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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_hankel_complex (s17dl)

Purpose

nag_specfun_hankel_complex (s17dl) 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 $z$, non-negative $\nu$ and $n=0,1,\dots ,N-1$, with an option for exponential scaling.

Syntax

[cy, nz, ifail] = s17dl(m, fnu, z, n, scal)
[cy, nz, ifail] = nag_specfun_hankel_complex(m, fnu, z, n, scal)

Description

nag_specfun_hankel_complex (s17dl) 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, non-negative order. The $N$-member sequence is generated for orders $\nu$, $\nu +1,\dots ,\nu +N-1$. 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
 $Hν m z=1pe-pνKνze-p,$
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 half-plane only. Continuation of ${K}_{\nu }\left(z\right)$ to the left half-plane 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 $\left|z\right|$ or $\left(\nu +N-1\right)$, argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller $\left|z\right|$ or $\left(\nu +N-1\right)$, the computation is performed but results are accurate to less than half of machine precision. If $\left|z\right|$ is very small, near the machine underflow threshold, or $\left(\nu +N-1\right)$ 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.

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

Parameters

Compulsory Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
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:     $\mathrm{fnu}$ – double scalar
$\nu$, the order of the first member of the sequence of functions.
Constraint: ${\mathbf{fnu}}\ge 0.0$.
3:     $\mathrm{z}$ – complex scalar
The argument $z$ of the functions.
Constraint: ${\mathbf{z}}\ne \left(0.0,0.0\right)$.
4:     $\mathrm{n}$int64int32nag_int scalar
$N$, the number of members required in the sequence ${H}_{\nu }^{\left({\mathbf{m}}\right)}\left(z\right),{H}_{\nu +1}^{\left({\mathbf{m}}\right)}\left(z\right),\dots ,{H}_{\nu +N-1}^{\left({\mathbf{m}}\right)}\left(z\right)$.
Constraint: ${\mathbf{n}}\ge 1$.
5:     $\mathrm{scal}$ – string (length ≥ 1)
The scaling option.
${\mathbf{scal}}=\text{'U'}$
The results are returned unscaled.
${\mathbf{scal}}=\text{'S'}$
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: ${\mathbf{scal}}=\text{'U'}$ or $\text{'S'}$.

None.

Output Parameters

1:     $\mathrm{cy}\left({\mathbf{n}}\right)$ – complex array
The $N$ required function values: ${\mathbf{cy}}\left(i\right)$ contains ${H}_{\nu +i-1}^{\left({\mathbf{m}}\right)}\left(z\right)$, for $\mathit{i}=1,2,\dots ,N$.
2:     $\mathrm{nz}$int64int32nag_int scalar
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 ${\mathbf{cy}}\left(1\right),{\mathbf{cy}}\left(2\right),\dots ,{\mathbf{cy}}\left({\mathbf{nz}}\right)$ are set to zero. In the complementary half-planes, nz simply states the number of underflows, and not which elements they are.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{m}}\ne 1$ and ${\mathbf{m}}\ne 2$, or ${\mathbf{fnu}}<0.0$, or ${\mathbf{z}}=\left(0.0,0.0\right)$, or ${\mathbf{n}}<1$, or ${\mathbf{scal}}\ne \text{'U'}$ or $\text{'S'}$.
${\mathbf{ifail}}=2$
No computation has been performed due to the likelihood of overflow, because $\mathrm{abs}\left({\mathbf{z}}\right)$ is less than a machine-dependent threshold value.
${\mathbf{ifail}}=3$
No computation has been performed due to the likelihood of overflow, because ${\mathbf{fnu}}+{\mathbf{n}}-1$ is too large – how large depends on z and the overflow threshold of the machine.
W  ${\mathbf{ifail}}=4$
The computation has been performed, but the errors due to argument reduction in elementary functions make it likely that the results returned by nag_specfun_hankel_complex (s17dl) are accurate to less than half of machine precision. This error exit may occur if either $\mathrm{abs}\left({\mathbf{z}}\right)$ or ${\mathbf{fnu}}+{\mathbf{n}}-1$ is greater than a machine-dependent threshold value.
${\mathbf{ifail}}=5$
No computation has been performed because the errors due to argument reduction in elementary functions mean that all precision in results returned by nag_specfun_hankel_complex (s17dl) would be lost. This error exit may occur when either of $\mathrm{abs}\left({\mathbf{z}}\right)$ or ${\mathbf{fnu}}+{\mathbf{n}}-1$ is greater than a machine-dependent threshold value.
${\mathbf{ifail}}=6$
No results are returned because the algorithm termination condition has not been met. This may occur because the arguments supplied to nag_specfun_hankel_complex (s17dl) would have caused overflow or underflow.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Accuracy

All constants in nag_specfun_hankel_complex (s17dl) 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 $p=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(t,18\right)$. Because of errors in argument reduction when computing elementary functions inside nag_specfun_hankel_complex (s17dl), the actual number of correct digits is limited, in general, by $p-s$, where $s\approx \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left|{\mathrm{log}}_{10}\left|z\right|\right|,\left|{\mathrm{log}}_{10}\nu \right|\right)$ represents the number of digits lost due to the argument reduction. Thus the larger the values of $\left|z\right|$ and $\nu$, the less the precision in the result. If nag_specfun_hankel_complex (s17dl) 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_specfun_hankel_complex (s17dl) 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.

The time taken for a call of nag_specfun_hankel_complex (s17dl) is approximately proportional to the value of n, plus a constant. In general it is much cheaper to call nag_specfun_hankel_complex (s17dl) with n greater than $1$, rather than to make $N$ separate calls to nag_specfun_hankel_complex (s17dl).
Paradoxically, for some values of $z$ and $\nu$, it is cheaper to call nag_specfun_hankel_complex (s17dl) 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.

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 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.
function s17dl_example

fprintf('s17dl example results\n\n');

n  = int64(2);
m  = int64([1     1         1         2             2]);
nu = [0             2.3       2.12      6             6];
z  = [0.3 + 0.4i;   2 + 0i;  -1 + 0i;   3.1 - 1.6i;   3.1 - 1.6i];
scal = {'U';        'U';      'U';      'U';          'S'};

fprintf(' m    nu           z         scaled?');
fprintf('     H_{nu+%d}(z) ',[0:n-1]);
fprintf('  nz\n');
for i=1:numel(nu)

[cy, nz, ifail] = s17dl(m(i), nu(i), complex(z(i)), n, scal{i});

fprintf('%2d %7.3f  %7.3f%+7.3fi', m(i), nu(i), real(z(i)), imag(z(i)));
if scal{i} == 'U'
fprintf('  unscaled');
else
fprintf('    scaled');
end
for j = 1:n
fprintf(' %7.3f%+8.3fi', real(cy(j)), imag(cy(j)));
end
fprintf('%3d\n',nz);
end

s17dl example results

m    nu           z         scaled?     H_{nu+0}(z)      H_{nu+1}(z)   nz
1   0.000    0.300 +0.400i  unscaled   0.347  -0.559i  -0.791  -0.818i  0
1   2.300    2.000 +0.000i  unscaled   0.272  -0.740i   0.089  -1.412i  0
1   2.120   -1.000 +0.000i  unscaled  -0.772  -1.693i   2.601  +6.527i  0
2   6.000    3.100 -1.600i  unscaled  -1.371  -1.280i  -1.491  -5.993i  0
2   6.000    3.100 -1.600i    scaled   7.050  +6.052i   8.614 +29.352i  0