NAG CL Interface
s17arc (bessel_​y1_​real_​vector)

Settings help

CL Name Style:

1 Purpose

s17arc returns an array of values of the Bessel function Y1(x).

2 Specification

#include <nag.h>
void  s17arc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s17arc, nag_specfun_bessel_y1_real_vector or nag_bessel_y1_vector.

3 Description

s17arc evaluates an approximation to the Bessel function of the second kind Y1(xi) for an array of arguments xi, for i=1,2,,n.
Note:  Y1(x) is undefined for x0 and the function will fail for such arguments.
The function is based on four Chebyshev expansions:
For 0<x8,
Y1 (x) = 2π lnx x8 r=0 ar Tr (t) - 2πx + x8 r=0 br Tr (t) ,   with ​ t = 2 (x8) 2 - 1 .  
For x>8,
Y1 (x) = 2πx {P1(x)sin(x-3π4)+Q1(x)cos(x-3π4)}  
where P1 (x) = r=0 cr Tr (t) ,
and Q1 (x) = 8x r=0 dr Tr (t) , with t = 2 (8x) 2 - 1 .
For x near zero, Y1 (x) - 2 πx . 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 - 2 πx 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 Y1(x); 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

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

5 Arguments

1: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x[n] const double Input
On entry: the argument xi of the function, for i=1,2,,n.
Constraint: x[i-1]>0.0, for i=1,2,,n.
3: f[n] double Output
On exit: Y1(xi), the function values.
4: ivalid[n] Integer Output
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
No error.
On entry, xi is too large. f[i-1] contains the amplitude of the Y1 oscillation, 2πxi .
On entry, xi0.0, Y1 is undefined. f[i-1] contains 0.0.
xi is too close to zero, there is a danger of overflow. On failure, f[i-1] contains the value of Y1(x) at the smallest valid argument.
5: 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.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
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, at least one value of x was invalid.
Check ivalid for more information.

7 Accuracy

Let δ be the relative error in the argument and E be the absolute error in the result. (Since Y1(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 precision (e.g., if δ is due to data errors etc.), then E and δ are approximately related by:
E |xY0(x)-Y1(x)| δ  
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor |xY0(x)-Y1(x)|.
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 sin(x- 3π 4 ) . The amplitude 2 πx can be calculated with reasonable accuracy for all x, but sin(x- 3π4) cannot. If x- 3π4 is written as 2Nπ+θ where N is an integer and 0θ<2π, then sin(x- 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 Y1(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.
s17arc is not threaded in any implementation.

9 Further Comments


10 Example

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

10.1 Program Text

Program Text (s17arce.c)

10.2 Program Data

Program Data (s17arce.d)

10.3 Program Results

Program Results (s17arce.r)