NAG Library Function Document

nag_bessel_k0 (s18acc)

1: x – doubleInput: On entry: the argument $x$ of the function.
Constraint: $x > 0.0$ .
2: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_REAL_ARG_LE: On entry, x must not be less than or equal to 0.0: $x = 〈value〉$ .
$k_{0}$ is undefined and the function returns zero.

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 ≃ |{x K}_{1} (x) / K_{0} (x)| δ

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

|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

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 math library function exp.

None.

The following program reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.