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: $fnu$ – double Input

On entry:

ν

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

Constraint:

fnu \geq 0.0

2: $z$ – Complex Input

On entry: the argument

z

of the functions.

Constraint:

z \neq (0.0, 0.0)

3: $n$ – Integer Input

On entry:

N

, the number of members required in the sequence

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

Constraint:

n \geq 1

4: $scal$ – Nag_ScaleResType Input

On entry: the scaling option.

$scal = Nag_UnscaleRes$: The results are returned unscaled.
$scal = Nag_ScaleRes$: The results are returned scaled by the factor $e^{z}$ .

Constraint:

scal = Nag_UnscaleRes

Nag_ScaleRes

5: $cy [n]$ – Complex Output

On exit: the

N

required function values:

cy [i - 1]

contains

K_{ν + i - 1} (z)

, for

i = 1, 2, \dots, N

6: $nz$ – Integer * Output

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

nz > 0

and

Re (z) \geq 0.0

, elements

cy [0], cy [1], \dots, cy [nz - 1]

are set to zero. If

Re (z) < 0.0

, nz simply states the number of underflows, and not which elements they are.

7: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_COMPLEX_ZERO: On entry, $z = (0.0, 0.0)$ .
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_OVERFLOW_LIKELY: No computation because $|z| = 〈value〉 < 〈value〉$ .

No computation because $fnu + n - 1 = 〈value〉$ is too large.
NE_REAL: On entry, $fnu = 〈value〉$ .
Constraint: $fnu \geq 0.0$ .
NE_TERMINATION_FAILURE: No computation – algorithm termination condition not met.
NE_TOTAL_PRECISION_LOSS: No computation because $|z| = 〈value〉 > 〈value〉$ .

No computation because $fnu + n - 1 = 〈value〉 > 〈value〉$ .
NW_SOME_PRECISION_LOSS: Results lack precision because $|z| = 〈value〉 > 〈value〉$ .

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

7 Accuracy

All constants in s18dcc 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 s18dcc, 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 s18dcc 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 s18dcc 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

s18dcc is not threaded in any implementation.

9 Further Comments

The time taken for a call of s18dcc is approximately proportional to the value of n, plus a constant. In general it is much cheaper to call s18dcc with n greater than

1

, rather than to make

N

separate calls to s18dcc.

Paradoxically, for some values of

z

and

ν

, it is cheaper to call s18dcc 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

K_{0} (x)

K_{1} (x)

, i.e.,

ν = 0.0

1.0

, where

x

is real and positive, and only a single function value is required, then it may be much cheaper to call s18acc, s18adc, s18ccc or s18cdc, depending on whether a scaled result is required or not.

10 Example

The example program 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.

Interfaces: FL CL AD

NAG CL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s18dc: FL CL AD

NAG CL Interfaces18dcc (bessel_​k_​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 CL Interface
s18dcc (bessel_k_complex)