Convergence Analysis of Restarted Krylov Subspace Eigensolvers

Abstract

The $A$-gradient minimization of the Rayleigh quotient allows one to construct robust and fast-convergent eigensolvers for the generalized eigenvalue problem for $(A,M)$ with symmetric and positive definite matrices. The $A$-gradient steepest descent iteration is the simplest case of more general restarted Krylov subspace iterations for the special case that all stepwise generated Krylov subspaces are two-dimensional. This paper contains a convergence analysis of restarted Krylov subspace iterations for the minimization of the Rayleigh quotient with Krylov subspaces of arbitrary dimensions. The eigenpair approximations, namely, the Ritz vector and the Ritz value, are extracted in each step of the iteration by the Rayleigh--Ritz procedure. The new convergence analysis provides a sharp Ritz vector estimate together with a Ritz value estimate. These results improve the classical estimates by Kaniel [Math. Comp., 20 (1966), pp. 369--378] and Saad [SIAM J. Numer. Anal., 17 (1980), pp. 687--706] and Parlett [The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980] and generalize a result from Knyazev [Russian J. Numer. Anal. Math. Modelling, 2 (1987), pp. 371--396].