NAG Library Function Document
nag_complex_hankel (s17dlc)
1
Purpose
nag_complex_hankel (s17dlc) returns a sequence of values for the Hankel functions ${H}_{\nu +n}^{\left(1\right)}\left(z\right)$ or ${H}_{\nu +n}^{\left(2\right)}\left(z\right)$ for complex $z$, nonnegative
$\nu $ and $n=0,1,\dots ,N1$, 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 ${H}_{\nu}^{\left(1\right)}\left(z\right)$ or ${H}_{\nu}^{\left(2\right)}\left(z\right)$, where $z$ is complex, $\pi <\mathrm{arg}z\le \pi $, and $\nu $ is the real, nonnegative order. The $N$member sequence is generated for orders $\nu $, $\nu +1,\dots ,\nu +N1$. Optionally, the sequence is scaled by the factor ${e}^{iz}$ if the function is ${H}_{\nu}^{\left(1\right)}\left(z\right)$ or by the factor ${e}^{iz}$ if the function is ${H}_{\nu}^{\left(2\right)}\left(z\right)$.
Note: although the function may not be called with $\nu $ less than zero, for negative orders the formulae ${H}_{\nu}^{\left(1\right)}\left(z\right)={e}^{\nu \pi i}{H}_{\nu}^{\left(1\right)}\left(z\right)$, and ${H}_{\nu}^{\left(2\right)}\left(z\right)={e}^{\nu \pi i}{H}_{\nu}^{\left(2\right)}\left(z\right)$ may be used.
The function is derived from the function CBESH in
Amos (1986). It is based on the relation
where
$p=\frac{i\pi}{2}$ if
$m=1$ and
$p=\frac{i\pi}{2}$ if
$m=2$, and the Bessel function
${K}_{\nu}\left(z\right)$ is computed in the right halfplane only. Continuation of
${K}_{\nu}\left(z\right)$ to the left halfplane is computed in terms of the Bessel function
${I}_{\nu}\left(z\right)$. 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 ${H}_{\nu}^{\left(m\right)}\left(z\right)$ are computed using recurrence relations.
For very large $\leftz\right$ or $\left(\nu +N1\right)$, argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller $\leftz\right$ or $\left(\nu +N1\right)$, the computation is performed but results are accurate to less than half of machine precision. If $\leftz\right$ is very small, near the machine underflow threshold, or $\left(\nu +N1\right)$ 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 nonnegative order ACM Trans. Math. Software 12 265–273
5
Arguments
 1:
$\mathbf{m}$ – IntegerInput

On entry: the kind of functions required.
 ${\mathbf{m}}=1$
 The functions are ${H}_{\nu}^{\left(1\right)}\left(z\right)$.
 ${\mathbf{m}}=2$
 The functions are ${H}_{\nu}^{\left(2\right)}\left(z\right)$.
Constraint:
${\mathbf{m}}=1$ or $2$.
 2:
$\mathbf{fnu}$ – doubleInput

On entry: $\nu $, the order of the first member of the sequence of functions.
Constraint:
${\mathbf{fnu}}\ge 0.0$.
 3:
$\mathbf{z}$ – ComplexInput

On entry: the argument $z$ of the functions.
Constraint:
${\mathbf{z}}\ne \left(0.0,0.0\right)$.
 4:
$\mathbf{n}$ – IntegerInput

On entry: $N$, the number of members required in the sequence ${H}_{\nu}^{\left({\mathbf{m}}\right)}\left(z\right),{H}_{\nu +1}^{\left({\mathbf{m}}\right)}\left(z\right),\dots ,{H}_{\nu +N1}^{\left({\mathbf{m}}\right)}\left(z\right)$.
Constraint:
${\mathbf{n}}\ge 1$.
 5:
$\mathbf{scal}$ – Nag_ScaleResTypeInput

On entry: the scaling option.
 ${\mathbf{scal}}=\mathrm{Nag\_UnscaleRes}$
 The results are returned unscaled.
 ${\mathbf{scal}}=\mathrm{Nag\_ScaleRes}$
 The results are returned scaled by the factor ${e}^{iz}$ when ${\mathbf{m}}=1$, or by the factor ${e}^{iz}$ when ${\mathbf{m}}=2$.
Constraint:
${\mathbf{scal}}=\mathrm{Nag\_UnscaleRes}$ or $\mathrm{Nag\_ScaleRes}$.
 6:
$\mathbf{cy}\left[{\mathbf{n}}\right]$ – ComplexOutput

On exit: the $N$ required function values: ${\mathbf{cy}}\left[i1\right]$ contains
${H}_{\nu +i1}^{\left({\mathbf{m}}\right)}\left(z\right)$, for $\mathit{i}=1,2,\dots ,N$.
 7:
$\mathbf{nz}$ – Integer *Output

On exit: the number of components of
cy that are set to zero due to underflow. If
${\mathbf{nz}}>0$, then if
$\mathrm{Im}\left(z\right)>0.0$ and
${\mathbf{m}}=1$, or
$\mathrm{Im}\left(z\right)<0.0$ and
${\mathbf{m}}=2$, elements
${\mathbf{cy}}\left[0\right],{\mathbf{cy}}\left[1\right],\dots ,{\mathbf{cy}}\left[{\mathbf{nz}}1\right]$ are set to zero. In the complementary halfplanes,
nz simply states the number of underflows, and not which elements they are.
 8:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_COMPLEX_ZERO

On entry, ${\mathbf{z}}=\left(0.0,0.0\right)$.
 NE_INT

On entry,
m has illegal value:
${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
 NE_OVERFLOW_LIKELY

No computation because $\left{\mathbf{z}}\right=\u2329\mathit{\text{value}}\u232a<\u2329\mathit{\text{value}}\u232a$.
No computation because ${\mathbf{fnu}}+{\mathbf{n}}1=\u2329\mathit{\text{value}}\u232a$ is too large.
 NE_REAL

On entry, ${\mathbf{fnu}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{fnu}}\ge 0.0$.
 NE_TERMINATION_FAILURE

No computation – algorithm termination condition not met.
 NE_TOTAL_PRECISION_LOSS

No computation because $\left{\mathbf{z}}\right=\u2329\mathit{\text{value}}\u232a>\u2329\mathit{\text{value}}\u232a$.
No computation because ${\mathbf{fnu}}+{\mathbf{n}}1=\u2329\mathit{\text{value}}\u232a>\u2329\mathit{\text{value}}\u232a$.
 NW_SOME_PRECISION_LOSS

Results lack precision because $\left{\mathbf{z}}\right=\u2329\mathit{\text{value}}\u232a>\u2329\mathit{\text{value}}\u232a$.
Results lack precision, ${\mathbf{fnu}}+{\mathbf{n}}1=\u2329\mathit{\text{value}}\u232a>\u2329\mathit{\text{value}}\u232a$.
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 floatingpoint arithmetic being used $t$, then clearly the maximum number of correct digits in the results obtained is limited by $p=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(t,18\right)$. 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 $ps$, where $s\approx \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left{\mathrm{log}}_{10}\leftz\right\right,\left{\mathrm{log}}_{10}\nu \right\right)$ represents the number of digits lost due to the argument reduction. Thus the larger the values of $\leftz\right$ and $\nu $, the less the precision in the result. If nag_complex_hankel (s17dlc) is called with ${\mathbf{n}}>1$, 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 $\nu $ and different ${\mathbf{n}}$, the computed values may not agree exactly. Empirical tests with modest values of $\nu $ and $z$ have shown that the discrepancy is limited to the least significant $3$ – $4$ digits of precision.
8
Parallelism and Performance
nag_complex_hankel (s17dlc) is not threaded in any implementation.
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 $\nu $, it is cheaper to call nag_complex_hankel (s17dlc) with a larger value of ${\mathbf{n}}$ 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 ${\mathbf{n}}$, 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
${\mathbf{n}}=2$ to evaluate the function for orders
fnu and
${\mathbf{fnu}}+1$, and it prints the results. The process is repeated until the end of the input data stream is encountered.
10.1
Program Text
Program Text (s17dlce.c)
10.2
Program Data
Program Data (s17dlce.d)
10.3
Program Results
Program Results (s17dlce.r)