,$
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:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array x.
$n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{f}\left({\mathbf{n}}\right)$ – double array
${\mathrm{Bi}}^{\prime }\left({x}_{i}\right)$, the function values.
2:     $\mathrm{ivalid}\left({\mathbf{n}}\right)$int64int32nag_int array
${\mathbf{ivalid}}\left(\mathit{i}\right)$ 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 nag_specfun_airy_bi_deriv (s17ak), 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 nag_specfun_airy_bi_deriv (s17ak), as defined in the Users' Note for your implementation.
3:     $\mathrm{ifail}$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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
Check ivalid for more information.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
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 x from a file, evaluates the function at each value of ${x}_{i}$ and prints the results.
```function s17ax_example

fprintf('s17ax example results\n\n');

x = [-10; -1; 0; 1; 5; 10; 20];

[f, ivalid, ifail] = s17ax(x);

fprintf('     x           Bi''(x)   ivalid\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end

```
```s17ax example results

x           Bi'(x)   ivalid
-1.000e+01   1.194e-01    0
-1.000e+00   5.924e-01    0
0.000e+00   4.483e-01    0
1.000e+00   9.324e-01    0
5.000e+00   1.436e+03    0
1.000e+01   1.429e+09    0
2.000e+01   9.382e+25    0
```

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