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1Purpose

s18adf returns the value of the modified Bessel function ${K}_{1}\left(x\right)$, via the function name.

2Specification

Fortran Interface
 Real (Kind=nag_wp) :: s18adf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s18adf_ (const double *x, Integer *ifail)
The routine may be called by the names s18adf or nagf_specfun_bessel_k1_real.

3Description

s18adf evaluates an approximation to the modified Bessel function of the second kind ${K}_{1}\left(x\right)$.
Note:  ${K}_{1}\left(x\right)$ is undefined for $x\le 0$ and the routine will fail for such arguments.
The routine is based on five Chebyshev expansions:
For $0,
 $K1(x)=1x+xln⁡x∑′r=0arTr(t)-x∑′r=0brTr(t), where ​ t=2x2-1.$
For $1,
 $K1(x)=e-x∑′r=0crTr(t), where ​t=2x-3.$
For $2,
 $K1(x)=e-x∑′r=0drTr(t), where ​t=x-3.$
For $x>4$,
 $K1(x)=e-xx ∑′r=0erTr(t), where ​t=9-x 1+x .$
For $x$ near zero, ${K}_{1}\left(x\right)\simeq \frac{1}{x}$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision. For very small $x$ on some machines, it is impossible to calculate $\frac{1}{x}$ without overflow and the routine must fail.
For large $x$, where there is a danger of underflow due to the smallness of ${K}_{1}$, the result is set exactly to zero.

4References

NIST Digital Library of Mathematical Functions

5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2: $\mathbf{ifail}$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, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}>0.0$.
${K}_{0}$ is undefined and the function returns zero.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}>⟨\mathit{\text{value}}⟩$.
x is too small, there is a danger of overflow and the function returns approximately the largest representable value.
${\mathbf{ifail}}=-99$
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:
 $ε≃ | x K0(x)- K1(x) K1(x) |δ.$
Figure 1 shows the behaviour of the error amplification factor
 $| xK0(x) - K1 (x) K1(x) |.$
However, if $\delta$ is of the same order as the machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.
For small $x$, $\epsilon \simeq \delta$ and there is no amplification of errors.
For large $x$, $\epsilon \simeq x\delta$ and we have strong amplification of the relative error. Eventually ${K}_{1}$, 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.

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.