Public Shared Function s19ad ( _ x As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double
public: static double s19ad( double x, [OutAttribute] int% ifail )
- Type: System..::..DoubleOn entry: the argument of the function.Constraint: .
s19ad evaluates an approximation to the Kelvin function .
Note: for the function is undefined, so we need only consider .
The method is based on several Chebyshev expansions:
where , and are expansions in the variable ;
where is an expansion in the variable ;
where , and and are expansions in the variable .
For , the function is undefined, and hence the method fails and returns zero.
When is sufficiently close to zero, the result is computed as
and when is even closer to zero simply as
For large , is asymptotically given by and this becomes so small that it cannot be computed without underflow and the method fails.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Errors or warnings detected by the method:
- On entry, : the function is undefined. On failure the method 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 method returns zero. Note that for large , the errors are dominated by those of the standard function exp.
Underflow may occur for a few values of close to the zeros of , below the limit which causes a failure with .
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.