NAG FL Interfaces18aqf (bessel_​k0_​real_​vector)

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1Purpose

s18aqf returns an array of values of the modified Bessel function ${K}_{0}\left(x\right)$.

2Specification

Fortran Interface
 Subroutine s18aqf ( n, x, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(n) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nag.h>
 void s18aqf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s18aqf or nagf_specfun_bessel_k0_real_vector.

3Description

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,
 $K0(x)=-ln⁡x∑′r=0arTr(t)+∑′r=0brTr(t), where ​t=2x2-1.$
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: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
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: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${K}_{0}\left({x}_{i}\right)$, the function values.
4: $\mathbf{ivalid}\left({\mathbf{n}}\right)$Integer array Output
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

8Parallelism and Performance

s18aqf 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 (s18aqfe.f90)

10.2Program Data

Program Data (s18aqfe.d)

10.3Program Results

Program Results (s18aqfe.r)