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arXiv · 1003.4042

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

Sou-Cheng T. Choi·Christopher C. Paige·Michael A. Saunders
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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.

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BibTeXRIS

Sou-Cheng T. Choi, Christopher C. Paige, Michael A. Saunders. 2010-03-22. MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems. https://doi.org/10.1137/100787921

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