MINRES-QLP: A Krylov Subspace Method for Indefinite or Singular Symmetric Systems

Abstract

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 uppertridiagonal). 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.

CG Versus MINRES: An Empirical Comparison

“…From (11) First, on the nonsingular indefinite system 2 1 1 0 More generally, we can gain an impression of the behavior of k x by recalling from (Choi et al, 2011) the connection between MINRES and MINRES-QLP. Both methods compute the iterates = …”

Application of Krylov based methods in calibration for radio astronomy

“…We will now show how this matrix vector product can be performed in an efficient way. We have chosen minresQLP because it is capable of handling singular matrices which adds robustness during the iterations [5].…”

“…In this paper we will show that the matrices used during the estimation process possess a strong Khatri-Rao structure which, combined with Krylov subspace based methods, like minresQLP [5], can be exploited to reduce the computational † This research was supported by NWO-TOP 2010, 614.00.005. costs and achieve accurate results with low complexity and fast convergence rate, without using an alternating approach. Another advantage of the proposed method is a tremendous reduction in the memory usage which could be desired in some applications.…”

MINRES-QLP Pack and Reliable Reproducible Research via Supportable Scientific Software

“…To carefully handle singularity and exploit various symmetry structures, we have designed a suite of MINRES-QLP [19,20,21,22] algorithms, which can constructively reveal (numerical) singularity and compatibility of a given linear system of equations; users do not have to know these properties a priori. (We are also currently developing two related algorithms known as GMRES-QLP and GMRES-URV for unsymmetric singular square systems; see [19,23].…”

“…She is grateful to Michael Saunders and Chris Paige, her co-authors of MINRES-QLP [20,21]. She also thanks Fred Hickernell for introducing her to Doctest [31], co-developing the definition of SSS and the principles of RRR via SSS during the development of GAIL [10,11] and the course Reliable Mathematical Software [29].…”