# NAG CL Interfaces18asc (bessel_​i0_​real_​vector)

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## 1Purpose

s18asc returns an array of values of the modified Bessel function ${I}_{0}\left(x\right)$.

## 2Specification

 #include
 void s18asc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s18asc, nag_specfun_bessel_i0_real_vector or nag_bessel_i0_vector.

## 3Description

s18asc 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 function 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 function 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: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left[{\mathbf{n}}\right]$double Output
On exit: ${I}_{0}\left({x}_{i}\right)$, the function values.
4: $\mathbf{ivalid}\left[{\mathbf{n}}\right]$Integer Output
On exit: ${\mathbf{ivalid}}\left[\mathit{i}-1\right]$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
${x}_{i}$ is too large. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains the approximate value of ${I}_{0}\left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_GT in s18aec , as defined in the Users' Note for your implementation.
5: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of x was invalid.

## 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 function 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

s18asc 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 (s18asce.c)

### 10.2Program Data

Program Data (s18asce.d)

### 10.3Program Results

Program Results (s18asce.r)