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NAG C Library Manual

NAG Library Function Documentnag_bessel_i0_vector (s18asc)

1  Purpose

nag_bessel_i0_vector (s18asc) returns an array of values of the modified Bessel function ${I}_{0}\left(x\right)$.

2  Specification

 #include #include
 void nag_bessel_i0_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3  Description

nag_bessel_i0_vector (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,
 $I0x=ex∑′r=0arTrt, where ​ t=2 x4 -1.$
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 function must fail because of the danger of overflow in calculating ${e}^{x}$.

4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5  Arguments

1:     nIntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:     x[n]const doubleInput
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:     f[n]doubleOutput
On exit: ${I}_{0}\left({x}_{i}\right)$, the function values.
4:     ivalid[n]IntegerOutput
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 NE_REAL_ARG_GT in nag_bessel_i0 (s18aec), as defined in the Users' Note for your implementation.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.
NW_IVALID
On entry, at least one value of x was invalid.

7  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:
 $ε≃ x I1x I0 x δ.$
Figure 1 shows the behaviour of the error amplification factor
 $xI1x I0x .$
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 $x$ 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.

None.

9  Example

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

9.1  Program Text

Program Text (s18asce.c)

9.2  Program Data

Program Data (s18asce.d)

9.3  Program Results

Program Results (s18asce.r)