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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_bessel_i1_real_vector (s18at)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_bessel_i1_real_vector (s18at) returns an array of values for the modified Bessel function

I_{1} (x)

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

I_{1} (x_{i})

for an array of arguments

x_{i}

, for

i = 1, 2, \dots, n

Note:

I_{1} (- x) = - I_{1} (x)

, so the approximation need only consider

x \geq 0

The function is based on three Chebyshev expansions:

For

0 < x \leq 4

I_{1} (x) = x \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t),   where ​ t = 2 {(\frac{x}{4})}^{2} - 1;

For

4 < x \leq 12

I_{1} (x) = e^{x} \underset{r = 0}{\sum^{'}} b_{r} T_{r} (t),   where ​ t = \frac{x - 8}{4};

For

x > 12

I_{1} (x) = \frac{e^{x}}{\sqrt{x}} \underset{r = 0}{\sum^{'}} c_{r} T_{r} (t),   where ​ t = 2 (\frac{12}{x}) - 1 .

For small

x

I_{1} (x) ≃ x

. 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

I_{1} (x)

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: The argument $x_{i}$ of the function, for $i = 1, 2, \dots, n$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array x.
$n$ , the number of points.

Constraint: $n \geq 0$ .

Output Parameters

1: $f (n)$ – double array

I_{1} (x_{i})

, the function values.

2: $ivalid (n)$ – int64int32nag_int array

ivalid (i)

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $x_{i}$ is too large. $f (i)$ contains the approximate value of $I_{1} (x_{i})$ at the nearest valid argument. The threshold value is the same as for $ifail = 1$ in nag_specfun_bessel_i1_real (s18af), as defined in the Users' Note for your implementation.

3: $ifail$ – int64int32nag_int scalar

ifail = 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 $ifail = 1$: On entry, at least one value of x was invalid.
Check ivalid for more information.

$ifail = 2$: Constraint: $n \geq 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Let

δ

and

ε

be the relative errors in the argument and result respectively.

δ

is somewhat larger than the machine precision (i.e., if

δ

is due to data errors etc.), then

ε

and

δ

are approximately related by:

ε ≃ |\frac{x I_{0} (x) - I_{1} (x)}{I_{1} (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x I_{0} (x) - I_{1} (x)}{I_{1} (x)}| .

Figure 1

However, if

δ

is of the same order as machine precision, then rounding errors could make

ε

slightly larger than the above relation predicts.

For small

x

ε ≃ δ

and there is no amplification of errors.

For large

x

ε ≃ x δ

and we have strong amplification of errors. However, for quite moderate values of

x

(

x > \hat{x}

, the threshold value), the function must fail because

I_{1} (x)

would overflow; hence in practice the loss of accuracy for

x

close to

\hat{x}

is not excessive and the errors will be dominated by those of the standard function exp.

Further Comments

None.

Example

This example reads values of x from a file, evaluates the function at each value of

x_{i}

and prints the results.

Open in the MATLAB editor: s18at_example

function s18at_example


fprintf('s18at example results\n\n');

x = [0; 0.5; 1; 3; 6; 8; 10; 15; 20; -1];

[f, ivalid, ifail] = s18at(x);

fprintf('     x           I_1(x)   ivalid\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end

s18at example results

     x           I_1(x)   ivalid
   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

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Chapter Contents

Chapter Introduction

NAG Toolbox