What is recorded here?
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. When measurements outnumber unknown quantities, they rarely all agree exactly. Least squares finds the vector that makes their combined mismatch as small as possible. For a huge sparse matrix, storing a dense factorization or explicitly forming the normal equations can be costly and can worsen numerical conditioning. LSQR builds a much smaller bidiagonal problem using only matrix actions and transpose actions, then solves that problem with orthogonal transformations. It also estimates useful norms and supplies stopping tests tied to the accuracy of the input data. This supports tasks such as large gravity-meter adjustments and numerical modeling. A small residual still needs interpretation: an ill-conditioned system can have a sensitive solution, and uncertainty estimates need additional assumptions. The method improves an important computational route without turning an approximate fit into a guarantee that the underlying model is correct.
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Scientific scope
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.
Read the original work
- Paige and Saunders, LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares; exact1982 original
- ACM-deposited exact LSQR1982 algorithm-paper identity
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People and systems
Christopher C. Paige; Michael A. Saunders
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English catalogue narrative
LSQR: stable sparse linear equations and least squares by bidiagonalization
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.
When measurements outnumber unknown quantities, they rarely all agree exactly. Least squares finds the vector that makes their combined mismatch as small as possible. For a huge sparse matrix, storing a dense factorization or explicitly forming the normal equations can be costly and can worsen numerical conditioning. LSQR builds a much smaller bidiagonal problem using only matrix actions and transpose actions, then solves that problem with orthogonal transformations. It also estimates useful norms and supplies stopping tests tied to the accuracy of the input data. This supports tasks such as large gravity-meter adjustments and numerical modeling. A small residual still needs interpretation: an ill-conditioned system can have a sensitive solution, and uncertainty estimates need additional assumptions. The method improves an important computational route without turning an approximate fit into a guarantee that the underlying model is correct.
Scientific scope
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.
People and systems
Christopher C. Paige; Michael A. Saunders