{"paper":{"title":"Inexact versions of several block-splitting preconditioners for indefinite least squares problems","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"Inexact block-splitting preconditioners confine all eigenvalues of the preconditioned matrix to the unit disk centered at 1 for indefinite least squares systems.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Davod Khojasteh Salkuyeh, Mohaddese Kaveh Shaldehi","submitted_at":"2026-02-28T02:37:08Z","abstract_excerpt":"This paper introduces inexact versions of several block-splitting preconditioners for solving the three-by-three block linear systems arising from a special class of indefinite least squares problems. We first establish the convergence conditions for the corresponding stationary iterative methods. Then, it follows that under these conditions, all eigenvalues of the preconditioned matrices are contained within a circle centered at $(1,0)$ with radius $1$. This property implies that these preconditioners are effective in accelerating the convergence of the GMRES method. Furthermore, we analyze t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"under these conditions, all eigenvalues of the preconditioned matrices are contained within a circle centered at (1,0) with radius 1. This property implies that these preconditioners are effective in accelerating the convergence of the GMRES method.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The convergence conditions established for the stationary iterative methods hold for the special class of indefinite least squares problems considered.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Inexact block-splitting preconditioners are introduced for indefinite least squares problems, with eigenvalue bounds that limit GMRES iterations and numerical tests confirming faster convergence.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Inexact block-splitting preconditioners confine all eigenvalues of the preconditioned matrix to the unit disk centered at 1 for indefinite least squares systems.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4a9ade4bc8cc8773936d1d4225460df03b53a78f2c2bd5cac94bbfa686d97a0a"},"source":{"id":"2603.00419","kind":"arxiv","version":2},"verdict":{"id":"5608b7bd-5f3e-4a50-8ee8-325772c0f850","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T18:51:12.936223Z","strongest_claim":"under these conditions, all eigenvalues of the preconditioned matrices are contained within a circle centered at (1,0) with radius 1. This property implies that these preconditioners are effective in accelerating the convergence of the GMRES method.","one_line_summary":"Inexact block-splitting preconditioners are introduced for indefinite least squares problems, with eigenvalue bounds that limit GMRES iterations and numerical tests confirming faster convergence.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The convergence conditions established for the stationary iterative methods hold for the special class of indefinite least squares problems considered.","pith_extraction_headline":"Inexact block-splitting preconditioners confine all eigenvalues of the preconditioned matrix to the unit disk centered at 1 for indefinite least squares systems."},"references":{"count":23,"sample":[{"doi":"","year":1994,"title":"Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994","work_id":"9eebae1f-b3a4-4251-ba9d-d0307998c04b","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"M. Benzi, G.H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl. 26 (2004) 20–41","work_id":"019900e6-6f64-4ccf-854a-a74f1bf7a49f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numer. 26 (2004) 1-137","work_id":"b60779cd-c97a-430e-bc3e-2f2980c27149","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"A. Bojanczyk, N.J. Higham, H. Patel, Solving the indefinite least squares problem by hyperbolic QR factorization, SIAM J. Matrix Anal. Appl. 24 (2003) 914–931","work_id":"78a405d8-853e-439f-af3b-ee60f7f69fdb","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Bojanczyk, Algorithms for indefinite linear least squares problems, Linear Algebra Appl","work_id":"32012592-7423-4ef8-8db1-ef5f7cfb33c4","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":23,"snapshot_sha256":"04708d92dbbdabee62ca2d2b4cc10391dea62c6d5c1856907554e0e4ab0e7659","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}