NAG FL Interface
s17arf (bessel_​y1_​real_​vector)

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1 Purpose

s17arf returns an array of values of the Bessel function Y1(x).

2 Specification

Fortran Interface
Subroutine s17arf ( n, x, f, ivalid, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ivalid(n)
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: f(n)
C Header Interface
#include <nag.h>
void  s17arf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s17arf or nagf_specfun_bessel_y1_real_vector.

3 Description

s17arf 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 routine will fail for such arguments.
The routine 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 routine will fail.
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 Y1(x); only the amplitude, 2πx , 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: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x(n) Real (Kind=nag_wp) array Input
On entry: the argument xi of the function, for i=1,2,,n.
Constraint: x(i)>0.0, for i=1,2,,n.
3: f(n) Real (Kind=nag_wp) array Output
On exit: Y1(xi), the function values.
4: ivalid(n) Integer array Output
On exit: ivalid(i) contains the error code for xi, for i=1,2,,n.
No error.
On entry, xi is too large. f(i) contains the amplitude of the Y1 oscillation, 2πxi .
On entry, xi0.0, Y1 is undefined. f(i) contains 0.0.
xi is too close to zero, there is a danger of overflow. On soft failure, f(i) contains the value of Y1(x) at the smallest valid argument.
5: 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, at least one value of x was invalid.
Check ivalid for more information.
On entry, n=value.
Constraint: n0.
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 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 routine must fail.
Figure 1
Figure 1

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s17arf 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 (s17arfe.f90)

10.2 Program Data

Program Data (s17arfe.d)

10.3 Program Results

Program Results (s17arfe.r)