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

Sou-Cheng Choi (scchoi***at***stanford.edu)
Christopher Paige (paige***at***cs.mcgill.ca)
Michael Saunders (saunders***at***stanford.edu)

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.

Keywords: MINRES, Krylov subspace method, Lanczos process, conjugate-gradient method, minimum-residual method, ill-posed problem, singular least-squares problem, sparse matrix

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Applications -- Science and Engineering (Statistics )

Category 3: Optimization Software and Modeling Systems

Citation: Systems Optimization Laboratory (SOL), Stanford University, March 2010

Download: [PDF]

Entry Submitted: 03/16/2010
Entry Accepted: 03/16/2010
Entry Last Modified: 04/03/2011

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