nag_kelvin_kei (s19adc) returns a value for the Kelvin function .

nag_kelvin_kei (s19adc) evaluates an approximation to the Kelvin function .

The function is based on several Chebyshev expansions.

For large ,  is so small that it cannot be computed without underflow and the function fails.

"NE_REAL_ARG_GT"

On entry, x must not be greater than : . x is too large, and the result underflows and the function returns zero.

"NE_REAL_ARG_LT"

On entry, x must not be less than 0.0: . The function is undefined and returns zero.

Let  be the absolute error in the result, and  be the relative error in the argument. If  is somewhat larger than the machine representation error, then we have .

For small , errors are attenuated by the function and hence are limited by the machine precision.

For medium and large , the error behaviour, like the function itself, is oscillatory and hence only absolute accuracy of the function can be maintained. For this range of , the amplitude of the absolute error decays like , which implies a strong attenuation of error. Eventually, , which is asymptotically given by , becomes so small that it cannot be calculated without causing underflow and therefore the function returns zero. Note that for large , the errors are dominated by those of the math library function exp.

Underflow may occur for a few values of  close to the zeros of , which causes failure NE_REAL_ARG_GT.