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NAG C Library Manual

# NAG Library Function Documentnag_bessel_k0 (s18acc)

## 1  Purpose

nag_bessel_k0 (s18acc) returns the value of the modified Bessel function ${K}_{0}\left(x\right)$.

## 2  Specification

 #include #include
 double nag_bessel_k0 (double x, NagError *fail)

## 3  Description

nag_bessel_k0 (s18acc) evaluates an approximation to the modified Bessel function of the second kind, ${K}_{0}\left(x\right)$.
The function is based on Chebyshev expansions.

## 4  References

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

## 5  Arguments

1:     xdoubleInput
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_REAL_ARG_LE
On entry, x must not be less than or equal to 0.0: ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
${k}_{0}$ is undefined and the function returns zero.

## 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 $\epsilon \simeq \left|{xK}_{1}\left(x\right)/{K}_{0}\left(x\right)\right|\delta$.
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 $\left|1/\mathrm{ln}x\right|$, 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 ${e}^{-x}/\sqrt{x}$, 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 math library function exp.

None.

## 9  Example

The following program reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

### 9.1  Program Text

Program Text (s18acce.c)

### 9.2  Program Data

Program Data (s18acce.d)

### 9.3  Program Results

Program Results (s18acce.r)