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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_airy_ai_deriv_vector (s17aw)

## Purpose

nag_specfun_airy_ai_deriv_vector (s17aw) returns an array of values of the derivative of the Airy function Ai(x)$\mathrm{Ai}\left(x\right)$.

## Syntax

[f, ivalid, ifail] = s17aw(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_airy_ai_deriv_vector(x, 'n', n)

## Description

nag_specfun_airy_ai_deriv_vector (s17aw) evaluates an approximation to the derivative of the Airy function Ai(xi)$\mathrm{Ai}\left({x}_{i}\right)$ for an array of arguments xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$. It is based on a number of Chebyshev expansions.
For x < 5$x<-5$,
 Ai′(x) = root(4, − x) [a(t)cosz + (b(t))/ζsinz] , $Ai′(x)=-x4 [a(t)cos⁡z+b(t)ζsin⁡z] ,$
where z = π/4 + ζ$z=\frac{\pi }{4}+\zeta$, ζ = (2/3)sqrt(x3)$\zeta =\frac{2}{3}\sqrt{-{x}^{3}}$ and a(t)$a\left(t\right)$ and b(t)$b\left(t\right)$ are expansions in variable t = 2(5/x)31$t=-2{\left(\frac{5}{x}\right)}^{3}-1$.
For 5x0$-5\le x\le 0$,
 Ai′(x) = x2f(t) − g(t), $Ai′(x)=x2f(t)-g(t),$
where f$f$ and g$g$ are expansions in t = 2(x/5)31$t=-2{\left(\frac{x}{5}\right)}^{3}-1$.
For 0 < x < 4.5$0,
 Ai′(x) = e − 11x / 8y(t), $Ai′(x)=e-11x/8y(t),$
where y(t)$y\left(t\right)$ is an expansion in t = 4 (x/9)1$t=4\left(\frac{x}{9}\right)-1$.
For 4.5x < 9$4.5\le x<9$,
 Ai′(x) = e − 5x / 2v(t), $Ai′(x)=e-5x/2v(t),$
where v(t)$v\left(t\right)$ is an expansion in t = 4 (x/9)3$t=4\left(\frac{x}{9}\right)-3$.
For x9$x\ge 9$,
 Ai′(x) = root(4,x) e − z u(t) , $Ai′(x) = x 4 e-z u(t) ,$
where z = (2/3)sqrt(x3)$z=\frac{2}{3}\sqrt{{x}^{3}}$ and u(t)$u\left(t\right)$ is an expansion in t = 2 (18/z)1$t=2\left(\frac{18}{z}\right)-1$.
For |x| < $|x|<\text{}$ the square of the machine precision, the result is set directly to Ai(0)${\mathrm{Ai}}^{\prime }\left(0\right)$. This both saves time and avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy and so the function must fail. This occurs for x < ((sqrt(π))/ε)4 / 7$x<-{\left(\frac{\sqrt{\pi }}{\epsilon }\right)}^{4/7}$, where ε$\epsilon$ is the machine precision.
For large positive arguments, where Ai${\mathrm{Ai}}^{\prime }$ decays in an essentially exponential manner, there is a danger of underflow 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:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
The argument xi${x}_{\mathit{i}}$ of the function, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
n$n$, the number of points.
Constraint: n0${\mathbf{n}}\ge 0$.

None.

### Output Parameters

1:     f(n) – double array
Ai(xi)${\mathrm{Ai}}^{\prime }\left({x}_{i}\right)$, the function values.
2:     ivalid(n) – int64int32nag_int array
ivalid(i)${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
ivalid(i) = 0${\mathbf{ivalid}}\left(i\right)=0$
No error.
ivalid(i) = 1${\mathbf{ivalid}}\left(i\right)=1$
xi${x}_{i}$ is too large and positive. f(i)${\mathbf{f}}\left(\mathit{i}\right)$ contains zero. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in nag_specfun_airy_ai_deriv (s17aj), as defined in the Users' Note for your implementation.
ivalid(i) = 2${\mathbf{ivalid}}\left(i\right)=2$
xi${x}_{i}$ is too large and negative. f(i)${\mathbf{f}}\left(\mathit{i}\right)$ contains zero. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{2}}$ in nag_specfun_airy_ai_deriv (s17aj), as defined in the Users' Note for your implementation.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ 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 ifail = 1${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
ifail = 2${\mathbf{ifail}}=2$
Constraint: n0${\mathbf{n}}\ge 0$.

## 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 in character and here relative error is needed. The absolute error, E$E$, and the relative error, ε$\epsilon$, are related in principle to the relative error in the argument, δ$\delta$, by
 E ≃ |x2Ai(x)|δ ε ≃ |( x2 Ai(x) )/( Ai′(x) )|δ. $E≃ | x2 Ai(x) |δ ε≃ | x2 Ai(x) Ai′(x) |δ.$
In practice, approximate equality is the best that can be expected. When δ$\delta$, ε$\epsilon$ or E$E$ is of the order of the machine precision, the errors in the result will be somewhat larger.
For small x$x$, positive or negative, errors are strongly attenuated by the function and hence will be roughly bounded by the machine precision.
For moderate to large negative x$x$, the error, like the function, is oscillatory; however the amplitude of the error grows like
 (|x|7 / 4)/(sqrt(π)). $|x|7/4π.$
Therefore it becomes impossible to calculate the function with any accuracy if |x|7 / 4 > (sqrt(π))/δ ${|x|}^{7/4}>\frac{\sqrt{\pi }}{\delta }$.
For large positive x$x$, the relative error amplification is considerable:
 ε/δ ≃ sqrt(x3). $εδ≃x3.$
However, very large arguments are not possible due to the danger of underflow. Thus in practice error amplification is limited.

None.

## Example

```function nag_specfun_airy_ai_deriv_vector_example
x = [-10; -1; 0; 1; 5; 10; 20];
[f, ivalid, ifail] = nag_specfun_airy_ai_deriv_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
```
```

X           Y
-1.000e+01   9.963e-01    0
-1.000e+00  -1.016e-02    0
0.000e+00  -2.588e-01    0
1.000e+00  -1.591e-01    0
5.000e+00  -2.474e-04    0
1.000e+01  -3.521e-10    0
2.000e+01  -7.586e-27    0

```
```function s17aw_example
x = [-10; -1; 0; 1; 5; 10; 20];
[f, ivalid, ifail] = s17aw(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
```
```

X           Y
-1.000e+01   9.963e-01    0
-1.000e+00  -1.016e-02    0
0.000e+00  -2.588e-01    0
1.000e+00  -1.591e-01    0
5.000e+00  -2.474e-04    0
1.000e+01  -3.521e-10    0
2.000e+01  -7.586e-27    0

```