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NAG C Library Manual

# NAG Library Function Documentnag_bessel_k0_vector (s18aqc)

## 1  Purpose

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

## 2  Specification

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

## 3  Description

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.

## 4  References

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

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:     x[n]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:     f[n]doubleOutput
On exit: ${K}_{0}\left({x}_{i}\right)$, the function values.
4:     ivalid[n]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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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.
NW_IVALID
On entry, at least one value of x was invalid.

## 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 function 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 (s18aqce.c)

### 9.2  Program Data

Program Data (s18aqce.d)

### 9.3  Program Results

Program Results (s18aqce.r)