Public Shared Function s19ab ( _ x As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double
public: static double s19ab( double x, [OutAttribute] int% ifail )
- Type: System..::..DoubleOn entry: the argument of the function.
s19ab evaluates an approximation to the Kelvin function .
Note: , so the approximation need only consider .
The method is based on several Chebyshev expansions:
where , ,
and , , , and are expansions in the variable .
When is sufficiently close to zero, the result is computed as . If this result would underflow, the result returned is .
For large , there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner; therefore the method must fail.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Errors or warnings detected by the method:
- On entry, is too large for an accurate result to be returned. On failure, the method returns zero. See also the Users' Note for your implementation.
Since the function is oscillatory, the absolute error rather than the relative error is important. Let be the absolute error in the function, and be the relative error in the argument. If is somewhat larger than the machine precision, then we have:
(provided is within machine bounds).
For small the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.
For medium and large , the error behaviour is oscillatory and its amplitude grows like . Therefore it is impossible to calculate the functions with any accuracy when . Note that this value of is much smaller than the minimum value of for which the function overflows.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.