Public Shared Function s18ac ( _ x As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double
public: static double s18ac( double x, [OutAttribute] int% ifail )
- Type: System..::..DoubleOn entry: the argument of the function.Constraint: .
s18ac evaluates an approximation to the modified Bessel function of the second kind .
Note: is undefined for and the method will fail for such arguments.
The method is based on five Chebyshev expansions:
For near zero, , where denotes Euler's constant. This approximation is used when is sufficiently small for the result to be correct to machine precision.
For large , where there is a danger of underflow due to the smallness of , the result is set exactly to zero.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Errors or warnings detected by the method:
- , is undefined. On failure the method returns zero.
Let and be the relative errors in the argument and result respectively.
If is somewhat larger than the machine precision (i.e., if is due to data errors etc.), then and are approximately related by:
Figure 1 shows the behaviour of the error amplification factor
However, if is of the same order as machine precision, then rounding errors could make slightly larger than the above relation predicts.
For small , the amplification factor is approximately , which implies strong attenuation of the error, but in general can never be less than the machine precision.
For large , and we have strong amplification of the relative error. Eventually , which is asymptotically given by , becomes so small that it cannot be calculated without underflow and hence the method will return zero. Note that for large the errors will be dominated by those of the standard function exp.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.