s18ae returns the value of the modified Bessel function ${I}_{0}\left(x\right)$.

# Syntax

C# |
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public static double s18ae( double x, out int ifail ) |

Visual Basic |
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Public Shared Function s18ae ( _ x As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double |

Visual C++ |
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public: static double s18ae( double x, [OutAttribute] int% ifail ) |

F# |
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static member s18ae : x : float * ifail : int byref -> float |

#### Parameters

- x
- Type: System..::..Double
*On entry*: the argument $x$ of the function.

- ifail
- Type: System..::..Int32%
*On exit*: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

s18ae returns the value of the modified Bessel function ${I}_{0}\left(x\right)$.

# Description

s18ae 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 method is based on three Chebyshev expansions:

For $0<x\le 4$,

For $4<x\le 12$,

For $x>12$,

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.

$${I}_{0}\left(x\right)={e}^{x}\underset{r=0}{{\sum}^{\prime}}\phantom{\rule{0.25em}{0ex}}{a}_{r}{T}_{r}\left(t\right)\text{, \hspace{1em} where}t=2\left(\frac{x}{4}\right)-1\text{.}$$ |

$${I}_{0}\left(x\right)={e}^{x}\underset{r=0}{{\sum}^{\prime}}\phantom{\rule{0.25em}{0ex}}{b}_{r}{T}_{r}\left(t\right)\text{, \hspace{1em} where}t=\frac{x-8}{4}\text{.}$$ |

$${I}_{0}\left(x\right)=\frac{{e}^{x}}{\sqrt{x}}\underset{r=0}{{\sum}^{\prime}}\phantom{\rule{0.25em}{0ex}}{c}_{r}{T}_{r}\left(t\right)\text{, \hspace{1em} where}t=2\left(\frac{12}{x}\right)-1\text{.}$$ |

For large $x$, the method must fail because of the danger of overflow in calculating ${e}^{x}$.

# References

Abramowitz M and Stegun I A (1972)

*Handbook of Mathematical Functions*(3rd Edition) Dover Publications# Error Indicators and Warnings

Errors or warnings detected by the method:

- ${\mathbf{ifail}}=1$
- $\left|{\mathbf{x}}\right|$ is too large. On failure the method returns the approximate value of ${I}_{0}\left(x\right)$ at the nearest valid argument. (see the Users' Note for your implementation for details)

# Accuracy

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:

Figure 1 shows the behaviour of the error amplification factor

$$\epsilon \simeq \left|\frac{x{I}_{1}\left(x\right)}{{I}_{0}\left(x\right)}\right|\delta \text{.}$$ |

$$\left|\frac{x{I}_{1}\left(x\right)}{{I}_{0}\left(x\right)}\right|\text{.}$$ |

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 method 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.

# Parallelism and Performance

None.

# Further Comments

None.

# Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

Example program (C#): s18aee.cs