AWhat changed
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.
Scientific scope
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. 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.
Evidence
Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.
Contributors
- Christopher C. Paige · SCIENCE: Original source-listed scientific author of this scoped work; byline does not determine effort fractions.
- Michael A. Saunders · SCIENCE: Original source-listed scientific author of this scoped work; byline does not determine effort fractions.
- Paige and Saunders, Solution of Sparse Indefinite Systems of Linear Equations; exact 1975 original
- SIAM-deposited exact 1975 Paige/Saunders identity
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