1Purpose

s19adc returns a value for the Kelvin function $\mathrm{kei}x$.

2Specification

 #include
 double s19adc (double x, NagError *fail)
The function may be called by the names: s19adc, nag_specfun_kelvin_kei or nag_kelvin_kei.

3Description

s19adc evaluates an approximation to the Kelvin function $\mathrm{kei}x$.
Note:  for $x<0$ the function is undefined, so we need only consider $x\ge 0$.
The function is based on several Chebyshev expansions:
For $0\le x\le 1$,
 $kei⁡x=-π4ft+x24-gtlogx+vt$
where $f\left(t\right)$, $g\left(t\right)$ and $v\left(t\right)$ are expansions in the variable $t=2{x}^{4}-1$;
For $1,
 $kei⁡x=exp-98x ut$
where $u\left(t\right)$ is an expansion in the variable $t=x-2$;
For $x>3$,
 $kei⁡x=π 2x e-x/2 1+1x ctsin⁡β+1xdtcos⁡β$
where $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$, and $c\left(t\right)$ and $d\left(t\right)$ are expansions in the variable $t=\frac{6}{x}-1$.
For $x<0$, the function is undefined, and hence the function fails and returns zero.
When $x$ is sufficiently close to zero, the result is computed as
 $kei⁡x=-π4+1-γ-logx2 x24$
and when $x$ is even closer to zero simply as
 $kei⁡x=-π4.$
For large $x$, $\mathrm{kei}x$ is asymptotically given by $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$ and this becomes so small that it cannot be computed without underflow and the function fails.

4References

NIST Digital Library of Mathematical Functions

5Arguments

1: $\mathbf{x}$double Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}\ge 0.0$.
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, ${\mathbf{x}}=〈\mathit{\text{value}}〉$. The function returns zero.
Constraint: ${\mathbf{x}}\le 〈\mathit{\text{value}}〉$.
x is too large, the result underflows and the function returns zero.
NE_REAL_ARG_LT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge 0.0$.
The function is undefined and returns zero.

7Accuracy

Let $E$ be the absolute error in the result, and $\delta$ be the relative error in the argument. If $\delta$ is somewhat larger than the machine representation error, then we have:
 $E≃ x2 - ker1⁡x+ kei1⁡x δ.$
For small $x$, errors are attenuated by the function and hence are limited by the machine precision.
For medium and large $x$, the error behaviour, like the function itself, is oscillatory and hence only absolute accuracy of the function can be maintained. For this range of $x$, the amplitude of the absolute error decays like $\sqrt{\frac{\pi x}{2}}{e}^{-x/\sqrt{2}}$, which implies a strong attenuation of error. Eventually, $\mathrm{kei}x$, which is asymptotically given by $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$,becomes so small that it cannot be calculated without causing underflow and therefore the function returns zero. Note that for large $x$, the errors are dominated by those of the standard math library function exp.

8Parallelism and Performance

Underflow may occur for a few values of $x$ close to the zeros of $\mathrm{kei}x$, below the limit which causes a failure with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_GT.

10Example

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