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SouCheng Choi (scchoistanford.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 leastsquares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a leastsquares solution but not necessarily the minimumlength (pseudoinverse) solution. This understanding motivates us to design a MINRESlike algorithm to compute minimumlength solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is uppertridiagonal). MINRESQLP uses a QLP decomposition (where rotations on the right reduce R to lowertridiagonal form). On illconditioned systems (singular or not), MINRESQLP can give more accurate solutions than MINRES. We derive preconditioned MINRESQLP, 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, conjugategradient method, minimumresidual method, illposed problem, singular leastsquares 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 Modify/Update this entry  
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