Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems
SIAM Journal on Scientific Computing
The symmetric eigenvalue problem
The symmetric eigenvalue problem
MINRES and MINERR Are Better than SYMMLQ in Eigenpair Computations
SIAM Journal on Scientific Computing
Matrix algorithms
An Inverse Free Preconditioned Krylov Subspace Method for Symmetric Generalized Eigenvalue Problems
SIAM Journal on Scientific Computing
Convergence Analysis of Inexact Rayleigh Quotient Iteration
SIAM Journal on Matrix Analysis and Applications
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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 initial stagnation 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.” A 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.