{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "paige-saunders-lsqr-1982", "title": "LSQR: stable sparse linear equations and least squares by bidiagonalization", "summary": "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.\n\nWhen 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.", "boundary": "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.", "area": "Numerical linear algebra", "topic": "Mathematics", "year": 1982, "lane": "HUMAN", "result": "ALGORITHM", "package_id": "paige-saunders-lsqr-1982", "target": { "record_id": "urn:uuid:3754dc64-dc14-4e2f-a574-0f9dea112185", "version": 1, "content_hash": "sha256:4ffe45c9631e2cb795a0f34b524536ce96e6cd5fd7320be479f62f6a6868a556" }, "humans": [ "Christopher C. Paige", "Michael A. Saunders" ], "authors_complete": true, "human_role_note": "Source-listed contributors to this scoped work. Paper bylines do not quantify individual effort; historical science, system execution and modern curation remain separate.", "agents": [], "related": [ "hestenes-stiefel-conjugate-gradients-1952", "saad-schultz-gmres-1986", "paige-saunders-minres-symmlq-1975" ], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "wrs24-paigesaunders-lsqr1982-original", "title": "Paige and Saunders, LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares; exact1982 original", "url": "https://web.stanford.edu/group/SOL/software/lsqr/lsqr-toms82a.pdf" }, { "id": "wrs24-paigesaunders-lsqr1982-crossref", "title": "ACM-deposited exact LSQR1982 algorithm-paper identity", "url": "https://api.crossref.org/works/10.1145%2F355984.355989" } ], "points": "180000", "published_grade": { "id": "paige-saunders-lsqr-1982", "entry_id": "paige-saunders-lsqr-1982", "band": "A", "modifier": null, "target": { "record_id": "urn:uuid:3754dc64-dc14-4e2f-a574-0f9dea112185", "version": 1, "content_hash": "sha256:4ffe45c9631e2cb795a0f34b524536ce96e6cd5fd7320be479f62f6a6868a556" }, "what_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.", "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. 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Saunders", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "formulator" ], "source": { "citation": "Paige and Saunders, LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares; exact1982 original", "url": "https://web.stanford.edu/group/SOL/software/lsqr/lsqr-toms82a.pdf" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "Paige and Saunders, LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares; exact1982 original", "url": "https://web.stanford.edu/group/SOL/software/lsqr/lsqr-toms82a.pdf" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-1", "kind": "source_reference", "source": { "citation": "ACM-deposited exact LSQR1982 algorithm-paper identity", "url": "https://api.crossref.org/works/10.1145%2F355984.355989" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "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.", "domain": "Numerical linear algebra" }, "sources": [ { "citation": "Paige and Saunders, LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares; exact1982 original", "url": "https://web.stanford.edu/group/SOL/software/lsqr/lsqr-toms82a.pdf" }, { "citation": "ACM-deposited exact LSQR1982 algorithm-paper identity", "url": "https://api.crossref.org/works/10.1145%2F355984.355989" } ], "statement": "# LSQR: stable sparse linear equations and least squares by bidiagonalization\n\nPaige 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.\n\nWhen 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.\n\n## Scientific scope\n\nExact 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.\n\n## People and systems\n\nChristopher C. Paige; Michael A. Saunders\n\n\n\n## Original sources\n\n- [Paige and Saunders, LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares; exact1982 original](https://web.stanford.edu/group/SOL/software/lsqr/lsqr-toms82a.pdf)\n- [ACM-deposited exact LSQR1982 algorithm-paper identity](https://api.crossref.org/works/10.1145%2F355984.355989)\n", "summary": "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.\n\nWhen 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.", "title": "LSQR: stable sparse linear equations and least squares by bidiagonalization" }, "record_id": "urn:uuid:3754dc64-dc14-4e2f-a574-0f9dea112185", "version": 1, "source_ref": { "record_id": "urn:uuid:3754dc64-dc14-4e2f-a574-0f9dea112185", "version": 1, "content_hash": "sha256:4ffe45c9631e2cb795a0f34b524536ce96e6cd5fd7320be479f62f6a6868a556" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:588c257def7c09d0fdf359ab2b15b378defa1d25ca1bbcfb78f1faca395c805e" }, "target": { "record_id": "urn:uuid:3754dc64-dc14-4e2f-a574-0f9dea112185", "version": 1, "content_hash": "sha256:4ffe45c9631e2cb795a0f34b524536ce96e6cd5fd7320be479f62f6a6868a556" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "paige-saunders-lsqr-1982", "entry_id": "paige-saunders-lsqr-1982", "revision": 1, "title": "LSQR: stable sparse linear equations and least squares by bidiagonalization", "context": "Numerical linear algebra", "track": "Numerical linear algebra", "band": "A", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-04", "evidence_cutoff": "2026-10-04", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_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.", "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_summary": "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.", "references": [ { "title": "Paige and Saunders, LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares; exact 1982 original", "url": "https://web.stanford.edu/group/SOL/software/lsqr/lsqr-toms82a.pdf", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." }, { "title": "ACM-deposited exact LSQR 1982 algorithm-paper identity", "url": "https://api.crossref.org/works/10.1145%2F355984.355989", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." } ], "roles": [ { "who": "Christopher C. Paige", "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.", "credit": "SCIENCE" }, { "who": "Michael A. 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