nag_bessel_y1 (s17adc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_bessel_y1 (s17adc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_bessel_y1 (s17adc) returns the value of the Bessel function Y1x.

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_bessel_y1 (double x, NagError *fail)

3  Description

nag_bessel_y1 (s17adc) evaluates an approximation to the Bessel function of the second kind Y1x.
Note:  Y1x is undefined for x0 and the function will fail for such arguments.
The function is based on four Chebyshev expansions:
For 0<x8,
Y1x=2π lnxx8r=0arTrt-2πx +x8r=0brTrt,   with ​t=2 x8 2-1.  
For x>8,
Y1x=2πx P1xsinx-3π4+Q1xcosx-3π4  
where P1x=r=0crTrt,
and Q1x= 8xr=0drTrt, with t=2 8x 2-1.
For x near zero, Y1x- 2πx . This approximation is used when x is sufficiently small for the result to be correct to machine precision. For extremely small x, there is a danger of overflow in calculating - 2πx  and for such arguments the function will fail.
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 Y1x; only the amplitude, 2πx , 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

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

5  Arguments

1:     x doubleInput
On entry: the argument x of the function.
Constraint: x>0.0.
2:     fail NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation 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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, x=value.
Constraint: xvalue.
x is too large, the function returns the amplitude of the Y1 oscillation, 2/πx.
On entry, x=value.
Constraint: x>0.0.
Y1 is undefined, the function returns zero.
x is too close to zero and there is danger of overflow, x=value.
Constraint: x>value.
The function returns the value of Y1x at the smallest valid argument.

7  Accuracy

Let δ be the relative error in the argument and E be the absolute error in the result. (Since Y1x oscillates about zero, absolute error and not relative error is significant, except for very small x.)
If δ is somewhat larger than the machine precision (e.g., if δ is due to data errors etc.), then E and δ are approximately related by:
E x Y0 x - Y1 x δ  
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor xY0x-Y1x.
However, if δ is of the same order as machine precision, then rounding errors could make E slightly larger than the above relation predicts.
For very small x, absolute error becomes large, but the relative error in the result is of the same order as δ.
For very large x, the above relation ceases to apply. In this region, Y1 x 2 πx sinx- 3π4 . The amplitude 2 πx  can be calculated with reasonable accuracy for all x, but sinx- 3π4 cannot. If x- 3π4  is written as 2Nπ+θ where N is an integer and 0θ<2π, then sinx- 3π4 is determined by θ only. If x>δ-1, θ cannot be determined with any accuracy at all. Thus if x is greater than, or of the order of, the inverse of the machine precision, it is impossible to calculate the phase of Y1x and the function must fail.
Figure 1
Figure 1

8  Parallelism and Performance

nag_bessel_y1 (s17adc) 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 (s17adce.c)

10.2  Program Data

Program Data (s17adce.d)

10.3  Program Results

Program Results (s17adce.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 −2 −1.5 −1 −0.5 0 0.5 0 5 10 15 20 25 30 35 40 45 50 Y1(x) x Example Program Returned Values for the Bessel Function Y1(x) gnuplot_plot_1

nag_bessel_y1 (s17adc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016