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Re: Mathematica 20x slower than Java at arithmetic/special functions, is

• To: mathgroup at smc.vnet.net
• Subject: [mg115900] Re: Mathematica 20x slower than Java at arithmetic/special functions, is
• From: AES <siegman at stanford.edu>
• Date: Tue, 25 Jan 2011 04:20:41 -0500 (EST)
• References: <ihiab8$m3b$1@smc.vnet.net> <ihjltm$t5p$1@smc.vnet.net>

Relevant to this thread -- maybe?  (I'm not qualified to say)

Some years ago I was doing some optical beam modal expansions involving 
numerical expansions of sums over Hermite-gaussian, Laguerre-gaussian, 
and Legendre functions with indexes running from n=0 up to, in extreme 
cases, n=60 or 70 -- just doing them as 'brute force' Mathematica 
Sum[]s: no compiling (and no checking on timing).

In some cases I wanted to observe effects of small variations of 
function arguments about special values like z = Pi, but without getting 
diverted into any attempts at doing first order analytical perturbations 
or expansions of the sums or their individual terms.  

Soon became apparent that if I evaluated, say, a 64-term sum for 
arguments z = 0.998 Pi, 0.999 Pi, Pi, 1.001 Pi, 1.002 Pi, I could see 
obvious numerical round-off errors in the results.

But if I did the exact same calculations except writing the arguments as 
z = 998 Pi/1000, 999 Pi/1000, Pi, 1001 Pi/1000, 1002 Pi/ 1000, I got 
results that appeared to be accurate to the final digit in the results.

I assumed this meant that Mathematica detected these exact inputs, and 
automatically switched to a (slower) exact arithmetical evaluation of 
the terms in the series (??).  In the cases at issue here, maybe 
Mathematic does something similar -- and Java doesn't?