FullSimplify problem on Mathematica 4.1.2.0

• To: mathgroup at smc.vnet.net
• Subject: [mg33549] FullSimplify problem on Mathematica 4.1.2.0
• From: uhap023 at alpha1.rhbnc.ac.uk (Tom Crane)
• Date: Fri, 29 Mar 2002 06:14:03 -0500 (EST)
• Organization: Dept. Physics, Royal Holloway, University of London
• Sender: owner-wri-mathgroup at wolfram.com

```Hello All,
I have encountered the following problem with FullSimplify[] on
Mathematica 4.1.2.0. The problem is to FullSimplify the eigenvalues of a simple
(real, and in this case symmetric), 3x3 matrix. as follows;

Rmat = {{Ra + \[Rho]ac, 0, -\[Rho]ca}, {0,  Rb + \[Rho]ac, -\[Rho]ca},
{-\[Rho]ac, -\[Rho]ac, Rc + 2*\[Rho]ca}}

{ra, rb, rc} = Eigenvalues[Rmat]

ra = FullSimplify[ra]

The FullSimplify[] operation never finishes. Under Linux with the virtual
memory ulimited to 300MB to prevent it completely running the system into the
ground, on a 333MHz AMD K6-2 machine with 192MB RAM, the job runs for about 8
hours and then exits with;

"No more memory available.
Mathematica kernel has shut down.
Try quitting other applications and then retry."

This calculation fails similarly to complete after running all night, on a
1.4GHz machine running the same version of Mathematica but under Windows98.

However, under Mathematica 4.0/Windows on only a 133MHz machine the calculation
completes in about 8hours with the following successful result;

\!\(1\/6\ \((2\ \((Ra + Rb + Rc + 2\ \[Rho]ac +
2\ \[Rho]ca)\) + \((2\ 2\^\(1/3\)\ \((Ra\^2 + Rb\^2 + Rc\^2
-
2\ Rc\ \[Rho]ac + \[Rho]ac\^2 +
Rb\ \((\(-Rc\) + \[Rho]ac - 2\ \[Rho]ca)\) +
4\ Rc\ \[Rho]ca + 2\ \[Rho]ac\ \[Rho]ca + 4\ \[Rho]ca\^2
-
Ra\ \((Rb + Rc - \[Rho]ac +
2\ \[Rho]ca)\))\))\)/\((2\ Ra\^3 + 2\ Rb\^3 + 2\ \
Rc\^3 - 6\ Rc\^2\ \[Rho]ac + 6\ Rc\ \[Rho]ac\^2 - 2\ \[Rho]ac\^3 + 12\
Rc\^2\ \
\[Rho]ca - 6\ Rc\ \[Rho]ac\ \[Rho]ca - 6\ \[Rho]ac\^2\ \[Rho]ca + 24\ Rc\
\
\[Rho]ca\^2 + 12\ \[Rho]ac\ \[Rho]ca\^2 + 16\ \[Rho]ca\^3 - 3\ Rb\^2\
\((Rc - \
\[Rho]ac + 2\ \[Rho]ca)\) - 3\ Ra\^2\ \((Rb + Rc - \[Rho]ac + 2\
\[Rho]ca)\) \
- 3\ Rb\ \((Rc\^2 - 2\ Rc\ \[Rho]ac + \[Rho]ac\^2 + 4\ Rc\ \[Rho]ca - \
\[Rho]ac\ \[Rho]ca + 4\ \[Rho]ca\^2)\) - 3\ Ra\ \((Rb\^2 + Rc\^2 - 2\ Rc\
\
\[Rho]ac + \[Rho]ac\^2 + 4\ Rc\ \[Rho]ca - \[Rho]ac\ \[Rho]ca + 4\ \
\[Rho]ca\^2 - 4\ Rb\ \((Rc - \[Rho]ac + 2\ \[Rho]ca)\))\) +
\[Sqrt]\((\(-4\)\ \
\((Ra\^2 + Rb\^2 + Rc\^2 - 2\ Rc\ \[Rho]ac + \[Rho]ac\^2 + Rb\ \((\(-Rc\)
+ \
\[Rho]ac - 2\ \[Rho]ca)\) + 4\ Rc\ \[Rho]ca + 2\ \[Rho]ac\ \[Rho]ca + 4\ \
\[Rho]ca\^2 - Ra\ \((Rb + Rc - \[Rho]ac + 2\ \[Rho]ca)\))\)\^3 +
\((\(-2\)\ \
Ra\^3 - 2\ Rb\^3 + 3\ Rb\^2\ \((Rc - \[Rho]ac + 2\ \[Rho]ca)\) + 3\ Ra\^2\
\
\((Rb + Rc - \[Rho]ac + 2\ \[Rho]ca)\) + 3\ Rb\ \((Rc\^2 - 2\ Rc\ \[Rho]ac
+ \
\[Rho]ac\^2 + 4\ Rc\ \[Rho]ca - \[Rho]ac\ \[Rho]ca + 4\ \[Rho]ca\^2)\) +
3\ \
Ra\ \((Rb\^2 + Rc\^2 - 2\ Rc\ \[Rho]ac + \[Rho]ac\^2 + 4\ Rc\ \[Rho]ca - \
\[Rho]ac\ \[Rho]ca + 4\ \[Rho]ca\^2 - 4\ Rb\ \((Rc - \[Rho]ac + 2\ \
\[Rho]ca)\))\) - 2\ \((Rc\^3 - \[Rho]ac\^3 - 3\ Rc\^2\ \((\[Rho]ac - 2\ \
\[Rho]ca)\) - 3\ \[Rho]ac\^2\ \[Rho]ca + 6\ \[Rho]ac\ \[Rho]ca\^2 + 8\ \
\[Rho]ca\^3 + 3\ Rc\ \((\[Rho]ac\^2 - \[Rho]ac\ \[Rho]ca + 4\
\[Rho]ca\^2)\))\
\))\)\^2)\))\)\^\(1/3\) +
2\^\(2/3\)\ \((2\ Ra\^3 + 2\ Rb\^3 + 2\ Rc\^3 - 6\ Rc\^2\ \[Rho]ac
+ \
6\ Rc\ \[Rho]ac\^2 - 2\ \[Rho]ac\^3 + 12\ Rc\^2\ \[Rho]ca - 6\ Rc\
\[Rho]ac\ \
\[Rho]ca - 6\ \[Rho]ac\^2\ \[Rho]ca + 24\ Rc\ \[Rho]ca\^2 + 12\ \[Rho]ac\
\
\[Rho]ca\^2 + 16\ \[Rho]ca\^3 - 3\ Rb\^2\ \((Rc - \[Rho]ac + 2\
\[Rho]ca)\) - \
3\ Ra\^2\ \((Rb + Rc - \[Rho]ac + 2\ \[Rho]ca)\) - 3\ Rb\ \((Rc\^2 - 2\
Rc\ \
\[Rho]ac + \[Rho]ac\^2 + 4\ Rc\ \[Rho]ca - \[Rho]ac\ \[Rho]ca + 4\ \
\[Rho]ca\^2)\) - 3\ Ra\ \((Rb\^2 + Rc\^2 - 2\ Rc\ \[Rho]ac + \[Rho]ac\^2 +
4\ \
Rc\ \[Rho]ca - \[Rho]ac\ \[Rho]ca + 4\ \[Rho]ca\^2 - 4\ Rb\ \((Rc -
\[Rho]ac \
+ 2\ \[Rho]ca)\))\) + \[Sqrt]\((\(-4\)\ \((Ra\^2 + Rb\^2 + Rc\^2 - 2\ Rc\
\
\[Rho]ac + \[Rho]ac\^2 + Rb\ \((\(-Rc\) + \[Rho]ac - 2\ \[Rho]ca)\) + 4\
Rc\ \
\[Rho]ca + 2\ \[Rho]ac\ \[Rho]ca + 4\ \[Rho]ca\^2 - Ra\ \((Rb + Rc -
\[Rho]ac \
+ 2\ \[Rho]ca)\))\)\^3 + \((\(-2\)\ Ra\^3 - 2\ Rb\^3 + 3\ Rb\^2\ \((Rc - \
\[Rho]ac + 2\ \[Rho]ca)\) + 3\ Ra\^2\ \((Rb + Rc - \[Rho]ac + 2\
\[Rho]ca)\) \
+ 3\ Rb\ \((Rc\^2 - 2\ Rc\ \[Rho]ac + \[Rho]ac\^2 + 4\ Rc\ \[Rho]ca - \
\[Rho]ac\ \[Rho]ca + 4\ \[Rho]ca\^2)\) + 3\ Ra\ \((Rb\^2 + Rc\^2 - 2\ Rc\
\
\[Rho]ac + \[Rho]ac\^2 + 4\ Rc\ \[Rho]ca - \[Rho]ac\ \[Rho]ca + 4\ \
\[Rho]ca\^2 - 4\ Rb\ \((Rc - \[Rho]ac + 2\ \[Rho]ca)\))\) - 2\ \((Rc\^3 -
\
\[Rho]ac\^3 - 3\ Rc\^2\ \((\[Rho]ac - 2\ \[Rho]ca)\) - 3\ \[Rho]ac\^2\ \
\[Rho]ca + 6\ \[Rho]ac\ \[Rho]ca\^2 + 8\ \[Rho]ca\^3 + 3\ Rc\
\((\[Rho]ac\^2 \
- \[Rho]ac\ \[Rho]ca + 4\ \[Rho]ca\^2)\))\))\)\^2)\))\)\^\(1/3\))\)\)

Has anybody else come across this problem? Is there a memory leak in version
4.2.1.0? Are there any fixes?

Regards
Tom.

--
Tom Crane, Dept. Physics, Royal Holloway, University of London, Egham Hill,
Egham, Surrey, TW20 0EX, England.
Email:	uhap023 at vms.rhbnc.ac.uk
SPAN:   19.875
Fax:    +44 (0) 1784 472794

```

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