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ALGORITHM & DOCUMENTATION: MINRES-QLP for Singular Symmetric and Hermitian Linear Equations and Least-Squares Problems

Sou-Cheng Choi(sctchoi***at***uchicago.edu)

Abstract: We describe algorithm MINRES-QLP and its FORTRAN 90 implementation for solving symmetric or Hermitian linear systems or least-squares problems. If the system is singular, MINRES-QLP computes the unique minimum-length solution (also known as the pseudoinverse solution), which generally eludes MINRES. In all cases, it overcomes a potential instability in the original MINRES algorithm. A positive-definite preconditioner may be supplied. Our FORTRAN 90 implementation illustrates a design pattern that allows users to make problem data known to the solver but hidden and secure from other program units. In particular, we circumvent the need for reverse communication. While we focus here on a FORTRAN 90 implementation, we also provide and maintain MATLAB versions of MINRES and MINRES-QLP.

Keywords: Krylov subspace method, Lanczos process, conjugate-gradient method, singular least-squares, linear equations, minimum-residual method, ill-posed problem, regression, sparse matrix, data encapsulation

Category 1: Optimization Software and Modeling Systems

Category 2: Applications -- Science and Engineering

Category 3: Applications -- OR and Management Sciences

Citation: @techreport{CS12b, AUTHOR = {Choi, Sou-Cheng T. and Saunders, Michael A.}, TITLE = {{ALGORITHM \& DOCUMENTATION: MINRES-QLP} for singular symmetric and {Hermitian} linear equations and least-squares problems}, INSTITUTION = {Computation Institute, University of Chicago}, ADDRESS = {IL}, TYPE = {Technical Report}, number = {ANL/MCS-P3027-0812}, YEAR = 2012 }

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Entry Submitted: 01/12/2013
Entry Accepted: 01/13/2013
Entry Last Modified: 01/12/2013

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