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 $z$, non-negative
$\nu $ and $n=0,1,\dots ,N-1$, with an option for exponential scaling.
The function may be called by the names: s17dlc, nag_specfun_hankel_complex or nag_complex_hankel.
3Description
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, 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
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 $(\nu +N-1)$, argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller $\left|z\right|$ or $(\nu +N-1)$, 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 $(\nu +N-1)$ 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.
Amos D E (1986) Algorithm 644: A portable package for Bessel functions of a complex argument and non-negative order ACM Trans. Math. Software12 265–273
5Arguments
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:
${\mathbf{z}}\ne (0.0,0.0)$.
4: $\mathbf{n}$ – IntegerInput
On entry: $N$, the number of members required in the sequence ${H}_{\nu}^{\left({\mathbf{m}}\right)}\left(z\right),{H}_{\nu +1}^{\left({\mathbf{m}}\right)}\left(z\right),\dots ,{H}_{\nu +N-1}^{\left({\mathbf{m}}\right)}\left(z\right)$.
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 $\mathrm{Nag\_ScaleRes}$.
On exit: the $N$ required function values: ${\mathbf{cy}}\left[i-1\right]$ contains
${H}_{\nu +i-1}^{\left({\mathbf{m}}\right)}\left(z\right)$, 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 half-planes, 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 7 in the Introduction to the NAG Library CL Interface).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_COMPLEX_ZERO
On entry, ${\mathbf{z}}=(0.0,0.0)$.
NE_INT
On entry, m has illegal value: ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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 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 $\left|{\mathbf{z}}\right|=\u27e8\mathit{\text{value}}\u27e9<\u27e8\mathit{\text{value}}\u27e9$.
No computation because ${\mathbf{fnu}}+{\mathbf{n}}-1=\u27e8\mathit{\text{value}}\u27e9$ is too large.
NE_REAL
On entry, ${\mathbf{fnu}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{fnu}}\ge 0.0$.
NE_TERMINATION_FAILURE
No computation – algorithm termination condition not met.
NE_TOTAL_PRECISION_LOSS
No computation because $\left|{\mathbf{z}}\right|=\u27e8\mathit{\text{value}}\u27e9>\u27e8\mathit{\text{value}}\u27e9$.
No computation because ${\mathbf{fnu}}+{\mathbf{n}}-1=\u27e8\mathit{\text{value}}\u27e9>\u27e8\mathit{\text{value}}\u27e9$.
NW_SOME_PRECISION_LOSS
Results lack precision because $\left|{\mathbf{z}}\right|=\u27e8\mathit{\text{value}}\u27e9>\u27e8\mathit{\text{value}}\u27e9$.
All constants in 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 $p=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(t,18)$. Because of errors in argument reduction when computing elementary functions inside s17dlc, the actual number of correct digits is limited, in general, by $p-s$, where $s\approx \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,\left|{\mathrm{log}}_{10}\left|z\right|\right|,\left|{\mathrm{log}}_{10}\nu \right|)$ 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 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 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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s17dlc is not threaded in any implementation.
9Further Comments
The time taken for a call of s17dlc is approximately proportional to the value of n, plus a constant. In general it is much cheaper to call s17dlc with n greater than $1$, rather than to make $N$ separate calls to s17dlc.
Paradoxically, for some values of $z$ and $\nu $, it is cheaper to call 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.
10Example
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.