complex eigenvalues

[Steven G Gilmour]

Dear All

I was talking to some numerical type people and this came up. I think a
routine for general matrices is being used, when perhaps there is a better
one for symmetric matrices. Apparently, the MatLab eig command first
checks if your matrix is symmetric or not, then uses the appropriate
numerical procedure. 

Steve

p.s. Octave is a free MatLab clone, which is quite easily compatible with
R, if anyone is interested.

On Fri, 18 Jun 2004, R A Bailey wrote:

> My student Rebecca Lodwick put this matrix into Maple and it produced
> all real eigenvalues.  Perhaps we arn't using the best algorithms for
> integer symmetric matrices?  She told me that if you use Mathematica
> then there is a command something like
> 
> Needs ["Miscellaneous `Real only`"]
> 
> 
> Rosemary
> 
> > 
> > Dear All,
> > 
> > As we are crunching out more and more designs it happens some times that
> > the numerical library used to compute the eigenvalues of the information
> > matrix returns complex numbers. Here is an example:
> > 
> >     complex eigenvalue found ...
> >     id: v7-b6-k3-109
> >     blocks:
> >     [[0, 1, 2], [0, 1, 2], [0, 1, 3], [2, 4, 6], [3, 4, 5], [3, 4, 5]]
> >     incidence_matrix:
> >     [[1 1 1 0 0 0]
> >      [1 1 1 0 0 0]
> >      [1 1 0 1 0 0]
> >      [0 0 1 0 1 1]
> >      [0 0 0 1 1 1]
> >      [0 0 0 0 1 1]
> >      [0 0 0 1 0 0]]
> >     concurrence_matrix:
> >     [[3 3 2 1 0 0 0]
> >      [3 3 2 1 0 0 0]
> >      [2 2 3 0 1 0 1]
> >      [1 1 0 3 2 2 0]
> >      [0 0 1 2 3 2 1]
> >      [0 0 0 2 2 2 0]
> >      [0 0 1 0 1 0 1]]
> >     eigenvalues:
> >     [  2.22044605e-16 +0.00000000e+00j   6.66666667e-01 -1.85103892e-16j
> >        6.66666667e-01 +1.85103892e-16j   2.33333333e+00 +0.00000000e+00j
> >        2.33333333e+00 +0.00000000e+00j   3.00000000e+00 +0.00000000e+00j
> >        3.00000000e+00 +0.00000000e+00j]
> >     ... skipped.
> > 
> > Of course, the imaginary parts are very small. Nevertheless, the
> > appearance of complex numbers causes problems in the computation of
> > statistical properties since the algorithms involved expect real
> > numbers.
> > 
> > To avoid these problems while maintaining as much numerical precision as
> > possible for the further computation, I have modified the block_design
> > Python module.  From now on, all eigenvalues of the information matrix
> > returned by the numerical library are turned into nonnegative reals by
> > taking their absolute values. While this is not the perfect solution, we
> > certainly can do this in a "numerical sense" since, in theory, all
> > eigenvalues are nonnegative real numbers.
> > 
> > Using this method the eigenvalues for the above example become:
> > 
> >     [2.2204460492503131e-16, 0.66666666666666674,
> >     0.66666666666666674, 2.3333333333333335,
> >     2.3333333333333335, 3.0000000000000004,
> > 			3.0000000000000027]
> > 
> > The perfect solution would be the exact computation, however writing
> > such code would take some time and money! ... you know what I mean :-)
> > 
> > Please comment on how you find this numerical solution for the time
> > being.
> > 
> > --             ,
> >     Peter Dobcsanyi
> > 
> > _______________________________________________
> > Developers mailing list
> > Developers at designtheory.org
> > http://designtheory.org/cgi-bin/mailman/listinfo/developers
> > 
> 
> 
> -- 
> R. A. Bailey
>        
> Snail:  School of Mathematical Sciences    Tel:  (+44) 20 7882 5517
>         Queen Mary, University of London
>         Mile End Road                      Email: r.a.bailey at qmul.ac.uk
>         London E1 4NS
>         U.K.
> 
> _______________________________________________
> Developers mailing list
> Developers at designtheory.org
> http://designtheory.org/cgi-bin/mailman/listinfo/developers
> 


Dr Steven G Gilmour

School of Mathematical Sciences
Queen Mary, University of London
Mile End Road
London E1 4NS
United Kingdom

Tel: +44 (0)20 7882 7833
Fax: +44 (0)20 8981 9587 (department fax, not private)

Web page: http://www.maths.qmul.ac.uk/~sgg