Algebraic properties of projected problems in LSQR

Abstract

AbstractLSQR represents a standard Krylov projection method for the solution of systems of linear algebraic equations, linear approximation problems or regularization of discrete inverse problem. Its convergence properties (residual norms, error norms, influence of finite precision arithmetic etc.) have been widely studied. It has been observed that the components of the solution of the projected bidiagonal problem typically increase and their sign alternates. This behavior is the core of approximation properties of LSQR and is observed also for hybrid LSQR with inner Tikhonov regularization. Here we provide rigorous analysis of sign changes and monotonicity of individual components of projected solutions and projected residuals in LSQR. The results hold also for Hybrid LSQR with a fixed inner regularization parameter. The derivations do not rely on maintaining orthogonality in Krylov bases determined by the bidiagonalization process. Numerical illustration is included.