NAG C Library Function Document

nag_kelvin_bei (s19abc)

nag_kelvin_bei (s19abc) returns a value for the Kelvin function $\mathrm{bei}x$.

nag_kelvin_bei (s19abc) evaluates an approximation to the Kelvin function $\mathrm{bei}x$.

The function is based on several Chebyshev expansions.

For large $x$, there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner; therefore the function must fail.

Since the function is oscillatory, the absolute error rather than the relative error is important. Let $E$ be the absolute error in the function, and $\delta$ be the relative error in the argument. If $\delta$ is somewhat larger than the machine precision , then we have $E\simeq \left|x\left(-{\mathrm{ber}}_{1}x+{\mathrm{bei}}_{1}x\right)/\sqrt{2}\right|\delta$ (provided $E$ is within machine bounds).

For small $x$ the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.

For medium and large $x$, the error behaviour is oscillatory and its amplitude grows like $\sqrt{x/2\pi }{e}^{x/\sqrt{2}}$. Therefore it is impossible to calculate the functions with any accuracy when $\sqrt{x}{e}^{x/\sqrt{2}}>\sqrt{2\pi }/\delta$. Note that this value of $x$ is much smaller than the minimum value of $x$ for which the function overflows.

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.