NAG CL Interface
s17dcc (bessel_​y_​complex)

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1 Purpose

s17dcc returns a sequence of values for the Bessel functions Yν+n(z) for complex z, non-negative ν and n=0,1,,N-1, with an option for exponential scaling.

2 Specification

#include <nag.h>
void  s17dcc (double fnu, Complex z, Integer n, Nag_ScaleResType scal, Complex cy[], Integer *nz, NagError *fail)
The function may be called by the names: s17dcc, nag_specfun_bessel_y_complex or nag_complex_bessel_y.

3 Description

s17dcc evaluates a sequence of values for the Bessel function Yν(z), where z is complex, -π<argzπ, and ν is the real, non-negative order. The N-member sequence is generated for orders ν, ν+1,,ν+N-1. Optionally, the sequence is scaled by the factor e-|Im(z)|.
Note:  although the function may not be called with ν less than zero, for negative orders the formula Y-ν(z)=Yν(z)cos(πν)+Jν(z)sin(πν) may be used (for the Bessel function Jν(z), see s17dec).
The function is derived from the function CBESY in Amos (1986). It is based on the relation Yν(z)= Hν (1) (z)-Hν (2) (z) 2i , where Hν (1) (z) and Hν (2) (z) are the Hankel functions of the first and second kinds respectively (see s17dlc).
When N is greater than 1, extra values of Yν(z) are computed using recurrence relations.
For very large |z| or (ν+N-1), argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller |z| or (ν+N-1), the computation is performed but results are accurate to less than half of machine precision. If |z| is very small, near the machine underflow threshold, or (ν+N-1) is too large, there is a risk of overflow and so no computation is performed. In all the above cases, a warning is given by the function.

4 References

NIST Digital Library of Mathematical Functions
Amos D E (1986) Algorithm 644: A portable package for Bessel functions of a complex argument and non-negative order ACM Trans. Math. Software 12 265–273

5 Arguments

1: fnu double Input
On entry: ν, the order of the first member of the sequence of functions.
Constraint: fnu0.0.
2: z Complex Input
On entry: z, the argument of the functions.
Constraint: z(0.0,0.0).
3: n Integer Input
On entry: N, the number of members required in the sequence Yν(z),Yν+1(z),,Yν+N-1(z).
Constraint: n1.
4: scal Nag_ScaleResType Input
On entry: the scaling option.
scal=Nag_UnscaleRes
The results are returned unscaled.
scal=Nag_ScaleRes
The results are returned scaled by the factor e-|Im(z)|.
Constraint: scal=Nag_UnscaleRes or Nag_ScaleRes.
5: cy[n] Complex Output
On exit: the N required function values: cy[i-1] contains Y ν+i-1 (z) , for i=1,2,,N.
6: nz Integer * Output
On exit: the number of components of cy that are set to zero due to underflow. The positions of such components in the array cy are arbitrary.
7: 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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_COMPLEX_ZERO
On entry, z=(0.0,0.0).
NE_INT
On entry, n=value.
Constraint: n1.
NE_INTERNAL_ERROR
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.
NE_NO_LICENCE
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.
NE_OVERFLOW_LIKELY
No computation because |z|=value<value.
No computation because fnu+n-1=value is too large.
No computation because z.re=value>value, scal=Nag_UnscaleRes.
NE_REAL
On entry, fnu=value.
Constraint: fnu0.0.
NE_TERMINATION_FAILURE
No computation – algorithm termination condition not met.
NE_TOTAL_PRECISION_LOSS
No computation because |z|=value>value.
No computation because fnu+n-1=value>value.
NW_SOME_PRECISION_LOSS
Results lack precision because |z|=value>value.
Results lack precision because fnu+n-1=value>value.

7 Accuracy

All constants in s17dcc are given to approximately 18 digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used t, then clearly the maximum number of correct digits in the results obtained is limited by p=min(t,18). Because of errors in argument reduction when computing elementary functions inside s17dcc, the actual number of correct digits is limited, in general, by p-s, where smax(1,,,|log10|z||,|log10ν|) represents the number of digits lost due to the argument reduction. Thus the larger the values of |z| and ν, the less the precision in the result. If s17dcc is called with n>1, then computation of function values via recurrence may lead to some further small loss of accuracy.
If function values which should nominally be identical are computed by calls to s17dcc with different base values of ν and different n, the computed values may not agree exactly. Empirical tests with modest values of ν and z have shown that the discrepancy is limited to the least significant 34 digits of precision.

8 Parallelism and Performance

s17dcc is not threaded in any implementation.

9 Further Comments

The time taken for a call of s17dcc is approximately proportional to the value of n, plus a constant. In general it is much cheaper to call s17dcc with n greater than 1, rather than to make N separate calls to s17dcc.
Paradoxically, for some values of z and ν, it is cheaper to call s17dcc with a larger value of n than is required, and then discard the extra function values returned. However, it is not possible to state the precise circumstances in which this is likely to occur. It is due to the fact that the base value used to start recurrence may be calculated in different regions for different n, and the costs in each region may differ greatly.
Note that if the function required is Y0(x) or Y1(x), i.e., ν=0.0 or 1.0, where x is real and positive, and only a single unscaled function value is required, then it may be much cheaper to call s17acc or s17adc respectively.

10 Example

This example prints a caption and then proceeds to read sets of data from the input data stream. The first datum is a value for the order fnu, the second is a complex value for the argument, z, and the third is a character value used as a flag to set the argument scal. The program calls the function with n=2 to evaluate the function for orders fnu and fnu+1, and it prints the results. The process is repeated until the end of the input data stream is encountered.

10.1 Program Text

Program Text (s17dcce.c)

10.2 Program Data

Program Data (s17dcce.d)

10.3 Program Results

Program Results (s17dcce.r)