nag_complex_bessel_i (s18dec) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_complex_bessel_i (s18dec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_complex_bessel_i (s18dec) returns a sequence of values for the modified Bessel functions Iν+nz 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_bessel_i (double fnu, Complex z, Integer n, Nag_ScaleResType scal, Complex cy[], Integer *nz, NagError *fail)

3  Description

nag_complex_bessel_i (s18dec) evaluates a sequence of values for the modified Bessel function Iνz, 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-Rez.
The function is derived from the function CBESI in Amos (1986).
Note:  although the function may not be called with ν less than zero, for negative orders the formula I-νz=Iνz+ 2π sinπνKνz may be used (for the Bessel function Kνz, see nag_complex_bessel_k (s18dcc)).
When N is greater than 1, extra values of Iνz 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 Rez is too large and the unscaled function is required, 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:     fnudoubleInput
On entry: ν, the order of the first member of the sequence of functions.
Constraint: fnu0.0.
2:     zComplexInput
On entry: the argument z of the functions.
3:     nIntegerInput
On entry: N, the number of members required in the sequence Iνz,Iν+1z,,Iν+N-1z.
Constraint: n1.
4:     scalNag_ScaleResTypeInput
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-Rez.
Constraint: scal=Nag_UnscaleRes or Nag_ScaleRes.
5:     cy[n]ComplexOutput
On exit: the N required function values: cy[i-1] contains I ν+i-1 z , for i=1,2,,N.
6:     nzInteger *Output
On exit: the number of components of cy that are set to zero due to underflow.
If nz>0, then elements cy[n-nz],cy[n-nz+1],,cy[n-1] are set to zero.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
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.
NE_OVERFLOW_LIKELY
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 absz=value>value.
No computation because fnu+n-1=value>value.
NW_SOME_PRECISION_LOSS
Results lack precision because absz=value>value.
Results lack precision because fnu+n-1=value>value.

7  Accuracy

All constants in nag_complex_bessel_i (s18dec) 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_bessel_i (s18dec), 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_bessel_i (s18dec) 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_bessel_i (s18dec) 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  Further Comments

The time taken for a call of nag_complex_bessel_i (s18dec) is approximately proportional to the value of n, plus a constant. In general it is much cheaper to call nag_complex_bessel_i (s18dec) with n greater than 1, rather than to make N separate calls to nag_complex_bessel_i (s18dec).
Paradoxically, for some values of z and ν, it is cheaper to call nag_complex_bessel_i (s18dec) 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 I0x or I1x, i.e., ν=0.0 or 1.0, where x is real and positive, and only a single function value is required, then it may be much cheaper to call nag_bessel_i0 (s18aec), nag_bessel_i1 (s18afc), nag_bessel_i0_scaled (s18cec) or nag_bessel_i1_scaled (s18cfc), depending on whether a scaled result is required or not.

9  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.

9.1  Program Text

Program Text (s18dece.c)

9.2  Program Data

Program Data (s18dece.d)

9.3  Program Results

Program Results (s18dece.r)


nag_complex_bessel_i (s18dec) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012