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. Spantini, Alessio; Cui, Tiangang; Willcox, Karen; Tenorio, Luis; Marzouk, Youssef: Goal-oriented optimal approximations of Bayesian linear inverse problems (2017)
  2. Duintjer Tebbens, Jurjen; Meurant, Gérard: On the convergence of Q-OR and Q-MR Krylov methods for solving nonsymmetric linear systems (2016)
  3. Greif, C.; Paige, C.C.; Titley-Peloquin, D.; Varah, J.M.: Numerical equivalences among Krylov subspace algorithms for skew-symmetric matrices (2016)
  4. Potschka, Andreas: Backward step control for global Newton-type methods (2016)
  5. 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)
  6. Vecharynski, Eugene; Knyazev, Andrew: Preconditioned steepest descent-like methods for symmetric indefinite systems (2016)
  7. Zhang, Jianhua; Dai, Hua: A transpose-free quasi-minimal residual variant of the CORS method for solving non-Hermitian linear systems (2015)
  8. Choi, Sou-Cheng T.; Saunders, Michael A.: Algorithm 937: MINRES-QLP for symmetric and Hermitian linear equations and least-squares problems (2014)
  9. Gould, Nick; Orban, Dominique; Rees, Tyrone: Projected Krylov methods for saddle-point systems (2014)
  10. Haelterman, R.; Petit, J.; Lauwens, B.; Bruyninckx, H.; Vierendeels, J.: On the non-singularity of the quasi-Newton-least squares method (2014)
  11. Moré, Jorge J.; Wild, Stefan M.: Do you trust derivatives or differences? (2014)
  12. Choi, Sou-Cheng T.; Paige, Christopher C.; Saunders, Michael A.: MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems (2011)
  13. Jiang, Xiaoye; Lim, Lek-Heng; Yao, Yuan; Ye, Yinyu: Statistical ranking and combinatorial Hodge theory (2011)
  14. Choi, Sou-Cheng T.; Paige, Christopher C.; Saunders, Michael A.: MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems (2010) ioport
  15. Olshanskii, M.A.; Simoncini, V.: Acquired clustering properties and solution of certain saddle point systems (2010)
  16. Chang, X.-W.; Paige, C.C.; Titley-Peloquin, D.: Stopping criteria for the iterative solution of linear least squares problems (2009)
  17. Curtis, Frank E.; Nocedal, Jorge; Wächter, Andreas: A matrix-free algorithm for equality constrained optimization problems with rank-deficient Jacobians (2009)
  18. Paige, C.C.; Saunders, M.A.: Solution of sparse indefinite systems of linear equations (1975)