NAG CL Interfaces18acc (bessel_​k0_​real)

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1Purpose

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

2Specification

 #include
 double s18acc (double x, NagError *fail)
The function may be called by the names: s18acc, nag_specfun_bessel_k0_real or nag_bessel_k0.

3Description

s18acc evaluates an approximation to the modified Bessel function of the second kind ${K}_{0}\left(x\right)$.
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,
 $K0(x)=-ln⁡x∑′r=0arTr(t)+∑′r=0brTr(t), where ​t=2x2-1.$
For $1,
 $K0(x)=e-x∑′r=0crTr(t), where ​t=2x-3.$
For $2,
 $K0(x)=e-x∑′r=0drTr(t), where ​t=x-3.$
For $x>4$,
 $K0(x)=e-xx ∑′r=0erTr(t),where ​ t=9-x 1+x .$
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: $\mathbf{x}$double Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_LE
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}>0.0$.
${K}_{0}$ is undefined and the function returns zero.

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
 $| x K1(x) K0 (x) |.$
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 $|\frac{1}{\mathrm{ln}x}|$, 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 math library function exp.

8Parallelism and Performance

s18acc is not threaded in any implementation.

None.

10Example

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

10.1Program Text

Program Text (s18acce.c)

10.2Program Data

Program Data (s18acce.d)

10.3Program Results

Program Results (s18acce.r)