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{\large Fei Xue \\
{\bf Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration.
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{\tt petrinet@math.umd.edu}\\
Howard C. Elman
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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". 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.
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