1: $x (n)$ – double array: The argument $x_{i}$ of the function, for $i = 1, 2, \dots, n$ .

Constraint: $x (i) > 0.0$ , 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

K_{0} (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.
$ivalid (i) = 1$: $x_{i} \leq 0.0$ , $K_{0} (x_{i})$ is undefined. $f (i)$ contains $0.0$ .

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 K_{1} (x)}{K_{0} (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x K_{1} (x)}{K_{0} (x)}| .

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

, the amplification factor is approximately

|\frac{1}{\ln x}|

, which implies strong attenuation of the error, but in general

ε

can never be less than the machine precision.

For large

x

ε ≃ x δ

and we have strong amplification of the relative error. Eventually

K_{0}

, which is asymptotically given by

\frac{e^{- x}}{\sqrt{x}}

, becomes so small that it cannot be calculated without underflow and hence the function will return zero. Note that for large

x

the errors will be dominated by those of the standard function exp.

Figure 1

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: s18aq_example

function s18aq_example


fprintf('s18aq example results\n\n');

x = [0.4; 0.6; 1.4; 1.6; 2.5; 3.5; 6; 8; 10; 1000];

[f, ivalid, ifail] = s18aq(x);

fprintf('     x           K_0(x)   ivalid\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end

s18aq example results

     x           K_0(x)   ivalid
   4.000e-01   1.115e+00    0
   6.000e-01   7.775e-01    0
   1.400e+00   2.437e-01    0
   1.600e+00   1.880e-01    0
   2.500e+00   6.235e-02    0
   3.500e+00   1.960e-02    0
   6.000e+00   1.244e-03    0
   8.000e+00   1.465e-04    0
   1.000e+01   1.778e-05    0
   1.000e+03   0.000e+00    0

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox