# NAG FL Interfaces19aqf (kelvin_​ker_​vector)

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## 1Purpose

s19aqf returns an array of values for the Kelvin function $\mathrm{ker}x$.

## 2Specification

Fortran Interface
 Subroutine s19aqf ( 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 s19aqf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s19aqf or nagf_specfun_kelvin_ker_vector.

## 3Description

s19aqf 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 routine is based on several Chebyshev expansions:
For $0,
 $ker⁡x=-f(t)log(x)+π16x2g(t)+y(t)$
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,
 $ker⁡x=exp(-1116x) q(t)$
where $q\left(t\right)$ is an expansion in the variable $t=x-2$.
For $x>3$,
 $ker⁡x=π 2x e-x/2 [(1+1xc(t))cos⁡β-1xd(t)sin⁡β]$
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
 $ker⁡x=-γ-log(x2)+(π-38x2) x216$
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 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: ${\mathbf{x}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $\mathrm{ker}{x}_{i}$, 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 s19acf , as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left(i\right)=2$
${x}_{i}\le 0.0$, 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, $\epsilon$ 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 $\frac{1}{|\mathrm{log}\left(x\right)|}$, 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 routine returns zero. Note that for large $x$ the errors are dominated by those of the standard function exp.

## 8Parallelism and Performance

s19aqf is not threaded in any implementation.

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

## 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 (s19aqfe.f90)

### 10.2Program Data

Program Data (s19aqfe.d)

### 10.3Program Results

Program Results (s19aqfe.r)