S18GKF (PDF version)
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NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

S18GKF returns a sequence of values for the Bessel functions Jα+n-1z or Jα-n+1z for complex z, non-negative α<1 and n=1,2,,N+1.

2  Specification

REAL (KIND=nag_wp)  A
COMPLEX (KIND=nag_wp)  Z, B(abs(NL)+1)

3  Description

S18GKF evaluates a sequence of values for the Bessel function of the first kind Jαz, where z is complex and nonzero and α is the order with 0α<1. The N+1-member sequence is generated for orders α,α+1,,α+N when N0. Note that + is replaced by - when N<0. For positive orders the routine may also be called with z=0, since Jq0=0 when q>0. For negative orders the formula
is used to generate the required sequence. The appropriate values of Jqz and Yqz are obtained by calls to S17DCF and S17DEF.

4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5  Parameters

1:     Z – COMPLEX (KIND=nag_wp)Input
On entry: the argument z of the function.
Constraint: Z0.0,0.0 when NL<0.
2:     A – REAL (KIND=nag_wp)Input
On entry: the order α of the first member in the required sequence of function values.
Constraint: 0.0A<1.0.
3:     NL – INTEGERInput
On entry: the value of N.
Constraint: absNL101.
4:     B(absNL+1) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: with IFAIL=0 or 3, the required sequence of function values: Bn contains J α+n-1 z  if NL0 and J α-n+1 z  otherwise, for n=1,2,,absNL+1.
5:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. 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,Z=0.0,0.0 when NL<0,
The computation has been abandoned due to the likelihood of overflow.
The computation has been completed but some precision has been lost.
The computation has been abandoned due to total loss of precision.
The computation has been abandoned due to failure to satisfy the termination condition.

7  Accuracy

All constants in S17DCF and S17DEF are specified to approximately 18 digits of precision. If t denotes the number of digits of precision in the floating point arithmetic being used, 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 S17DCF and S17DEF, the actual number of correct digits is limited, in general, by p-s, where s max1,log10z,log10α  represents the number of digits lost due to the argument reduction. Thus the larger the values of z and α, the less the precision in the result.

8  Further Comments


9  Example

This example evaluates J0z,J1z,J2z and J3z at z=0.6-0.8i, and prints the results.

9.1  Program Text

Program Text (s18gkfe.f90)

9.2  Program Data

Program Data (s18gkfe.d)

9.3  Program Results

Program Results (s18gkfe.r)

S18GKF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

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