# NAG FL Interfaces18aef (bessel_​i0_​real)

## 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 <nag.h>
 double s18aef_ (const double *x, Integer *ifail)
The routine may be called by the names s18aef or nagf_specfun_bessel_i0_real.

## 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

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
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: $\left|{\mathbf{x}}\right|\le 〈\mathit{\text{value}}〉$.
$\left|{\mathbf{x}}\right|$ is too large and the function returns the approximate value of ${I}_{0}\left(x\right)$ at the nearest valid argument.
${\mathbf{ifail}}=-99$
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:
 $ε≃ x I1x I0 x δ.$
Figure 1 shows the behaviour of the error amplification factor
 $xI1x I0x .$
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)