AWhat changed
Paige and Saunders present LSQR, using Golub-Kahan bidiagonalization and orthogonal transformations to solve sparse rectangular linear equations and least-squares problems using matrix and transpose actions, with inexpensive norm estimates and data-aware stopping criteria.
Scientific scope
Paige and Saunders present LSQR, using Golub-Kahan bidiagonalization and orthogonal transformations to solve sparse rectangular linear equations and least-squares problems using matrix and transpose actions, with inexpensive norm estimates and data-aware stopping criteria. Exact March 1982 ACM TOMS 8(1),43-71 algorithm paper; companion Algorithm 583 is a separate June 1982 software article. Real rectangular matrices need not have fullcolumnrank; exact zero-start vectors remain in range(A^T), giving minimum-length solutions on completed undamped solves. Floating-point rank-deficient iterations can instead grow rapidly; this is explicitly discussed. Algebraic CGLS equivalence is not numerical identity or automatic stability. Exact residual minimization does not make the normal-residual norm ||A^Tr|| monotonically decrease. Residual-based backward-error acceptance is not a forward-error or scientific-correctness certificate. Some norm/condition estimates rely on ideal orthogonality and visible singular directions. Standard-error estimates assume fullcolumnrank, m>n and sufficient convergence; source warns their accuracy cannot be guaranteed. CONLIM regularization is heuristic. Damping selection is outside this paper, and original comparisons do not establish universal solver superiority. Later LSMR/software/preconditioning retain separate credit. Original experimental data/figures and companion code were not independently replayed.
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, LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares; exact 1982 original
- ACM-deposited exact LSQR 1982 algorithm-paper identity
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