# NAG FL Interfaces18aff (bessel_​i1_​real)

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## 1Purpose

s18aff returns a value for the modified Bessel function ${I}_{1}\left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s18aff ( x,
 Real (Kind=nag_wp) :: s18aff Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s18aff_ (const double *x, Integer *ifail)
The routine may be called by the names s18aff or nagf_specfun_bessel_i1_real.

## 3Description

s18aff evaluates an approximation to the modified Bessel function of the first kind ${I}_{1}\left(x\right)$.
Note:  ${I}_{1}\left(-x\right)=-{I}_{1}\left(x\right)$, so the approximation need only consider $x\ge 0$.
The routine is based on three Chebyshev expansions:
For $0,
 $I1(x)=x∑′r=0arTr(t), where ​t=2 (x4) 2-1;$
For $4,
 $I1(x)=ex∑′r=0brTr(t), where ​t=x-84;$
For $x>12$,
 $I1(x)=exx ∑′r=0crTr(t), where ​t =2⁢ (12x) -1.$
For small $x$, ${I}_{1}\left(x\right)\simeq x$. 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 ${I}_{1}\left(x\right)$ cannot be represented without overflow.
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: $|{\mathbf{x}}|\le ⟨\mathit{\text{value}}⟩$.
$|{\mathbf{x}}|$ is too large and the function returns the approximate value of ${I}_{1}\left(x\right)$ 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:
 $ε≃ | xI0(x)- I1(x) I1 (x) |δ.$
Figure 1 shows the behaviour of the error amplification factor
 $| xI0(x) - I1(x) I1(x) |.$ 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 $x$, $\epsilon \simeq \delta$ and there is no amplification of errors.
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 ${I}_{1}\left(x\right)$ 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

s18aff 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 (s18affe.f90)

### 10.2Program Data

Program Data (s18affe.d)

### 10.3Program Results

Program Results (s18affe.r)