NAG Library Function Document
nag_bessel_j1 (s17afc) returns the value of the Bessel function .
||nag_bessel_j1 (double x,
nag_bessel_j1 (s17afc) evaluates an approximation to the Bessel function of the first kind .
The function 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 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. The range for which this occurs is roughly related to the 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
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.)
If is somewhat larger than 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 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, 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 (s17afce.c)
9.2 Program Data
Program Data (s17afce.d)
9.3 Program Results
Program Results (s17afce.r)