nag_complex_hankel (s17dlc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_complex_hankel (s17dlc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_complex_hankel (s17dlc) returns a sequence of values for the Hankel functions Hν+n 1 z or Hν+n 2 z for complex z, non-negative ν and n=0,1,,N-1, with an option for exponential scaling.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_complex_hankel (Integer m, double fnu, Complex z, Integer n, Nag_ScaleResType scal, Complex cy[], Integer *nz, NagError *fail)

3  Description

nag_complex_hankel (s17dlc) evaluates a sequence of values for the Hankel function Hν 1 z or Hν 2 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-iz if the function is Hν 1 z or by the factor eiz if the function is Hν 2 z.
Note:  although the function may not be called with ν less than zero, for negative orders the formulae H-ν 1 z=eνπiHν 1 z, and H-ν 2 z=e-νπiHν 2 z may be used.
The function is derived from the function CBESH in Amos (1986). It is based on the relation
Hν m z=1pe-pνKνze-p,  
where p= iπ2  if m=1 and p=- iπ2  if m=2, and the Bessel function Kνz is computed in the right half-plane only. Continuation of Kνz to the left half-plane is computed in terms of the Bessel function Iνz. These functions are evaluated using a variety of different techniques, depending on the region under consideration.
When N is greater than 1, extra values of Hν m 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

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
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:     m IntegerInput
On entry: the kind of functions required.
The functions are Hν 1 z.
The functions are Hν 2 z.
Constraint: m=1 or 2.
2:     fnu doubleInput
On entry: ν, the order of the first member of the sequence of functions.
Constraint: fnu0.0.
3:     z ComplexInput
On entry: the argument z of the functions.
Constraint: z0.0,0.0.
4:     n IntegerInput
On entry: N, the number of members required in the sequence Hν m z,Hν+1 m z,,Hν+N-1 m z.
Constraint: n1.
5:     scal Nag_ScaleResTypeInput
On entry: the scaling option.
The results are returned unscaled.
The results are returned scaled by the factor e-iz when m=1, or by the factor eiz when m=2.
Constraint: scal=Nag_UnscaleRes or Nag_ScaleRes.
6:     cy[n] ComplexOutput
On exit: the N required function values: cy[i-1] contains H ν+i-1 m z , for i=1,2,,N.
7:     nz Integer *Output
On exit: the number of components of cy that are set to zero due to underflow. If nz>0, then if Imz>0.0 and m=1, or Imz<0.0 and m=2, elements cy[0],cy[1],,cy[nz-1] are set to zero. In the complementary half-planes, nz simply states the number of underflows, and not which elements they are.
8:     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.
On entry, argument value had an illegal value.
On entry, z=0.0,0.0.
On entry, m has illegal value: m=value.
On entry, n=value.
Constraint: n1.
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.
No computation because z=value<value.
No computation because fnu+n-1=value is too large.
On entry, fnu=value.
Constraint: fnu0.0.
No computation – algorithm termination condition not met.
No computation because z=value>value.
No computation because fnu+n-1=value>value.
Results lack precision because z=value>value.
Results lack precision, fnu+n-1=value>value.

7  Accuracy

All constants in nag_complex_hankel (s17dlc) 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=mint,18. Because of errors in argument reduction when computing elementary functions inside nag_complex_hankel (s17dlc), the actual number of correct digits is limited, in general, by p-s, where s max1, log10z , 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 nag_complex_hankel (s17dlc) 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 nag_complex_hankel (s17dlc) 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 3 – 4 digits of precision.

8  Parallelism and Performance

nag_complex_hankel (s17dlc) is not threaded in any implementation.

9  Further Comments

The time taken for a call of nag_complex_hankel (s17dlc) is approximately proportional to the value of n, plus a constant. In general it is much cheaper to call nag_complex_hankel (s17dlc) with n greater than 1, rather than to make N separate calls to nag_complex_hankel (s17dlc).
Paradoxically, for some values of z and ν, it is cheaper to call nag_complex_hankel (s17dlc) 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.

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 kind of function, m, the second is a value for the order fnu, the third is a complex value for the argument, z, and the fourth 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 (s17dlce.c)

10.2  Program Data

Program Data (s17dlce.d)

10.3  Program Results

Program Results (s17dlce.r)

nag_complex_hankel (s17dlc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

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