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NAG C Library Manual

# NAG Library Function Documentnag_kelvin_ker_vector (s19aqc)

## 1  Purpose

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

## 2  Specification

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

## 3  Description

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,
 $ker⁡x=-ftlog⁡x+π16x2gt+yt$
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.

## 4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:     x[n]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:     f[n]doubleOutput
On exit: $\mathrm{ker}{x}_{i}$, the function values.
4:     ivalid[n]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 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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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.
NW_IVALID
On entry, at least one value of x was invalid.

## 7  Accuracy

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 $\frac{1}{\left|\mathrm{log}x\right|}$, 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.

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 NW_IVALID.

## 9  Example

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

### 9.1  Program Text

Program Text (s19aqce.c)

### 9.2  Program Data

Program Data (s19aqce.d)

### 9.3  Program Results

Program Results (s19aqce.r)