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NAG Library Manual

# NAG Library Function Documentnag_kelvin_kei_vector (s19arc)

## 1  Purpose

nag_kelvin_kei_vector (s19arc) returns an array of values for the Kelvin function $\mathrm{kei}x$.

## 2  Specification

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

## 3  Description

nag_kelvin_kei_vector (s19arc) evaluates an approximation to the Kelvin function $\mathrm{kei}{x}_{i}$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Note:  for $x<0$ the function is undefined, 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.

## 4  References

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

## 5  Arguments

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: ${\mathbf{x}}\left[\mathit{i}-1\right]\ge 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:    $\mathbf{f}\left[{\mathbf{n}}\right]$doubleOutput
On exit: $\mathrm{kei}{x}_{i}$, 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_kei (s19adc), as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left[i-1\right]=2$
${x}_{i}<0.0$, 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 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NW_IVALID
On entry, at least one value of x was invalid.

## 7  Accuracy

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 function exp.

## 8  Parallelism and Performance

Not applicable.

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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_IVALID.

## 10  Example

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

### 10.1  Program Text

Program Text (s19arce.c)

### 10.2  Program Data

Program Data (s19arce.d)

### 10.3  Program Results

Program Results (s19arce.r)