NAG Library Function Document
nag_complex_hankel (s17dlc)
1
Purpose
nag_complex_hankel (s17dlc) returns a sequence of values for the Hankel functions or for complex , non-negative
and , 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 or , where is complex, , and is the real, non-negative order. The -member sequence is generated for orders , . Optionally, the sequence is scaled by the factor if the function is or by the factor if the function is .
Note: although the function may not be called with less than zero, for negative orders the formulae , and may be used.
The function is derived from the function CBESH in
Amos (1986). It is based on the relation
where
if
and
if
, and the Bessel function
is computed in the right half-plane only. Continuation of
to the left half-plane is computed in terms of the Bessel function
. These functions are evaluated using a variety of different techniques, depending on the region under consideration.
When is greater than , extra values of are computed using recurrence relations.
For very large or , argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller or , the computation is performed but results are accurate to less than half of machine precision. If is very small, near the machine underflow threshold, or 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:
– IntegerInput
-
On entry: the kind of functions required.
- The functions are .
- The functions are .
Constraint:
or .
- 2:
– doubleInput
-
On entry: , the order of the first member of the sequence of functions.
Constraint:
.
- 3:
– ComplexInput
-
On entry: the argument of the functions.
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of members required in the sequence .
Constraint:
.
- 5:
– Nag_ScaleResTypeInput
-
On entry: the scaling option.
- The results are returned unscaled.
- The results are returned scaled by the factor when , or by the factor when .
Constraint:
or .
- 6:
– ComplexOutput
-
On exit: the required function values: contains
, for .
- 7:
– Integer *Output
-
On exit: the number of components of
cy that are set to zero due to underflow. If
, then if
and
, or
and
, elements
are set to zero. In the complementary half-planes,
nz simply states the number of underflows, and not which elements they are.
- 8:
– 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 had an illegal value.
- NE_COMPLEX_ZERO
-
On entry, .
- NE_INT
-
On entry,
m has illegal value:
.
On entry, .
Constraint: .
- 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 .
No computation because is too large.
- NE_REAL
-
On entry, .
Constraint: .
- NE_TERMINATION_FAILURE
-
No computation – algorithm termination condition not met.
- NE_TOTAL_PRECISION_LOSS
-
No computation because .
No computation because .
- NW_SOME_PRECISION_LOSS
-
Results lack precision because .
Results lack precision, .
7
Accuracy
All constants in nag_complex_hankel (s17dlc) are given to approximately digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used , then clearly the maximum number of correct digits in the results obtained is limited by . 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 , where represents the number of digits lost due to the argument reduction. Thus the larger the values of and , the less the precision in the result. If nag_complex_hankel (s17dlc) is called with , 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 and different , the computed values may not agree exactly. Empirical tests with modest values of and have shown that the discrepancy is limited to the least significant – 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
, rather than to make
separate calls to
nag_complex_hankel (s17dlc).
Paradoxically, for some values of and , it is cheaper to call nag_complex_hankel (s17dlc) with a larger value of 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 , 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
to evaluate the function for orders
fnu and
, 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)