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

TL;DR

MINRES-QLP is a new Krylov subspace method designed to compute minimum-length solutions for singular symmetric systems, improving accuracy over MINRES especially on ill-conditioned problems.

Contribution

The paper introduces MINRES-QLP, a novel algorithm that extends MINRES with QLP decomposition to handle singular systems and provide more accurate solutions.

Findings

MINRES-QLP computes more accurate solutions on ill-conditioned systems.

It provides better estimates of residuals, norms, and condition numbers.

Preconditioned versions and new stopping rules enhance its practical utility.

Abstract

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 understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.