NAG Library Routine Document

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	nz
Real (Kind=nag_wp), Intent (In)	::	fnu
Complex (Kind=nag_wp), Intent (In)	::	z
Complex (Kind=nag_wp), Intent (Out)	::	cy(n)
Character (1), Intent (In)	::	scal

C Header Interface

#include <nagmk26.h>

void	s17def_ (const double fnu, const Complex z, const Integer n, const char scal, Complex cy[], Integer nz, Integer ifail, const Charlen length_scal)

3

Description

s17def evaluates a sequence of values for the Bessel function

J_{ν} (z)

, where

z

is complex,

- π < \arg z \leq π

, and

ν

is the real, non-negative order. The

N

-member sequence is generated for orders

ν

ν + 1, \dots, ν + N - 1

. Optionally, the sequence is scaled by the factor

e^{- |Im (z)|}

Note: although the routine may not be called with

ν

less than zero, for negative orders the formula

J_{- ν} (z) = J_{ν} (z) \cos (π ν) - Y_{ν} (z) \sin (π ν)

may be used (for the Bessel function

Y_{ν} (z)

, see s17dcf).

The routine is derived from the routine CBESJ in Amos (1986). It is based on the relations

J_{ν} (z) = e^{ν π i / 2} I_{ν} (- i z)

Im (z) \geq 0.0

, and

J_{ν} (z) = e^{- ν π i / 2} I_{ν} (i z)

Im (z) < 0.0

The Bessel function

I_{ν} (z)

is computed using a variety of techniques depending on the region under consideration.

When

N

is greater than

1

, extra values of

J_{ν} (z)

are computed using recurrence relations.

For very large

|z|

(ν + N - 1)

, argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller

|z|

(ν + N - 1)

, the computation is performed but results are accurate to less than half of machine precision. If

Im (z)

is large, there is a risk of overflow and so no computation is performed. In all the above cases, a warning is given by the routine.

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: $fnu$ – Real (Kind=nag_wp)Input

On entry:

ν

, the order of the first member of the sequence of functions.

Constraint:

fnu \geq 0.0

2: $z$ – Complex (Kind=nag_wp)Input

On entry: the argument

z

of the functions.

3: $n$ – IntegerInput

On entry:

N

, the number of members required in the sequence

J_{ν} (z), J_{ν + 1} (z), \dots, J_{ν + N - 1} (z)

Constraint:

n \geq 1

4: $scal$ – Character(1)Input

On entry: the scaling option.

$scal ='U'$: The results are returned unscaled.
$scal ='S'$: The results are returned scaled by the factor $e^{- |Im (z)|}$ .

Constraint:

scal ='U'

'S'

5: $cy (n)$ – Complex (Kind=nag_wp) arrayOutput

On exit: the

N

required function values:

cy (i)

contains

J_{ν + i - 1} (z)

, for

i = 1, 2, \dots, N

6: $nz$ – IntegerOutput

On exit: the number of components of cy that are set to zero due to underflow. If

nz > 0

, then elements

cy (n - nz + 1)

cy (n - nz + 2), \dots, cy (n)

are set to zero.

7: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $fnu = 〈value〉$ .
Constraint: $fnu \geq 0.0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, scal has an illegal value: $scal = 〈value〉$ .

$ifail = 2$: No computation because $Im (z) = 〈value〉 > 〈value〉$ , $scal ='U'$ .

$ifail = 3$: Results lack precision because $|z| = 〈value〉 > 〈value〉$ .

Results lack precision because $fnu + n - 1 = 〈value〉 > 〈value〉$ .

$ifail = 4$: No computation because $|z| = 〈value〉 > 〈value〉$ .

No computation because $fnu + n - 1 = 〈value〉 > 〈value〉$ .

$ifail = 5$: No computation – algorithm termination condition not met.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

All constants in s17def 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 s17def, the actual number of correct digits is limited, in general, by

p - s

, where

s \approx \max (1, |\log_{10} |z||, |\log_{10} ν|)

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 s17def 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 s17def 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

s17def is not threaded in any implementation.

9

Further Comments

The time taken for a call of s17def is approximately proportional to the value of

n

, plus a constant. In general it is much cheaper to call s17def with

n

greater than

1

, rather than to make

N

separate calls to s17def.

Paradoxically, for some values of

z

and

ν

, it is cheaper to call s17def 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

J_{0} (x)

J_{1} (x)

, i.e.,

ν = 0.0

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 s17aef or s17aff 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 to set the argument scal. The program calls the routine 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.

NAG Library Routine Document

s17def (bessel_j_complex)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results