# NAG Library Routine Document

## 1Purpose

s18aef returns the value of the modified Bessel function ${I}_{0}\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s18aef ( x,
 Real (Kind=nag_wp) :: s18aef Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include nagmk26.h
 double s18aef_ (const double *x, Integer *ifail)

## 3Description

s18aef evaluates an approximation to the modified Bessel function of the first kind ${I}_{0}\left(x\right)$.
Note:  ${I}_{0}\left(-x\right)={I}_{0}\left(x\right)$, so the approximation need only consider $x\ge 0$.
The routine is based on three Chebyshev expansions:
For $0,
 $I0x=ex∑′r=0arTrt, where ​ t=2 x4 -1.$
For $4,
 $I0x=ex∑′r=0brTrt, where ​ t=x-84.$
For $x>12$,
 $I0x=exx ∑′r=0crTrt, where ​ t=2 12x -1.$
For small $x$, ${I}_{0}\left(x\right)\simeq 1$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.
For large $x$, the routine must fail because of the danger of overflow in calculating ${e}^{x}$.

## 4References

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

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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:
${\mathbf{ifail}}=1$
$\left|{\mathbf{x}}\right|$ is too large. On soft failure the routine returns the approximate value of ${I}_{0}\left(x\right)$ at the nearest valid argument. (see the Users' Note for your implementation for details)
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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:
 $ε≃ x I1x I0 x δ.$
Figure 1 shows the behaviour of the error amplification factor
 $xI1x I0x .$
Figure 1
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 $\frac{{x}^{2}}{2}$, 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 errors. However the routine must fail for quite moderate values of $x$, because ${I}_{0}\left(x\right)$ would overflow; hence in practice the loss of accuracy for large $x$ is not excessive. Note that for large $x$ the errors will be dominated by those of the standard function exp.

## 8Parallelism and Performance

s18aef 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 (s18aefe.f90)

### 10.2Program Data

Program Data (s18aefe.d)

### 10.3Program Results

Program Results (s18aefe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017