# NAG FL Interfaces17akf (airy_​bi_​deriv)

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## 1Purpose

s17akf returns a value for the derivative of the Airy function $\mathrm{Bi}\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s17akf ( x,
 Real (Kind=nag_wp) :: s17akf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s17akf_ (const double *x, Integer *ifail)
The routine may be called by the names s17akf or nagf_specfun_airy_bi_deriv.

## 3Description

s17akf 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 $x<-5$,
 $Bi′(x)=-x4 [-a(t)sin⁡z+b(t)ζcos⁡z] ,$
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 $-5\le x\le 0$,
 $Bi′(x)=3(x2f(t)+g(t)),$
where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^{3}-1$.
For $0,
 $Bi′(x)=e3x/2y(t),$
where $y\left(t\right)$ is an expansion in $t=4x/9-1$.
For $4.5\le x<9$,
 $Bi′(x)=e21x/8u(t),$
where $u\left(t\right)$ is an expansion in $t=4x/9-3$.
For $x\ge 9$,
 $Bi′(x)=x4ezv(t),$
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 $|x|<\text{}$ 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 routine 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 routine must fail.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
2: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\le ⟨\mathit{\text{value}}⟩$.
x is too large and positive. The function returns zero.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\ge ⟨\mathit{\text{value}}⟩$.
x is too large and negative. The function returns zero.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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
 $E≃ |x2Bi(x)|δ ε≃ | x2 Bi(x) Bi′(x) |δ.$
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 $\frac{{|x|}^{7/4}}{\sqrt{\pi }}$. Therefore, it becomes impossible to calculate the function with any accuracy if ${|x|}^{7/4}>\frac{\sqrt{\pi }}{\delta }$.
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.

## 8Parallelism and Performance

s17akf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s17akfe.f90)

### 10.2Program Data

Program Data (s17akfe.d)

### 10.3Program Results

Program Results (s17akfe.r)