# NAG CL Interfaces17awc (airy_​ai_​deriv_​vector)

## 1Purpose

s17awc returns an array of values of the derivative of the Airy function $\mathrm{Ai}\left(x\right)$.

## 2Specification

 #include
 void s17awc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s17awc, nag_specfun_airy_ai_deriv_vector or nag_airy_ai_deriv_vector.

## 3Description

s17awc evaluates an approximation to the derivative of the Airy function $\mathrm{Ai}\left({x}_{i}\right)$ 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 $x<-5$,
 $Ai′x=-x4 atcos⁡z+btζsin⁡z ,$
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 variable $t=-2{\left(\frac{5}{x}\right)}^{3}-1$.
For $-5\le x\le 0$,
 $Ai′x=x2ft-gt,$
where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^{3}-1$.
For $0,
 $Ai′x=e-11x/8yt,$
where $y\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-1$.
For $4.5\le x<9$,
 $Ai′x=e-5x/2vt,$
where $v\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-3$.
For $x\ge 9$,
 $Ai′x = x 4 e-z ut ,$
where $z=\frac{2}{3}\sqrt{{x}^{3}}$ and $u\left(t\right)$ 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 ${\mathrm{Ai}}^{\prime }\left(0\right)$. 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 $x<-{\left(\frac{\sqrt{\pi }}{\epsilon }\right)}^{4/7}$, where $\epsilon$ is the machine precision.
For large positive arguments, where ${\mathrm{Ai}}^{\prime }$ decays in an essentially exponential manner, there is a danger of underflow so the function must fail.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left[{\mathbf{n}}\right]$double Output
On exit: ${\mathrm{Ai}}^{\prime }\left({x}_{i}\right)$, the function values.
4: $\mathbf{ivalid}\left[{\mathbf{n}}\right]$Integer Output
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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_GT in s17ajc, 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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_LT in s17ajc, as defined in the Users' Note for your implementation.
5: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of x was invalid.

## 7Accuracy

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, $E$, and the relative error, $\epsilon$, are related in principle to the relative error in the argument, $\delta$, by
 $E≃ x2 Aix δ ε≃ x2 Aix Ai′x δ.$
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 be roughly bounded by the machine precision.
For moderate to large negative $x$, the error, like the function, is oscillatory; however, the amplitude of the error grows like
 $x7/4π.$
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:
 $εδ≃x3.$
However, very large arguments are not possible due to the danger of underflow. Thus in practice error amplification is limited.

## 8Parallelism and Performance

s17awc is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s17awce.c)

### 10.2Program Data

Program Data (s17awce.d)

### 10.3Program Results

Program Results (s17awce.r)