Implementation of a conjugate-gradient type method for solving sparse linear equations: Solve Ax=b or (A−sI)x=b. The matrix A−sI must be symmetric but it may be definite or indefinite or singular. The scalar s is a shifting parameter -- it may be any number. The method is based on Lanczos tridiagonalization. You may provide a preconditioner, but it must be positive definite. MINRES is really solving one of the least-squares problems minimize ||Ax−b|| or ||(A−sI)x−b||. If A is singular (and s=0), MINRES returns a least-squares solution with small ||Ar|| (where r=b−Ax), but in general it is not the minimum-length solution. To get the min-length solution, use MINRES-QLP [2,3]. Similarly if A−sI is singular.

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  1. Bergou, El Houcine; Diouane, Youssef; Kungurtsev, Vyacheslav; Royer, Clément W.: A nonmonotone matrix-free algorithm for nonlinear equality-constrained least-squares problems (2021)
  2. Claeys, Xavier; Giacomel, Lorenzo; Hiptmair, Ralf; Urzúa-Torres, Carolina: Quotient-space boundary element methods for scattering at complex screens (2021)
  3. Frye, Charles G.; Simon, James; Wadia, Neha S.; Ligeralde, Andrew; Deweese, Michael R.; Bouchard, Kristofer E.: Critical point-finding methods reveal gradient-flat regions of deep network losses (2021)
  4. Liu, Yang; Roosta, Fred: Convergence of Newton-MR under inexact Hessian information (2021)
  5. Montoison, Alexis; Orban, Dominique: TriCG and TriMR: two iterative methods for symmetric quasi-definite systems (2021)
  6. Al-Baali, Mehiddin; Caliciotti, Andrea; Fasano, Giovanni; Roma, Massimo: A class of approximate inverse preconditioners based on Krylov-subspace methods for large-scale nonconvex optimization (2020)
  7. He, Qinglong; Wang, Yanfei: Inexact Newton-type methods based on Lanczos orthonormal method and application for full waveform inversion (2020)
  8. Il’in, V. P.: On moment methods in Krylov subspaces (2020)
  9. Kalantzis, Vassilis: A domain decomposition Rayleigh-Ritz algorithm for symmetric generalized eigenvalue problems (2020)
  10. Lim, Lek-Heng: Hodge Laplacians on graphs (2020)
  11. Montoison, Alexis; Orban, Dominique: BiLQ: an iterative method for nonsymmetric linear systems with a quasi-minimum error property (2020)
  12. Schork, Lukas; Gondzio, Jacek: Implementation of an interior point method with basis preconditioning (2020)
  13. Zuo, Qian; He, Ying: Preconditioned GMRES method for a class of Toeplitz linear systems in fractional eigenvalue problems (2020)
  14. Dahito, Marie-Ange; Orban, Dominique: The conjugate residual method in linesearch and trust-region methods (2019)
  15. Estrin, Ron; Orban, Dominique; Saunders, Michael: Euclidean-norm error bounds for SYMMLQ and CG (2019)
  16. Estrin, Ron; Orban, Dominique; Saunders, Michael A.: LNLQ: an iterative method for least-norm problems with an error minimization property (2019)
  17. Estrin, Ron; Orban, Dominique; Saunders, Michael A.: LSLQ: an iterative method for linear least-squares with an error minimization property (2019)
  18. Manguoğlu, Murat; Mehrmann, Volker: A robust iterative scheme for symmetric indefinite systems (2019)
  19. Paige, Christopher C.: Accuracy of the Lanczos process for the eigenproblem and solution of equations (2019)
  20. Hallman, Eric; Gu, Ming: LSMB: minimizing the backward error for least-squares problems (2018)

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