Matlab's eigs not converging unless number of eigenvalues computed is large

Question

I am trying to compute the eigenvalues, λ (lambda), of a damped structure with the following equations of motion:

(λ²M + λC + K) x = 0,

where M, C, and K are sparse matrices. Using MATLAB's polyeig function works, but I would like to go to larger systems and take advantage of the sparsity of my matrices. I have used a state space linearization to obtain a generalized eigenvalue problem as follows:

(A - λB) z = 0,

with

A = [K , 0 ; 0, -M],

B = [-C , -M ; -M, 0],

z = [x ; λx]

Solving this with MATLAB's eigs function:

lambda = eigs(A,B,10,'sm')

Produces the following output:

lambda =
   1.0e+03 *
  -0.2518 - 1.3138i
  -0.2518 + 1.3138i
  -0.4690 - 1.7360i
  -0.4690 + 1.7360i
  -0.4690 - 1.7360i
  -0.4690 + 1.7360i
  -0.5387 - 1.8352i
  -0.5387 + 1.8352i
      NaN +    NaNi
      NaN +    NaNi

The first eight eigenvalues are correct, but it seems as though the last two eigenvalues were not able to converge. Increasing the number of Lanczos basis vectors does not seem to improve the problem.

Strangely however, increasing the number of eigenvalues computed (k) allows more and more eigenvalues to converge:

• k = 10: Number of lambdas converged = 8
• k = 20: Number of lambdas converged = 8
• k = 50: Number of lambdas converged = 8
• k = 100: Number of lambdas converged = 20
• k = 120: Number of lambdas converged = 80
• k = 150: Number of lambdas converged = 150

It may also be worth mentioning that many of the eigenvalues that do not converge with lower values of k appear to be degenerate or at least very closely spaced.

I was wondering if anybody can think of an explanation for this behavior? If so, is there any way to make all of the eigenvalues converge without making k very large? Thank you!

Answer 1

This is old, but still unanswered. Without the actual matrices it is difficult to be certain. This is my best guess:

eigs calls ARPACK routines. ARPACK exploits iterative methods (Arnoldi) to converge to, e.g., the eigenvalues with smallest magnitude (option sm). As for any iterative method, the user can specify options like the convergence Tolerance and the MaxIterations before the iterative process stops. The NaNs indicate eigenvalues that have not converged when MaxIterations is reached.

An important option for Arnoldi methods is the dimension of the Krylov subspace used to approximate the solutions. This can be specified by the option SubspaceDimension in eigs. The default value is max(2*k,20), so increasing k effectively increases the dimension of the Krylov subspace. If your problem requires a relatively large Krylov subspace for converging some eigenvalues to the desired Tolerance, this could explain why increasing k yields convergence of a larger number of eigenvalues.

To verify if my guess is correct you could either have a less restrictive Tolerance (e-6 may be sufficient?) or increase the value of SubspaceDimension while keeping k constant.