# NAG FL Interfaces19arf (kelvin_​kei_​vector)

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## 1Purpose

s19arf returns an array of values for the Kelvin function $\mathrm{kei}x$.

## 2Specification

Fortran Interface
 Subroutine s19arf ( n, x, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(n) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nag.h>
 void s19arf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s19arf or nagf_specfun_kelvin_kei_vector.

## 3Description

s19arf 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 routine is based on several Chebyshev expansions:
For $0\le x\le 1$,
 $kei⁡x=-π4f(t)+x24[-g(t)log(x)+v(t)]$
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,
 $kei⁡x=exp(-98x) u(t)$
where $u\left(t\right)$ is an expansion in the variable $t=x-2$;
For $x>3$,
 $kei⁡x=π 2x e-x/2 [(1+1x)c(t)sin⁡β+1xd(t)cos⁡β]$
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 routine fails and returns zero.
When $x$ is sufficiently close to zero, the result is computed as
 $kei⁡x=-π4+(1-γ-log(x2)) x24$
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 routine fails.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $\mathrm{kei}{x}_{i}$, the function values.
4: $\mathbf{ivalid}\left({\mathbf{n}}\right)$Integer array Output
On exit: ${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${x}_{i}$ is too large, the result underflows. ${\mathbf{f}}\left(\mathit{i}\right)$ contains zero. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in s19adf , as defined in the the Users' Note for your implementation.
${\mathbf{ivalid}}\left(i\right)=2$
${x}_{i}<0.0$, the function is undefined. ${\mathbf{f}}\left(\mathit{i}\right)$ contains $0.0$.
5: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 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 routine returns zero. Note that for large $x$, the errors are dominated by those of the standard function exp.

## 8Parallelism and Performance

s19arf is not threaded in any implementation.

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 ${\mathbf{ifail}}={\mathbf{1}}$.

## 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 (s19arfe.f90)

### 10.2Program Data

Program Data (s19arfe.d)

### 10.3Program Results

Program Results (s19arfe.r)