# NAG C Library Function Document

## 1Purpose

nag_bessel_k0_vector (s18aqc) returns an array of values of the modified Bessel function ${K}_{0}\left(x\right)$.

## 2Specification

 #include #include
 void nag_bessel_k0_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

## 3Description

nag_bessel_k0_vector (s18aqc) 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 function will fail for such arguments.
The function 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.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

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: ${\mathbf{x}}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:    $\mathbf{f}\left[{\mathbf{n}}\right]$doubleOutput
On exit: ${K}_{0}\left({x}_{i}\right)$, 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.
${\mathbf{ivalid}}\left[i-1\right]=1$
${x}_{i}\le 0.0$, ${K}_{0}\left({x}_{i}\right)$ is undefined. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains $0.0$.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID
On entry, at least one value of x was invalid.

## 7Accuracy

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 function will return zero. Note that for large $x$ the errors will be dominated by those of the standard function exp.
Figure 1

## 8Parallelism and Performance

nag_bessel_k0_vector (s18aqc) is not threaded in any implementation.

None.

## 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 (s18aqce.c)

### 10.2Program Data

Program Data (s18aqce.d)

### 10.3Program Results

Program Results (s18aqce.r)