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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_bessel_i1_real_vector (s18at)

## Purpose

nag_specfun_bessel_i1_real_vector (s18at) returns an array of values for the modified Bessel function I1(x)${I}_{1}\left(x\right)$.

## Syntax

[f, ivalid, ifail] = s18at(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_i1_real_vector(x, 'n', n)

## Description

nag_specfun_bessel_i1_real_vector (s18at) evaluates an approximation to the modified Bessel function of the first kind I1(xi)${I}_{1}\left({x}_{i}\right)$ for an array of arguments xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
Note:  I1(x) = I1(x)${I}_{1}\left(-x\right)=-{I}_{1}\left(x\right)$, so the approximation need only consider x0$x\ge 0$.
The function is based on three Chebyshev expansions:
For 0 < x4$0,
 I1(x) = x ∑′ arTr(t),   where ​t = 2(x/4)2 − 1; r = 0
$I1(x)=x∑′r=0arTr(t), where ​t=2 (x4) 2-1;$
For 4 < x12$4,
 I1(x) = ex ∑′ brTr(t),   where ​t = (x − 8)/4; r = 0
$I1(x)=ex∑′r=0brTr(t), where ​t=x-84;$
For x > 12$x>12$,
 I1(x) = (ex)/(sqrt(x)) ∑′ crTr(t),   where ​t = 2(12/x) − 1. r = 0
$I1(x)=exx ∑′r=0crTr(t), where ​t=2 (12x) -1.$
For small x$x$, I1(x)x${I}_{1}\left(x\right)\simeq x$. This approximation is used when x$x$ is sufficiently small for the result to be correct to machine precision.
For large x$x$, the function must fail because I1(x)${I}_{1}\left(x\right)$ cannot be represented without overflow.

## References

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

## Parameters

### Compulsory Input Parameters

1:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
The argument xi${x}_{\mathit{i}}$ of the function, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
n$n$, the number of points.
Constraint: n0${\mathbf{n}}\ge 0$.

None.

### Output Parameters

1:     f(n) – double array
I1(xi)${I}_{1}\left({x}_{i}\right)$, the function values.
2:     ivalid(n) – int64int32nag_int array
ivalid(i)${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
ivalid(i) = 0${\mathbf{ivalid}}\left(i\right)=0$
No error.
ivalid(i) = 1${\mathbf{ivalid}}\left(i\right)=1$
xi${x}_{i}$ is too large. f(i)${\mathbf{f}}\left(\mathit{i}\right)$ contains the approximate value of I1(xi)${I}_{1}\left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in nag_specfun_bessel_i1_real (s18af), as defined in the Users' Note for your implementation.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail = 1${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
ifail = 2${\mathbf{ifail}}=2$
Constraint: n0${\mathbf{n}}\ge 0$.

## Accuracy

Let δ$\delta$ and ε$\epsilon$ be the relative errors in the argument and result respectively.
If δ$\delta$ is somewhat larger than the machine precision (i.e., if δ$\delta$ is due to data errors etc.), then ε$\epsilon$ and δ$\delta$ are approximately related by:
 ε ≃ |( xI0(x) − I1(x) )/( I1 (x))|δ. $ε≃ | xI0(x)- I1(x) I1 (x) |δ.$
Figure 1 shows the behaviour of the error amplification factor
 |( xI0(x) − I1(x) )/(I1(x))|. $| xI0(x) - I1(x) I1(x) |.$
Figure 1
However, if δ$\delta$ is of the same order as machine precision, then rounding errors could make ε$\epsilon$ slightly larger than the above relation predicts.
For small x$x$, εδ$\epsilon \simeq \delta$ and there is no amplification of errors.
For large x$x$, εxδ$\epsilon \simeq x\delta$ and we have strong amplification of errors. However, for quite moderate values of x$x$ (x > $x>\stackrel{^}{x}$, the threshold value), the function must fail because I1(x)${I}_{1}\left(x\right)$ would overflow; hence in practice the loss of accuracy for x$x$ close to $\stackrel{^}{x}$ is not excessive and the errors will be dominated by those of the standard function exp.

None.

## Example

function nag_specfun_bessel_i1_real_vector_example
x = [0; 0.5; 1; 3; 6; 8; 10; 15; 20; -1];
[f, ivalid, ifail] = nag_specfun_bessel_i1_real_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end

X           Y
0.000e+00   0.000e+00    0
5.000e-01   2.579e-01    0
1.000e+00   5.652e-01    0
3.000e+00   3.953e+00    0
6.000e+00   6.134e+01    0
8.000e+00   3.999e+02    0
1.000e+01   2.671e+03    0
1.500e+01   3.281e+05    0
2.000e+01   4.245e+07    0
-1.000e+00  -5.652e-01    0

function s18at_example
x = [0; 0.5; 1; 3; 6; 8; 10; 15; 20; -1];
[f, ivalid, ifail] = s18at(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end

X           Y
0.000e+00   0.000e+00    0
5.000e-01   2.579e-01    0
1.000e+00   5.652e-01    0
3.000e+00   3.953e+00    0
6.000e+00   6.134e+01    0
8.000e+00   3.999e+02    0
1.000e+01   2.671e+03    0
1.500e+01   3.281e+05    0
2.000e+01   4.245e+07    0
-1.000e+00  -5.652e-01    0