1: $x$ – double Input: On entry: the argument $x$ of the function.

Constraint: $x > 0.0$ .
2: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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.
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.
NE_REAL_ARG_LE: On entry, $x = ⟨ value ⟩$ .
Constraint: $x > 0.0$ .
$K_{0}$ is undefined and the function returns zero.
NE_REAL_ARG_TOO_SMALL: On entry, $x = ⟨ value ⟩$ .
Constraint: $x > ⟨ value ⟩$ .
x is too small, there is a danger of overflow and the function returns approximately the largest representable value.

7 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_{0} (x) - K_{1} (x)}{K_{1} (x)} | δ .

Figure 1 shows the behaviour of the error amplification factor

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

However, if

δ

is of the same order as the 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 the relative error. Eventually

K_{1}

, 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

s18adc is not threaded in any implementation.

None.

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

s18ad: FL CL CPP AD