NAG FL Interface
s17acf (bessel_​y0_​real)

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1 Purpose

s17acf returns the value of the Bessel function Y0(x), via the function name.

2 Specification

Fortran Interface
Function s17acf ( x, ifail)
Real (Kind=nag_wp) :: s17acf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s17acf_ (const double *x, Integer *ifail)
The routine may be called by the names s17acf or nagf_specfun_bessel_y0_real.

3 Description

s17acf evaluates an approximation to the Bessel function of the second kind Y0(x).
Note:  Y0(x) is undefined for x0 and the routine will fail for such arguments.
The routine 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 routine 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 soft failure. The range for which this occurs is roughly related to machine precision; the routine 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 Real (Kind=nag_wp) Input
On entry: the argument x of the function.
Constraint: x>0.0.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, x=value.
Constraint: xvalue.
x is too large. On soft failure the routine returns the amplitude of the Y0 oscillation, 2/(πx).
On entry, x=value.
Constraint: x>0.0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 routine 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 routine must fail.
Figure 1
Figure 1

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s17acf 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 (s17acfe.f90)

10.2 Program Data

Program Data (s17acfe.d)

10.3 Program Results

Program Results (s17acfe.r)
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 −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 Example Program Returned Values for the Bessel Function Y0(x) gnuplot_plot_1