On 07/10/2012 11:50 AM, Steve Kargl wrote:
On Tue, Jul 10, 2012 at 11:39:59AM -0500, Stephen Montgomery-Smith wrote:
On 07/09/2012 12:02 AM, Steve Kargl wrote:
Yep. Another example is the use of upward recurion to compute
Bessel functions where the argument is larger than the order.
The algorithm is known to be unstable.
By upward recursion, do you mean equation (1) in
http://mathworld.wolfram.com/BesselFunction.html?
Yes.
So what do people use. Maybe something like
http://en.wikipedia.org/wiki/Bessel_function#Asymptotic_forms (second
set of equations), but finding some asymptotics with a few extra terms
in them?
They use downward recursion, which is known to be stable.
NIST has revised Abramowitz and Stegun, and it is available
on line. For Bessel function computations, look at
http://dlmf.nist.gov/10.74
and more importantly example 1 under the following link
http://dlmf.nist.gov/3.6#v
These algorithms are going to be subject to the same problems as using
Taylor's series to evaluate exp(x) for x>0. The computation will
require several floating point operations. Even if the method is not
unstable, I would think getting a ULP of about 1 is next to impossible,
that is, unless one performs all the computations in a higher precision,
and then rounds at the end.
Whereas as a scientist, having a bessel function computed to within 10
ULP would be just fine.
I am looking at the man page of j0 for FreeBSD-8.3. It says in conforms
to IEEE Std 1003.1-2001. Does that specify a certain ULP? I am
searching around in this document, and I am finding nothing. Whereas
the IEEE-754 document seems rather rigid, but on the other hand it
doesn't specifically talk about math functions other than sqrt.
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