NAG CL Interface
s17acc (bessel_​y0_​real)

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1 Purpose

s17acc returns the value of the Bessel function Y0(x).

2 Specification

#include <nag.h>
double  s17acc (double x, NagError *fail)
The function may be called by the names: s17acc, nag_specfun_bessel_y0_real or nag_bessel_y0.

3 Description

s17acc evaluates an approximation to the Bessel function of the second kind Y0(x).
Note:  Y0(x) is undefined for x0 and the function will fail for such arguments.
The function is based on four Chebyshev expansions:
For 0<x8,
Y0(x)=2π lnxr=0arTr(t)+r=0brTr(t),   with ​t=2 (x8) 2-1.  
For x>8,
Y0(x)=2πx {P0(x)sin(x-π4)+Q0(x)cos(x-π4)}  
where P0(x)=r=0crTr(t),
and Q0(x)= 8xr=0drTr(t),with ​ t=2 ( 8x) 2-1.
For x near zero, Y0(x)2π (ln(x2)+γ) , where γ denotes Euler's constant. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For very large x, 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 Y0(x); only the amplitude, 2πn , 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 x1/machine precision (see the Users' Note for your implementation for details).

4 References

NIST Digital Library of Mathematical Functions
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

5 Arguments

1: x double Input
On entry: the argument x of the function.
Constraint: x>0.0.
2: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, x=value.
Constraint: xvalue.
x is too large. On failure the function returns the amplitude of the Y0 oscillation, 2/(πx).
On entry, x=value.
Constraint: x>0.0.

7 Accuracy

Let δ be the relative error in the argument and E be the absolute error in the result. (Since Y0(x) oscillates about zero, absolute error and not relative error is significant, except for very small x.)
If δ is somewhat larger than the machine representation error (e.g., if δ is due to data errors etc.), then E and δ are approximately related by
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor |xY1(x)|.
However, if δ is of the same order as the machine representation errors, then rounding errors could make E slightly larger than the above relation predicts.
For very small x, the errors are essentially independent of δ and the function should provide relative accuracy bounded by the machine precision.
For very large x, the above relation ceases to apply. In this region, Y0(x) 2πx sin(x- π4). The amplitude 2πx can be calculated with reasonable accuracy for all x, but sin(x-π4) cannot. If x- π4 is written as 2Nπ+θ where N is an integer and 0θ<2π, then sin(x- π4) is determined by θ only. If xδ-1, θ cannot be determined with any accuracy at all. Thus if x is greater than, or of the order of the inverse of machine precision, it is impossible to calculate the phase of Y0(x) and the function must fail.
Figure 1
Figure 1

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s17acc is not threaded in any implementation.

9 Further Comments


10 Example

This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

10.1 Program Text

Program Text (s17acce.c)

10.2 Program Data

Program Data (s17acce.d)

10.3 Program Results

Program Results (s17acce.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0 5 10 15 20 25 30 35 40 45 50 Y0(x) x "s17acfe.r" Example Program Returned Values for the Bessel Function Y0(x)