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Re: Automatic method in Eigenvalues

  • To: mathgroup at smc.vnet.net
  • Subject: [mg108985] Re: Automatic method in Eigenvalues
  • From: Mike Bryniarski <melondisco at gmail.com>
  • Date: Fri, 9 Apr 2010 03:32:10 -0400 (EDT)
  • References: <hpkgm2$82t$1@smc.vnet.net>

Actually Mathematica can in some sense compute eigenvalues of larger
matrices analytically, but it has to return them as a Root
expressions.
Example:
m ={{10, 96, 50, 62, 13, 99, 44}, {78, 41, 73, 69, 9, 69, 58}, {9,
53,
  97, 27, 95, 15, 97}, {22, 60, 100, 84, 28, 76, 33}, {56, 16, 62, 45,
   47, 64, 65}, {8, 73, 71, 33, 44, 61, 20}, {30, 90, 98, 60, 31, 29,
  96}};
 Eigenvalues[m]

but when working with numerical data I believe Eigenvalues calls
LAPACK to do the computation.
When using sparse arrays Eigenvalues uses Arnoldi iteration from
ARPACK.
Some information on how Mathematica does things is available at:
http://reference.wolfram.com/mathematica/note/SomeNotesOnInternalImplementation.html

for more specific information than this you would need to look at
documentation for the implimentation of LAPACK or ARPACK

-Mike
On Apr 8, 8:03 am, "Fischer, Egil" <egil.fisc... at wur.nl> wrote:
> Hi,
>
> With Mathematica it is easy to obtain the Eigenvalues of a (large)
> matrix (Function Eigenvalues). Currently I am writing a paper and I
> would like to know which algorithm is used by Mathematica for a n x n
> matrix (n >= 5). The eigenvalues cannot be determined analytical, thus I
> am interested in which numerical method Mathematica uses. Using Options
> I can only find that the Method is Automatic (which does not help).
>
> Kind regards,
>
> ir. Egil A.J. Fischer
> Researcher
> Quantitative Veterinary Epidemiology and Risk Analysis (QVE-RA)
> Central Veterinary Institute of Wageningen UR
> Absent on fridays
> P.O.Box 65, 8200 AB Lelystad.
> Visiting address: Houtribweg 39, Lelystad.
> Phone: +31 (0)320 238370
> e-mail: egil.fisc... at wur.nl <blocked::mailto:egil.fisc... at wur.nl>
> internet:www.cvi.wur.nl<blocked::http://www.cvi.wur.nl/>
>
> KvK: de Animal Sciences Group is bij de Kamer van Koophandel in Lelystad
> geregistreerd als 'ID-Lelystad, Instituut voor Dierhouderij en
> Diergezondheid B.V.' onder nummer 39073054
> KvK: het Centraal Veterinair Instituut is bij de Kamer van Koophandel in
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> 09098104
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>
> P Please consider the environment before printing this e-mail



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