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Re: Linear algebra and machine precision

• To: mathgroup at smc.vnet.net
• Subject: [mg77163] Re: Linear algebra and machine precision
• From: David Bailey <dave at Remove_Thisdbailey.co.uk>
• Date: Tue, 5 Jun 2007 06:36:21 -0400 (EDT)
• References: <f40hja$6lh$1@smc.vnet.net>

Dmitry Garanin wrote:
> Hi,
> 
> I have Mathematica 5.0 on my laptop (Win XP pro 32 bit) and Mathematica 5.2 on my workstation (Win XP pro 64 bit). I have a program operating with eigensystems of huge matrices with nearly degenerate eigenvalues. The results become inaccurate on my laptop when I increase the size of the matrix while I always obtain accurate results on the workstation.
> 
> My initial thought was that on the workstation Mathematica uses longer machine numbers and that is why the results are accurate. However, $MachinePrecision outputs a number below 16 on both machines. Also computing 1-Sqrt(1+AVerySmallNumber) gives the same result on both machines. So that evidently Mathematica cannot really use those 64 bits. Or I understand something wrong?
> 
> Another explanation would be that linear algebra in Mathematica 5.0 is buggy.
> 
> BTW my program solving partial differential equations works well with Mathematica 5.0 but gives unstable results with Mathematica 5.2, so here the situation seems to be reversed.
> 
> What do you think?
> 
> Best regards,
> 
> Dmitry
> 
> 
The phrase "64-bit" really refers to the addressing capabilities of the 
machine, data can be of various lengths. Machine precision Reals are 
stored with 64-bit precision in both environments.

David Bailey
http://www.dbaileyconsultancy.co.uk