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. Dahito, Marie-Ange; Orban, Dominique: The conjugate residual method in linesearch and trust-region methods (2019)
  2. Estrin, Ron; Orban, Dominique; Saunders, Michael: Euclidean-norm error bounds for SYMMLQ and CG (2019)
  3. Estrin, Ron; Orban, Dominique; Saunders, Michael A.: LNLQ: an iterative method for least-norm problems with an error minimization property (2019)
  4. Estrin, Ron; Orban, Dominique; Saunders, Michael A.: LSLQ: an iterative method for linear least-squares with an error minimization property (2019)
  5. Manguoğlu, Murat; Mehrmann, Volker: A robust iterative scheme for symmetric indefinite systems (2019)
  6. Paige, Christopher C.: Accuracy of the Lanczos process for the eigenproblem and solution of equations (2019)
  7. Hallman, Eric; Gu, Ming: LSMB: minimizing the backward error for least-squares problems (2018)
  8. Spantini, Alessio; Cui, Tiangang; Willcox, Karen; Tenorio, Luis; Marzouk, Youssef: Goal-oriented optimal approximations of Bayesian linear inverse problems (2017)
  9. Duintjer Tebbens, Jurjen; Meurant, Gérard: On the convergence of Q-OR and Q-MR Krylov methods for solving nonsymmetric linear systems (2016)
  10. Greif, C.; Paige, C. C.; Titley-Peloquin, D.; Varah, J. M.: Numerical equivalences among Krylov subspace algorithms for skew-symmetric matrices (2016)
  11. Potschka, Andreas: Backward step control for global Newton-type methods (2016)
  12. Trinath, G.; Babu, V.: On the solution of the Neumann Poisson problem arising from a compact differencing scheme using the full multi-grid method (2016)
  13. Vecharynski, Eugene; Knyazev, Andrew: Preconditioned steepest descent-like methods for symmetric indefinite systems (2016)
  14. Zhang, Jianhua; Dai, Hua: A transpose-free quasi-minimal residual variant of the CORS method for solving non-Hermitian linear systems (2015)
  15. Choi, Sou-Cheng T.; Saunders, Michael A.: Algorithm 937: MINRES-QLP for symmetric and Hermitian linear equations and least-squares problems (2014)
  16. Gould, Nick; Orban, Dominique; Rees, Tyrone: Projected Krylov methods for saddle-point systems (2014)
  17. Haelterman, R.; Petit, J.; Lauwens, B.; Bruyninckx, H.; Vierendeels, J.: On the non-singularity of the quasi-Newton-least squares method (2014)
  18. Moré, Jorge J.; Wild, Stefan M.: Do you trust derivatives or differences? (2014)
  19. Choi, Sou-Cheng T.; Paige, Christopher C.; Saunders, Michael A.: MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems (2011)
  20. Jiang, Xiaoye; Lim, Lek-Heng; Yao, Yuan; Ye, Yinyu: Statistical ranking and combinatorial Hodge theory (2011)

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