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In article <17229@sophia.inria.fr> falzon@sophia.inria.fr writes:
>
> Does anybody have the general forms of eigenvalues and eigenvectors of an NxN
> pseudo-tridiagonal matrix of the form :
> 
> For example with N=4;
> 
>
>	[(1+1/a) -1/a       0         0 ]
>  A = 	[-1/a    (1+2/a)  -1/a        0 ]
>	[ 0       -1/a    (1+2/a)  -1/a ]
>	[ 0         0      -1/a  (1+1/a)] 
>
> Thanks in advance.
>
>
>      Fred.
>-----------------------------------------------------------------------------------------


Let me take this moment to recommend one of those old books lying
on the hard-to reach shelves of your library. 1966 "A collection of 
Matrices for Testing Computational Algorithms" by Robert T. Gregory and
David L. Karney by Wiley-Interscience. The book is dedicated to
A. S. Householder and it lives up to its title. 

I am certain (100%) that you will find the eigenvalues and eigenfunctions
to the matrix which is apparently from a second order central difference
finite difference discretization. It is a circulant matrix and have eigen
solutions which can be expressed in terms of bunch of cosines and sines. 
Contact me if you are still at loss. 

Ferhat 

-- 
Ferhat F. Hatay                        || Disclamer: |||||||||||||||||
Aerospace Engineering Sciences         || I speak for myself unless || 
University of Colorado, Boulder        || otherwise indicated.      ||
e-mail: hatay@spot.colorado.edu        |||||||||||||||||||||||||||||||

I am having difficulties to reach Eurepan dudes (dudesses!) by mail, 
somewhat they keep rejecting my mail. So, please forgive my overuse
of network to communicate the following -deep thoughts- to them. 
I don't claim any liability for my suggestions, anyone is welcomed 
to find any holes in my reasoning. feel free to poke my failing eyes. 
 
In article <17229@sophia.inria.fr> falzon@sophia.inria.fr writes:
>
> Does anybody have the general forms of eigenvalues and eigenvectors of an NxN
> pseudo-tridiagonal matrix of the form :
> 
> For example with N=4;
> 
>
>	[(1+1/a) -1/a       0         0 ]
>  A = 	[-1/a    (1+2/a)  -1/a        0 ]
>	[ 0       -1/a    (1+2/a)  -1/a ]
>	[ 0         0      -1/a  (1+1/a)] 
>
> Thanks in advance.
>
>
>      Fred.
>-----------------------------------------------------------------------------------------

Hmm, The problem adds to its lure when we consider it with 
some boundary conditions. It stops being circulant and
turns to a headache. Assuming that this is a solution
to a second order pde like Helmholtz or Poisson, let me 
suggest the following:

For dirichlet boundary conditions we can fold them 
onto (or into?) a system of order of N-2. The 
resulting -nonsingular- system (matrix) will again
have a circulant structure and a different right 
hand side. 

For neumann boundary conditions: I use my eigenvalue solver!

I still think there is an analytical solution for the problem
mentioned here. If we solve the eigenproblem for a matrix A:

    A = Q D Qt where D is diagonal matrix of eigenvalues.

    Q A Qt = D 

what would be the solution for

    B = A + S ? where S is a diagonal matrix. 

let's try:

   Q B Qt = Q A Qt + Q S Qt
    
   Q B Qt =  D + Q S Qt 

then what? eigenvectors of matrix B are Q same as the original 
matrix A and eigenvalues are given by D + S?

Do you see any holes in the reasoning above? It was rather
spontaneous so I should have probably checked myself.
  
cheers, 

Ferhat


-- 
Ferhat F. Hatay                        || Disclamer: |||||||||||||||||
Aerospace Engineering Sciences         || I speak for myself unless || 
University of Colorado, Boulder        || otherwise indicated.      ||
e-mail: hatay@spot.colorado.edu        |||||||||||||||||||||||||||||||