Aleksander:

If all you want is the smallest eigenvalue and the associated eigenvector, then the inverse power method suggested by others is a straightforward iterative approach that costs only n2 flops per iteration (due to the matrix-vector multipy at each iteration.  I suspect you are interested in such a 'cheap method' because to find all the eigenvalues and eigenvectors of an nxn matrix costs n3 flops (such as occurs when implementing eig( ) in MATLAB).  Since you mentioned that you were keen to find the smallest eigenvalues (i.e., multiple eigen-pairs), then I wanted to add to the discussion that another iterative method that will give estimates of multiple eigenvalues, that also costs only n2 flops per iteration, is the Lanczos method.  It is easily implemeted in MATLAB via the command eigs( ).   It will find estimates of the "well-separated" (including the biggest and smallest) eigenvalues of a matrix.   

You might use a package like scipy

python

>>> from scipy.linalg import eigh

>>> help (eigh)

Dear Peter,

I mean that I have some matrix with dimension nxn in which occurs an n variables, e. g. d_i (i=1,..,n). Also, there is constraint in the form d_1>d_2>....>d_n>0. Because there is this type of constraint I suppose that it is possible do determine which if the eigenvalues is smallest.

Can you exploit the structure of your matrix ?

Dear Nils,

I need to solve eigenvalue problem for this diff. eq. M*q''+Kq=0, where M is diagonal, and K is symetrical.

Both matrix are of nxn dimensions, and d_i (i = 1, ..., n) appear as a parameter.

One way is to apply the power method to A^(-1). This gives what is called the inverse power method. Now we do not calculate the inverse of A explicitly but we only solve linear algebraic systems  at each iteration alter we do A=LU. See the attached file.

What matrix depends on the parameters d_i ?

Is the stiffness matrix K tridiagonal ?

Nils, matrix K is in the attachment.

Both matrices (M and K) are dependent on the parameters d_i.

Aleksander:

If all you want is the smallest eigenvalue and the associated eigenvector, then the inverse power method suggested by others is a straightforward iterative approach that costs only n2 flops per iteration (due to the matrix-vector multipy at each iteration.  I suspect you are interested in such a 'cheap method' because to find all the eigenvalues and eigenvectors of an nxn matrix costs n3 flops (such as occurs when implementing eig( ) in MATLAB).  Since you mentioned that you were keen to find the smallest eigenvalues (i.e., multiple eigen-pairs), then I wanted to add to the discussion that another iterative method that will give estimates of multiple eigenvalues, that also costs only n2 flops per iteration, is the Lanczos method.  It is easily implemeted in MATLAB via the command eigs( ).   It will find estimates of the "well-separated" (including the biggest and smallest) eigenvalues of a matrix.   

I wish to make the following contribution on the discussion. As mentioned by Peter Breuer it is not possible to determine symbolically the smallest eigenvalue because it will depend in a complex manner on the relative values of the d_i's. In addition, as mentioned by Mohamed Hamdaoui and Benhammouda Brahim the smallest eigenvalue of a stiffness matrix can be determined numerically by using the inverse iteration or inverse power method whereby the power method is applied to K^-1.

The inverse of K matrix will not of course be determined by the explicit inversion procedure but by LU factorization followed by matrix reductions. However, Aleksandar's question remain unanswered, namely: What is the formula for the smallest eigenvalue in terms of d1, d2, ...............dn? In my own opinion, since the problem can be handled numerically only, it would be preferable to use Matlab to determine all the eigenvalues of the problem and then select the smallest eigenvalue for each combination of d1, d2, ............dn. This should be repeated for many combinations of the d_i's. A statistical package should then be used subsequently to derive a formula relating the smallest eigenvalue to the di's. This procedure may not be efficient but would certainly provide Aleksandar with a practical, though inelegant solution, to his problem. If the matrix size is large it will be more efficient to determine the smallest eigenvalue for each combination of d_i's using the inverse power method.