singular least-squares problem – Optimization Online

Minimal Residual Methods for Complex Symmetric, Skew Symmetric, and Skew Hermitian Systems

Published: 2013/05/07

Convex Optimization, Optimization Software and Modeling Systems, Robust Optimization complex symmetric, conjugate gradient method, krylov subspace method, lanczos process, minimum-residual method, minres, minres-qlp, preconditioner, singular least-squares problem, skew hermitian, skew symmetric, sparse matrix, structured matrices

While there is no lack of efficient Krylov subspace solvers for Hermitian systems, there are few for complex symmetric, skew symmetric, or skew Hermitian systems, which are increasingly important in modern applications including quantum dynamics, electromagnetics, and power systems. For a large consistent complex symmetric system, one may apply a non-Hermitian Krylov subspace method disregarding … Read more

MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems

Published: 2010/03/16, Updated: 2011/04/03

Sou-Cheng Choi

Christopher Paige

Michael A. Saunders

Convex Optimization, Optimization Software and Modeling Systems, Statistics conjugate gradient method, ill--posed problem, krylov subspace method, lanczos process, minimum-residual method, minres, singular least-squares problem, sparse matrix

CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ’s solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This … Read more