Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices

It is shown that the method of Arnoldi can be successfully used for solvinglarge unsymmetric eigenproblems. Like the symmetric Lanczos method, Arnoldi's algorithm realizes a projection process onto the Krylov subspace K_m spanned by v₁,Av₁,...,A^m-1v₁, where v₁ is the initial vector. We therefore study the convergence of the approximate eigenelements obtained by such a process. In particular, when the eigenvalues of A are real, we obtain bounds for the rates of convergence similar to those for the symmetric Lanczos algorithm. Some practical methods are presented in addition to that of Arnoldi, and several numerical experiments are described.