NAG CL Interfaces17akc (airy_​bi_​deriv)

1Purpose

s17akc returns a value for the derivative of the Airy function $\mathrm{Bi}\left(x\right)$.

2Specification

 #include
 double s17akc (double x, NagError *fail)
The function may be called by the names: s17akc, nag_specfun_airy_bi_deriv or nag_airy_bi_deriv.

3Description

s17akc 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.

4References

NIST Digital Library of Mathematical Functions

5Arguments

1: $\mathbf{x}$double Input
On entry: the argument $x$ of the function.
2: $\mathbf{fail}$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.
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.
NE_REAL_ARG_GT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
x is too large and positive. The function returns zero.
NE_REAL_ARG_LT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
x is too large and negative. The function returns zero.

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
 $E≃ x2 Bix δ ε≃ x2 Bix Bi′x δ.$
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.

8Parallelism and Performance

s17akc 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 (s17akce.c)

10.2Program Data

Program Data (s17akce.d)

10.3Program Results

Program Results (s17akce.r)