Fastest quotient iteration with variational principles for self-adjoint eigenvalue problems

Abstract

For the generalized eigenvalue problem, a quotient function is devised for estimating eigenvalues in terms of an approximate eigenvector. This gives rise to an infinite family of quotients, all entirely arguable to be used in estimation. Although the Rayleigh quotient is among them, one can suggest using it only in an auxiliary manner for choosing the quotient for near optimal results. In normal eigenvalue problems, for any approximate eigenvector, there always exists a "perfect" quotient exactly giving an eigenvalue. For practical estimates in the self-adjoint case, an approximate midpoint of the spectrum is a good choice for reformulating the eigenvalue problem yielding apparently the fastest quotient iterative method there exists. No distinction is made between estimating extreme or interior eigenvalues. Preconditioning from the left results in changing the inner-product and affects the estimates accordingly. Preconditioning from the right preserves self-adjointness and can hence be performed without any restrictions. It is used in variational methods for optimally computing approximate eigenvectors.