NAG Library Function Document

nag_complex_hankel (s17dlc)

+− 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

nag_complex_hankel (s17dlc) returns a sequence of values for the Hankel functions

H_{ν + n}^{(1)} (z)

H_{ν + n}^{(2)} (z)

for complex

z

, non-negative

ν

and

n = 0, 1, \dots, N - 1

, with an option for exponential scaling.

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_complex_hankel (Integer m, double fnu, Complex z, Integer n, Nag_ScaleResType scal, Complex cy[], Integer nz, NagError fail)

3 Description

nag_complex_hankel (s17dlc) evaluates a sequence of values for the Hankel function

H_{ν}^{(1)} (z)

H_{ν}^{(2)} (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^{- i z}

if the function is

H_{ν}^{(1)} (z)

or by the factor

e^{i z}

if the function is

H_{ν}^{(2)} (z)

Note: although the function may not be called with

ν

less than zero, for negative orders the formulae

H_{- ν}^{(1)} (z) = e^{ν π i} H_{ν}^{(1)} (z)

, and

H_{- ν}^{(2)} (z) = e^{- ν π i} H_{ν}^{(2)} (z)

may be used.

The function is derived from the function CBESH in Amos (1986). It is based on the relation

H_{ν}^{(m)} (z) = \frac{1}{p} e^{- p ν} K_{ν} (z e^{- p}),

where

p = \frac{i π}{2}

m = 1

and

p = - \frac{i π}{2}

m = 2

, and the Bessel function

K_{ν} (z)

is computed in the right half-plane only. Continuation of

K_{ν} (z)

to the left half-plane is computed in terms of the Bessel function

I_{ν} (z)

. These functions are evaluated using a variety of different techniques, depending on the region under consideration.

When

N

is greater than

1

, extra values of

H_{ν}^{(m)} (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

|z|

is very small, near the machine underflow threshold, or

(ν + N - 1)

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

4 References

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

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: m – IntegerInput

On entry: the kind of functions required.

$m = 1$: The functions are $H_{ν}^{(1)} (z)$ .
$m = 2$: The functions are $H_{ν}^{(2)} (z)$ .

Constraint:

m = 1

2

2: fnu – doubleInput

On entry:

ν

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

Constraint:

fnu \geq 0.0

3: z – ComplexInput

On entry: the argument

z

of the functions.

Constraint:

z \neq (0.0, 0.0)

4: n – IntegerInput

On entry:

N

, the number of members required in the sequence

H_{ν}^{(m)} (z), H_{ν + 1}^{(m)} (z), \dots, H_{ν + N - 1}^{(m)} (z)

Constraint:

n \geq 1

5: scal – Nag_ScaleResTypeInput

On entry: the scaling option.

$scal = Nag_UnscaleRes$: The results are returned unscaled.
$scal = Nag_ScaleRes$: The results are returned scaled by the factor $e^{- i z}$ when $m = 1$ , or by the factor $e^{i z}$ when $m = 2$ .

Constraint:

scal = Nag_UnscaleRes

Nag_ScaleRes

6: cy[n] – ComplexOutput

On exit: the

N

required function values:

cy [i - 1]

contains

H_{ν + i - 1}^{(m)} (z)

, for

i = 1, 2, \dots, N

7: nz – Integer *Output

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

nz > 0

, then if

Im (z) > 0.0

and

m = 1

, or

Im (z) < 0.0

and

m = 2

, elements

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

are set to zero. In the complementary half-planes, nz simply states the number of underflows, and not which elements they are.

8: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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, m has illegal value: $m = ⟨value⟩$ .
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.
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, $fnu + n - 1 = ⟨value⟩ > ⟨value⟩$ .

7 Accuracy

All constants in nag_complex_hankel (s17dlc) 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 nag_complex_hankel (s17dlc), 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 nag_complex_hankel (s17dlc) 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 nag_complex_hankel (s17dlc) 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

Not applicable.

9 Further Comments

The time taken for a call of nag_complex_hankel (s17dlc) is approximately proportional to the value of n, plus a constant. In general it is much cheaper to call nag_complex_hankel (s17dlc) with n greater than

1

, rather than to make

N

separate calls to nag_complex_hankel (s17dlc).

Paradoxically, for some values of

z

and

ν

, it is cheaper to call nag_complex_hankel (s17dlc) 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.

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 kind of function, m, the second is a value for the order fnu, the third is a complex value for the argument, z, and the fourth 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.

NAG Library Function Documentnag_complex_hankel (s17dlc)

+− 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 Library Function Document

nag_complex_hankel (s17dlc)