Convex quadratic semidefinite programming (QSDP) has been widely applied in solving engineering and scientific problems such as nearest correlation problems and nearest Euclidean distance matrix problems. In this paper, we study an inexact primal-dual infeasible path-following algorithm for QSDP problems of the form: minX{12X • Q(X) + C •X : A(X) = b, X 0}, where Q is a self-adjoint positive se...

We consider primal-dual IP methods where the linear system arising at each iteration is formulated in the reduced (augmented) form and solved approximately. Focusing on the iterates close to a solution, we analyze the accuracy of the so-called inexact step, i.e., the step that solves the unreduced system, when combining the effects of both different levels of accuracy in the inexact computation...

We consider possibly stiff and implicit systems of ordinary differential equations (ODEs). The major difficulty and computational bottleneck in the implementation of fully implicit Runge–Kutta (IRK) methods resides in the numerical solution of the resulting systems of nonlinear equations. To solve those systems we show that the use of inexact simplified Newton methods is efficient. Linear syste...

We propose a class of regularized Hermitian and skew-Hermitian splitting methods for the solution of large, sparse linear systems in saddle-point form. These methods can be used as stationary iterative solvers or as preconditioners for Krylov subspace methods. We establish unconditional convergence of the stationary iterations and we examine the spectral properties of the corresponding precondi...

A exible version of the QMR algorithm is presented which allows for the use of a diierent preconditioner at each step of the algorithm. In particular, inexact solutions of the preconditioned equations are allowed, as well as the use of some (inner) iterative method as a preconditioner. Several theorems are presented relating the norm of the residual of the new method with the norm of the residu...

We study inexact subspace iteration for solving generalized non-Hermitian eigenvalue problems with spectral transformation, with focus on a few strategies that help accelerate preconditioned iterative solution of the linear systems of equations arising in this context. We provide new insights into a special type of preconditioner with “tuning” that has been studied for this algorithm applied to...

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 main...

The object of this paper is to introduce range-space variants of standard Krylov iterative solvers for unsymmetric and symmetric linear systems, and to discuss how inexact matrix-vector products may be used in this context. The new range-space variants are characterized by possibly much lower storage and computational costs than their full-space counterparts, which is crucial in data assimilati...