NAG Library Function Document
nag_bessel_y0_vector (s17aqc) returns an array of values of the Bessel function .
||nag_bessel_y0_vector (Integer n,
const double x,
nag_bessel_y0_vector (s17aqc) evaluates an approximation to the Bessel function of the second kind for an array of arguments , for .
Note: is undefined for and the function will fail for such arguments.
The function 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 Section 7
), 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 on failure. The range for which this occurs is roughly related to machine precision
; the function 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
n – IntegerInput
, the number of points.
x[n] – const doubleInput
On entry: the argument of the function, for .
, for .
f[n] – doubleOutput
On exit: , the function values.
ivalid[n] – IntegerOutput
contains the error code for
- No error.
|On entry,|| is too large. contains the amplitude of the oscillation, .|
|On entry,||, is undefined. contains .|
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
On entry, argument had an illegal value.
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
On entry, at least one value of x
for more information.
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 .)
is somewhat larger than the machine representation error (e.g., if
is due to data errors etc.), then
are approximately related by
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 function 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 function must fail.
This example reads values of x
from a file, evaluates the function at each value of
and prints the results.
9.1 Program Text
Program Text (s17aqce.c)
9.2 Program Data
Program Data (s17aqce.d)
9.3 Program Results
Program Results (s17aqce.r)