NAG C Library Function Document

1Purpose

nag_kelvin_ker_vector (s19aqc) returns an array of values for the Kelvin function $\mathrm{ker}x$.

2Specification

 #include #include
 void nag_kelvin_ker_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3Description

nag_kelvin_ker_vector (s19aqc) evaluates an approximation to the Kelvin function $\mathrm{ker}{x}_{i}$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Note:  for $x<0$ the function is undefined and at $x=0$ it is infinite so we need only consider $x>0$.
The function is based on several Chebyshev expansions:
For $0,
 $ker⁡x=-ftlogx+π16x2gt+yt$
where $f\left(t\right)$, $g\left(t\right)$ and $y\left(t\right)$ are expansions in the variable $t=2{x}^{4}-1$.
For $1,
 $ker⁡x=exp-1116x qt$
where $q\left(t\right)$ is an expansion in the variable $t=x-2$.
For $x>3$,
 $ker⁡x=π 2x e-x/2 1+1xct cos⁡β-1xdtsin⁡β$
where $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$, and $c\left(t\right)$ and $d\left(t\right)$ are expansions in the variable $t=\frac{6}{x}-1$.
When $x$ is sufficiently close to zero, the result is computed as
 $ker⁡x=-γ-logx2+π-38x2 x216$
and when $x$ is even closer to zero, simply as $\mathrm{ker}x=-\gamma -\mathrm{log}\left(\frac{x}{2}\right)$.
For large $x$, $\mathrm{ker}x$ 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:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{x}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:    $\mathbf{f}\left[{\mathbf{n}}\right]$doubleOutput
On exit: $\mathrm{ker}{x}_{i}$, the function values.
4:    $\mathbf{ivalid}\left[{\mathbf{n}}\right]$IntegerOutput
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, the result underflows. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains zero. The threshold value is the same as for ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_GT in nag_kelvin_ker (s19acc), as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left[i-1\right]=2$
${x}_{i}\le 0.0$, the function is undefined. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains $0.0$.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID
On entry, at least one value of x was invalid.
Check ivalid for more information.

7Accuracy

Let $E$ be the absolute error in the result, $\epsilon$ be the relative error in the result and $\delta$ be the relative error in the argument. If $\delta$ is somewhat larger than the machine precision, then we have:
 $E≃ x2 ker1⁡x+ kei1⁡x δ,$
 $ε ≃ x2 ker1⁡x + kei1⁡x ker⁡x δ.$
For very small $x$, the relative error amplification factor is approximately given by $\frac{1}{\left|\mathrm{log}\left(x\right)\right|}$, which implies a strong attenuation of relative error. However, $\epsilon$ in general cannot be less than the machine precision.
For small $x$, errors are damped 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 the absolute accuracy for 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, $\mathrm{ker}x$, which asymptotically behaves like $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$, becomes so small that it cannot be calculated without causing underflow, and the function returns zero. Note that for large $x$ the errors are dominated by those of the standard function exp.

8Parallelism and Performance

nag_kelvin_ker_vector (s19aqc) is not threaded in any implementation.

9Further Comments

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

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 (s19aqce.c)

10.2Program Data

Program Data (s19aqce.d)

10.3Program Results

Program Results (s19aqce.r)