Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration
Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration
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Date
2008-01
Authors
Xue, Fei
Elman, Howard C.
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Abstract
We present a detailed convergence analysis of preconditioned MINRES for
approximately solving the linear systems that arise when Rayleigh Quotient
Iteration is used to compute the lowest eigenpair of a symmetric positive
definite matrix. We provide insight into the ``slow start'' of MINRES
iteration in both a qualitative and quantitative way, and show that the
convergence of MINRES mainly depends on how quickly the unique negative
eigenvalue of the preconditioned shifted coefficient matrix is
approximated by its corresponding harmonic Ritz value. By exploring when
the negative Ritz value appears in MINRES iteration, we obtain a better
understanding of the limitation of preconditioned MINRES in this context
and the virtue of a new type of preconditioner with ``tuning''. Comparison
of MINRES with SYMMLQ in this context is also given. Finally we show that
tuning based on a rank-2 modification can be applied with little
additional cost to guarantee positive definiteness of the tuned
preconditioner.