Guaranteed Extended Precision For Transcendental Functions

29/01/2011 - 23:56 von Dan | Report spam
Hi,

My request may be a bit unusual, but I need some help. I have been
developing a X86/87 function library that guarantees floating point
extended precision (IEEE Standard 754, 80-bit floating point), and I
seek evaluation-time comparisons.

For this purpose, I have published a table of evaluation-time
benchmarks for the Bessel function of the first kind for guaranteed 64
bits (mantissa) precision, at http://www.iging.com/NumericalAnaly...essel.htm.

If you have a comparable procedure (that calculates the Bessel
function of the 1st kind with extended precision), please evaluate it
for a few different parameter pairs, and let me know the execution
times.

The library file "XP87.LIB" that contains the Bessel function is
available upon request; free, for non commercial use.

D. Baruth
x87[at]iging[dot]com
 

Lesen sie die antworten

#1 Chip Eastham
09/02/2011 - 15:19 | Warnen spam
On Jan 29, 5:56 pm, Dan wrote:
Hi,

My request may be a bit unusual, but I need some help.  I have been
developing a X86/87 function library that guarantees floating point
extended precision (IEEE Standard 754, 80-bit floating point), and I
seek evaluation-time comparisons.

For this purpose, I have published a table of evaluation-time
benchmarks for the Bessel function of the first kind for guaranteed 64
bits (mantissa) precision, athttp://www.iging.com/NumericalAna...essel.htm.

If you have a comparable procedure (that calculates the Bessel
function of the 1st kind with extended precision), please evaluate it
for a few different parameter pairs, and let me know the execution
times.

The library file "XP87.LIB" that contains the Bessel function is
available upon request; free, for non commercial use.

D. Baruth
x87[at]iging[dot]com



Thanks for posting this. The extended precision
evaluation of transcendental functions is to me
an interesting challenge. I don't have a ready
comparison to give you. The last time I looked
at the Bessel functions was for a purpose of
evaluating multiple orders (at the same argument),
and IIRC continued fractions were an element of
the computation.

You allow for the return of a complex result,
but if I understand the approach, with real
orders and arguments this is of limited use.
You quote Abramowitz and Stegun's opus Handbook
of Mathematical Functions (Ch. 9 was prepared
by FWJ Olver) for this alternating series:

J_n(z)=(z/2)^n SUM (-z^2/4)^k/(k!\Gamma(n+k+1))

where the sum is k = 0 to +oo.

The calling sequence appears (based on stack
usage) to permit only real values for order n
and argument z. So complex values arise only
from the factor (z/2)^n, when z is negative
and n is not integer. The series portion of
the formula is the Bessel-Clifford function
evaluated at -z^2/4:

http://en.wikipedia.org/wiki/Bessel...d_function

and is real valued (for real n and z) and
entire.

So perhaps your challenge can be reframed as
finding rapid methods of guaranteed precision
for evaluating the Bessel-Clifford function.

regards, chip

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