NAG Library Function Document
nag_bessel_y1 (s17adc) returns the value of the Bessel function .
||nag_bessel_y1 (double x,
nag_bessel_y1 (s17adc) evaluates the Bessel function of the second kind, , .
The approximation is based on Chebyshev expansions.
For near zero, . This approximation is used when is sufficiently small for the result to be correct to machine precision. For extremely small , there is a danger of overflow in calculating and for such arguments the function will fail.
For very large
, it becomes impossible to provide results with any reasonable accuracy (see Section 8
), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of
, only the amplitude,
, can be determined and this is returned. The range for which this occurs is roughly related to machine precision
; the function will fail if
/ machine precision
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
x – doubleInput
On entry: the argument of the function.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
is too large, the function returns the amplitude of the
On entry, x
must not be less than or equal to 0.0:
is undefined, the function returns zero.
On entry, x
must be greater than
is too close to zero, there is a danger of overflow, the function returns the value of
at the smallest valid argument.
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 precision (e.g., if is due to data errors etc.), then and are approximately related by: (provided is also within machine bounds).
However, if is of the same order as machine precision, then rounding errors could make slightly larger than the above relation predicts.
For very small , absolute error becomes large, but the relative error in the result is of the same order as .
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 the machine precision, it is impossible to calculate the phase of and the function must fail.
The following program reads values of the argument from a file, evaluates the function at each value of and prints the results.
9.1 Program Text
Program Text (s17adce.c)
9.2 Program Data
Program Data (s17adce.d)
9.3 Program Results
Program Results (s17adce.r)