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

Abstract. CG, SYMMLQ, and MINRES are Krylov subspace methods for solving large symmetric systems of linear equations. CG (the conjugate-gradient method) is reliable on positive-definite systems, while SYMMLQ and MINRES are designed for indefinite systems. 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 solution (often called the 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). Our algorithm uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form), and so we call it MINRES-QLP. On singular or nonsingular systems, MINRES-QLP can give ...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Sou-Cheng T. Choi, Christopher C. Paige, Michael A. Saunders
• Keywords: CORR 2010, Education, Krylov Subspace Method, Solution, Systems