# NAG FL Interfaces18asf (bessel_​i0_​real_​vector)

## ▸▿ Contents

Settings help

FL Name Style:

FL Specification Language:

## 1Purpose

s18asf returns an array of values of the modified Bessel function ${I}_{0}\left(x\right)$.

## 2Specification

Fortran Interface
 Subroutine s18asf ( 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 s18asf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s18asf or nagf_specfun_bessel_i0_real_vector.

## 3Description

s18asf evaluates an approximation to the modified Bessel function of the first kind ${I}_{0}\left({x}_{i}\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
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,
 $I0(x)=ex∑′r=0arTr(t), where ​ t = 2 ⁢ (x4) -1.$
For $4,
 $I0(x)=ex∑′r=0brTr(t), where ​ t=x-84.$
For $x>12$,
 $I0(x)=exx ∑′r=0crTr(t), where ​ t=2⁢(12x) -1.$
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: $\mathbf{n}$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 $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${I}_{0}\left({x}_{i}\right)$, 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.
${\mathbf{ivalid}}\left(i\right)=1$
${x}_{i}$ is too large. ${\mathbf{f}}\left(\mathit{i}\right)$ contains the approximate value of ${I}_{0}\left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in s18aef , as defined in the Users' Note for your implementation.
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:
 $ε≃ | x I1(x) I0 (x) |δ.$
Figure 1 shows the behaviour of the error amplification factor
 $| xI1(x) I0(x) |.$
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 $\frac{{x}^{2}}{2}$, 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, for quite moderate values of $x$ ($x>\stackrel{^}{x}$, the threshold value), the routine must fail because ${I}_{0}\left(x\right)$ would overflow; hence in practice the loss of accuracy for $x$ close to $\stackrel{^}{x}$ is not excessive and the errors will be dominated by those of the standard function exp.

## 8Parallelism and Performance

s18asf 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 (s18asfe.f90)

### 10.2Program Data

Program Data (s18asfe.d)

### 10.3Program Results

Program Results (s18asfe.r)