MINRES-QLP: A KRYLOV SUBSPACE METHOD FOR INDEFINITE OR SINGULAR SYMMETRIC SYSTEMS

• Published in: SIAM journal on scientific computing Vol. 33; no. 3-4; pp. 1810 - 1836
• Main Authors: CHOI, Sou-Cheng T, PAIGE, Christopher C, SAUNDERS, Michael A
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Combinatorics; Combinatorics. Ordered structures; Decomposition; Designs and configurations; Eigenvalues; Exact sciences and technology; Grants; Mathematics; Methods; Methods of scientific computing (including symbolic computation, algebraic computation); Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; MINRES; Decomposition method; Krylov subspace method; minimum-residual method; Non linear equation; 65F50; Lanczos process; 65F10; singular least-squares problem; Least squares method; 93E24; Sparse matrix; 65F35; Conjugate gradient method; Transcendental equation; Numerical linear algebra; Least squares problem; conjugate-gradient method; Pseudoinverse; 15A06; Algorithm; Equation system; Numerical analysis; Scientific computation; overdetermined system; 65F22; Symmetric system; 65F20; Algebraic equation; Condition number; 65F25; Tridiagonal matrix; Preconditioning
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/100787921

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.