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

J_{α} (z)

, where

z

is complex and nonzero and

α

is the order with

0 \leq α < 1

. The

(| N | + 1)

-member sequence is generated for orders

α, α + 1, \dots, α + | N |

when

N \geq 0

. Note that

+

is replaced by

-

when

N < 0

. For positive orders the routine may also be called with

z = 0

, since

J_{q} (0) = 0

when

q > 0

. For negative orders the formula

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

is used to generate the required sequence. The appropriate values of

J_{q} (z)

and

Y_{q} (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: $z \neq (0.0, 0.0)$ 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: $0.0 \leq a < 1.0$ .
3: $nl$ – Integer Input: On entry: the value of $N$ .

Constraint: $abs (nl) \leq 101$ .
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 $nl \geq 0$ and $J_{α - n + 1} (z)$ otherwise, for $n = 1, 2, \dots, abs (nl) + 1$ .
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

−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, $| nl | = ⟨ value ⟩$ .
Constraint: $| nl | \leq 101$ .

On entry, $a = ⟨ value ⟩$ .
Constraint: $a < 1.0$ .

On entry, $a = ⟨ value ⟩$ .
Constraint: $a \geq 0.0$ .

On entry, $nl = ⟨ value ⟩$ .
Constraint: when $nl < 0$ , $z \neq (0.0, 0.0)$ .

$ifail = 2$: Computation abandoned due to the likelihood of overflow.

$ifail = 3$: Computation completed but some precision has been lost.

$ifail = 4$: Computation abandoned due to total loss of precision.

$ifail = 5$: Computation abandoned due to failure to satisfy the termination condition.

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: 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 \approx \max (1, | \log_{10} | z | |, | \log_{10} | 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

s18gkf is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example evaluates

J_{0} (z), J_{1} (z), J_{2} (z)

and

J_{3} (z)

z = 0.6 - 0.8 i

, and prints the results.

s18gk: FL CL CPP AD

NAG FL Interfaces18gkf (bessel_​j_​seq_​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

NAG FL Interface
s18gkf (bessel_j_seq_complex)