NAG CL Interface
s18dec (bessel_i_complex)
1
Purpose
s18dec returns a sequence of values for the modified Bessel functions ${I}_{\nu +n}\left(z\right)$ for complex $z$, nonnegative
$\nu $ and $n=0,1,\dots ,N1$, with an option for exponential scaling.
2
Specification
void 
s18dec (double fnu,
Complex z,
Integer n,
Nag_ScaleResType scal,
Complex cy[],
Integer *nz,
NagError *fail) 

The function may be called by the names: s18dec, nag_specfun_bessel_i_complex or nag_complex_bessel_i.
3
Description
s18dec evaluates a sequence of values for the modified Bessel function ${I}_{\nu}\left(z\right)$, 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 ${e}^{\left\mathrm{Re}\left(z\right)\right}$.
The function is derived from the function CBESI in
Amos (1986).
Note: although the function may not be called with
$\nu $ less than zero, for negative orders the formula
${I}_{\nu}\left(z\right)={I}_{\nu}\left(z\right)+\frac{2}{\pi}\mathrm{sin}\left(\pi \nu \right){K}_{\nu}\left(z\right)$ may be used (for the Bessel function
${K}_{\nu}\left(z\right)$, see
s18dcc).
When $N$ is greater than $1$, extra values of ${I}_{\nu}\left(z\right)$ 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 $\mathrm{Re}\left(z\right)$ is too large and the unscaled function is required, 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
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:
$\mathbf{fnu}$ – double
Input

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

2:
$\mathbf{z}$ – Complex
Input

On entry: the argument $z$ of the functions.

3:
$\mathbf{n}$ – Integer
Input

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

4:
$\mathbf{scal}$ – Nag_ScaleResType
Input

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 ${e}^{\left\mathrm{Re}\left(z\right)\right}$.
Constraint:
${\mathbf{scal}}=\mathrm{Nag\_UnscaleRes}$ or $\mathrm{Nag\_ScaleRes}$.

5:
$\mathbf{cy}\left[{\mathbf{n}}\right]$ – Complex
Output

On exit: the $N$ required function values: ${\mathbf{cy}}\left[i1\right]$ contains
${I}_{\nu +i1}\left(z\right)$, for $\mathit{i}=1,2,\dots ,N$.

6:
$\mathbf{nz}$ – Integer *
Output

On exit: the number of components of
cy that are set to zero due to underflow.
If ${\mathbf{nz}}>0$, then elements ${\mathbf{cy}}\left[{\mathbf{n}}{\mathbf{nz}}\right],{\mathbf{cy}}\left[{\mathbf{n}}{\mathbf{nz}}+1\right],\dots ,{\mathbf{cy}}\left[{\mathbf{n}}1\right]$ are set to zero.

7:
$\mathbf{fail}$ – 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

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 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 ${\mathbf{z}}\mathbf{.}\mathbf{re}=\u2329\mathit{\text{value}}\u232a>\u2329\mathit{\text{value}}\u232a$, ${\mathbf{scal}}=\mathrm{Nag\_UnscaleRes}$.
 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 because ${\mathbf{fnu}}+{\mathbf{n}}1=\u2329\mathit{\text{value}}\u232a>\u2329\mathit{\text{value}}\u232a$.
7
Accuracy
All constants in s18dec 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 s18dec, 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 s18dec 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 s18dec 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
s18dec is not threaded in any implementation.
The time taken for a call of s18dec is approximately proportional to the value of ${\mathbf{n}}$, plus a constant. In general it is much cheaper to call s18dec with ${\mathbf{n}}$ greater than $1$, rather than to make $N$ separate calls to s18dec.
Paradoxically, for some values of $z$ and $\nu $, it is cheaper to call s18dec 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.
Note that if the function required is
${I}_{0}\left(x\right)$ or
${I}_{1}\left(x\right)$, i.e.,
$\nu =0.0$ or
$1.0$, where
$x$ is real and positive, and only a single function value is required, then it may be much cheaper to call
s18aec,
s18afc,
s18cec or
s18cfc, depending on whether a scaled result is required or not.
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 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
${\mathbf{n}}=2$ 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
10.2
Program Data
10.3
Program Results