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    "title": "MINRES and SYMMLQ: solving sparse symmetric indefinite systems",
    "summary": "Paige and Saunders derive the Lanczos-based MINRES residual-minimizing and SYMMLQ LQ-factorization methods for large sparse real symmetric linear systems, avoiding the intermediate factorization failures that can defeat ordinary conjugate gradients on indefinite matrices.\n\nMany scientific calculations ask for a vector x whose matrix action matches measured or modeled data b. Ordinary conjugate gradients works particularly well when the matrix is symmetric positive definite. An indefinite matrix has both positive and negative spectral directions, and intermediate CG steps can become undefined even when the full system has a solution. Paige and Saunders built two alternatives from Lanczos vectors and stable orthogonal factorizations: SYMMLQ and MINRES. These methods use matrix-vector products and a small number of working vectors, which suits large sparse problems. MINRES chooses steps that reduce the residual, the mismatch between Ax and b. That mismatch can be small without x being uniquely determined or accurate, so symmetry, conditioning, the true residual and the chosen solution criterion remain essential.",
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        "description": "Exact September 1975 SIAM Journal on Numerical Analysis 12(4),617-629 contribution. Symmetry is required; arbitrary nonsymmetric systems are outside scope. SYMMLQ as developed requires a consistent system, while MINRES can address inconsistent symmetric equations. Positive definiteness is not required, but conditioning and roundoff still matter. The general-theory minimum-length conclusion assumes trial vectors span the range of A; residual minimization alone and a b-starting Lanczos chain do not universally select the minimum-length solution of a singular inconsistent system. Exact-arithmetic finite termination is separate from numerical stopping: computed residual estimates can diverge from true residuals and cause premature stopping, explicitly reported in Section 8. Full rounding-error analysis is deferred to later work. Later SPD preconditioning, reorthogonalization, MINRES-QLP, updated stopping criteria and software contributors retain separate version/credit. Original experiments and plotted data were not independently reproduced.",
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      "statement": "# MINRES and SYMMLQ: solving sparse symmetric indefinite systems\n\nPaige and Saunders derive the Lanczos-based MINRES residual-minimizing and SYMMLQ LQ-factorization methods for large sparse real symmetric linear systems, avoiding the intermediate factorization failures that can defeat ordinary conjugate gradients on indefinite matrices.\n\nMany scientific calculations ask for a vector x whose matrix action matches measured or modeled data b. Ordinary conjugate gradients works particularly well when the matrix is symmetric positive definite. An indefinite matrix has both positive and negative spectral directions, and intermediate CG steps can become undefined even when the full system has a solution. Paige and Saunders built two alternatives from Lanczos vectors and stable orthogonal factorizations: SYMMLQ and MINRES. These methods use matrix-vector products and a small number of working vectors, which suits large sparse problems. MINRES chooses steps that reduce the residual, the mismatch between Ax and b. That mismatch can be small without x being uniquely determined or accurate, so symmetry, conditioning, the true residual and the chosen solution criterion remain essential.\n\n## Scientific scope\n\nExact September 1975 SIAM Journal on Numerical Analysis 12(4),617-629 contribution. Symmetry is required; arbitrary nonsymmetric systems are outside scope. SYMMLQ as developed requires a consistent system, while MINRES can address inconsistent symmetric equations. Positive definiteness is not required, but conditioning and roundoff still matter. The general-theory minimum-length conclusion assumes trial vectors span the range of A; residual minimization alone and a b-starting Lanczos chain do not universally select the minimum-length solution of a singular inconsistent system. Exact-arithmetic finite termination is separate from numerical stopping: computed residual estimates can diverge from true residuals and cause premature stopping, explicitly reported in Section 8. Full rounding-error analysis is deferred to later work. Later SPD preconditioning, reorthogonalization, MINRES-QLP, updated stopping criteria and software contributors retain separate version/credit. Original experiments and plotted data were not independently reproduced.\n\n## People and systems\n\nChristopher C. Paige; Michael A. Saunders\n\n\n\n## Original sources\n\n- [Paige and Saunders, Solution of Sparse Indefinite Systems of Linear Equations; exact1975 original](https://web.stanford.edu/group/SOL/software/symmlq/PS75.pdf)\n- [SIAM-deposited exact1975 Paige/Saunders identity](https://api.crossref.org/works/10.1137%2F0712047)\n",
      "summary": "Paige and Saunders derive the Lanczos-based MINRES residual-minimizing and SYMMLQ LQ-factorization methods for large sparse real symmetric linear systems, avoiding the intermediate factorization failures that can defeat ordinary conjugate gradients on indefinite matrices.\n\nMany scientific calculations ask for a vector x whose matrix action matches measured or modeled data b. Ordinary conjugate gradients works particularly well when the matrix is symmetric positive definite. An indefinite matrix has both positive and negative spectral directions, and intermediate CG steps can become undefined even when the full system has a solution. Paige and Saunders built two alternatives from Lanczos vectors and stable orthogonal factorizations: SYMMLQ and MINRES. These methods use matrix-vector products and a small number of working vectors, which suits large sparse problems. MINRES chooses steps that reduce the residual, the mismatch between Ax and b. That mismatch can be small without x being uniquely determined or accurate, so symmetry, conditioning, the true residual and the chosen solution criterion remain essential.",
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