NAG FL Interface
s18gkf (bessel_​j_​seq_​complex)

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1 Purpose

s18gkf returns a sequence of values for the Bessel functions Jα+n-1(z) or Jα-n+1(z) for complex z, non-negative α<1 and n=1,2,,|N|+1.

2 Specification

Fortran Interface
Subroutine s18gkf ( z, a, nl, b, ifail)
Integer, Intent (In) :: nl
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: a
Complex (Kind=nag_wp), Intent (In) :: z
Complex (Kind=nag_wp), Intent (Out) :: b(abs(nl)+1)
C Header Interface
#include <nag.h>
void  s18gkf_ (const Complex *z, const double *a, const Integer *nl, Complex b[], Integer *ifail)
The routine may be called by the names s18gkf or nagf_specfun_bessel_j_seq_complex.

3 Description

s18gkf evaluates a sequence of values for the Bessel function of the first kind Jα(z), 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 Jq(0)=0 when q>0. For negative orders the formula
is used to generate the required sequence. The appropriate values of Jq(z) and Yq(z) are obtained by calls to s17dcf and s17def.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: z Complex (Kind=nag_wp) Input
On entry: the argument z of the function.
Constraint: z(0.0,0.0) 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 Integer Input
On entry: the value of N.
Constraint: abs(nl)101.
4: b(abs(nl)+1) Complex (Kind=nag_wp) array Output
On exit: with ifail=0 or 3, the required sequence of function values: b(n) contains J α+n-1 (z) if nl0 and J α-n+1 (z) otherwise, for n=1,2,,abs(nl)+1.
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, |nl|=value.
Constraint: |nl|101.
On entry, a=value.
Constraint: a<1.0.
On entry, a=value.
Constraint: a0.0.
On entry, nl=value.
Constraint: when nl<0, z(0.0,0.0).
Computation abandoned due to the likelihood of overflow.
Computation completed but some precision has been lost.
Computation abandoned due to total loss of precision.
Computation abandoned due to failure to satisfy the termination condition.
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

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=min(t,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 max(1,|log10|z||,|log10|α||) 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 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s18gkf is not threaded in any implementation.

9 Further Comments


10 Example

This example evaluates J0(z),J1(z),J2(z) and J3(z) at z=0.6-0.8i, and prints the results.

10.1 Program Text

Program Text (s18gkfe.f90)

10.2 Program Data

Program Data (s18gkfe.d)

10.3 Program Results

Program Results (s18gkfe.r)