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NAG Library Manual

# NAG Library Routine DocumentS18AQF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

S18AQF returns an array of values of the modified Bessel function ${K}_{0}\left(x\right)$.

## 2  Specification

 SUBROUTINE S18AQF ( N, X, F, IVALID, IFAIL)
 INTEGER N, IVALID(N), IFAIL REAL (KIND=nag_wp) X(N), F(N)

## 3  Description

S18AQF evaluates an approximation to the modified Bessel function of the second kind ${K}_{0}\left({x}_{i}\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Note:  ${K}_{0}\left(x\right)$ is undefined for $x\le 0$ and the routine will fail for such arguments.
The routine is based on five Chebyshev expansions:
For $0,
 $K0x=-ln⁡x∑′r=0arTrt+∑′r=0brTrt, where ​t=2x2-1.$
For $1,
 $K0x=e-x∑′r=0crTrt, where ​t=2x-3.$
For $2,
 $K0x=e-x∑′r=0drTrt, where ​t=x-3.$
For $x>4$,
 $K0x=e-xx ∑′r=0erTrt,where ​ t=9-x 1+x .$
For $x$ near zero, ${K}_{0}\left(x\right)\simeq -\gamma -\mathrm{ln}\left(\frac{x}{2}\right)$, where $\gamma$ denotes Euler's constant. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.
For large $x$, where there is a danger of underflow due to the smallness of ${K}_{0}$, the result is set exactly to zero.

## 4  References

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

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{N}}\ge 0$.
2:     X(N) – REAL (KIND=nag_wp) arrayInput
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:     F(N) – REAL (KIND=nag_wp) arrayOutput
On exit: ${K}_{0}\left({x}_{i}\right)$, the function values.
4:     IVALID(N) – INTEGER arrayOutput
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.
${\mathbf{IVALID}}\left(i\right)=1$
${x}_{i}\le 0.0$, ${K}_{0}\left({x}_{i}\right)$ is undefined. ${\mathbf{F}}\left(\mathit{i}\right)$ contains $0.0$.
5:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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: ${\mathbf{N}}\ge 0$.

## 7  Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is somewhat larger than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by:
 $ε≃ x K1 x K0 x δ.$
Figure 1 shows the behaviour of the error amplification factor
 $x K1x K0 x .$
However, if $\delta$ is of the same order as machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.
For small $x$, the amplification factor is approximately $\left|\frac{1}{\mathrm{ln}x}\right|$, which implies strong attenuation of the error, but in general $\epsilon$ can never be less than the machine precision.
For large $x$, $\epsilon \simeq x\delta$ and we have strong amplification of the relative error. Eventually ${K}_{0}$, which is asymptotically given by $\frac{{e}^{-x}}{\sqrt{x}}$, becomes so small that it cannot be calculated without underflow and hence the routine will return zero. Note that for large $x$ the errors will be dominated by those of the standard function exp.
Figure 1

None.

## 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 (s18aqfe.f90)

### 9.2  Program Data

Program Data (s18aqfe.d)

### 9.3  Program Results

Program Results (s18aqfe.r)