Public Shared Function s17ac ( _ x As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double
public: static double s17ac( double x, [OutAttribute] int% ifail )
- Type: System..::..DoubleOn entry: the argument of the function.Constraint: .
s17ac evaluates an approximation to the Bessel function of the second kind .
Note: is undefined for and the method will fail for such arguments.
The method is based on four 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 very large , it becomes impossible to provide results with any reasonable accuracy (see [Accuracy]), hence the method fails. Such arguments contain insufficient information to determine the phase of oscillation of ; only the amplitude, , can be determined and this is returned on failure. The range for which this occurs is roughly related to machine precision; the method will fail if (see the Users' Note for your implementation for details).
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO
Errors or warnings detected by the method:
- x is too large. On failure the method returns the amplitude of the oscillation, .
- , is undefined. On failure the method returns zero.
Let be the relative error in the argument and be the absolute error in the result. (Since oscillates about zero, absolute error and not relative error is significant, except for very small .)
If is somewhat larger than the machine representation error (e.g., if is due to data errors etc.), then and are approximately related by
(provided is also within machine bounds). Figure 1 displays the behaviour of the amplification factor .
However, if is of the same order as the machine representation errors, then rounding errors could make slightly larger than the above relation predicts.
For very small , the errors are essentially independent of and the method should provide relative accuracy bounded by the machine precision.
For very large , the above relation ceases to apply. In this region, . The amplitude can be calculated with reasonable accuracy for all , but cannot. If is written as where is an integer and , then is determined by only. If , cannot be determined with any accuracy at all. Thus if is greater than, or of the order of the inverse of machine precision, it is impossible to calculate the phase of and the method must fail.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.