Public Shared Function s18ad ( _ x As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double
public: static double s18ad( double x, [OutAttribute] int% ifail )
- Type: System..::..DoubleOn entry: the argument of the function.Constraint: .
s18ad 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, . This approximation is used when is sufficiently small for the result to be correct to machine precision. For very small on some machines, it is impossible to calculate without overflow and the method must fail.
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 the machine precision, then rounding errors could make slightly larger than the above relation predicts.
For small , and there is no amplification of errors.
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.