{"paper":{"title":"Classification of Double Saddle-Point Systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Symmetric double saddle-point systems divide into block-arrow and block-tridiagonal matrix forms that control their invertibility, spectra, and preconditioners.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Chen Greif, Susanne Bradley","submitted_at":"2026-05-13T22:16:17Z","abstract_excerpt":"We offer a classification of a broad and practically relevant class of symmetric double saddle-point system. At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms. We describe relevant applications, invertibility conditions, spectral properties, and block preconditioners. Our discussion is kept within a general framework rather than tailored to specific applications."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The systems under consideration are symmetric double saddle-point systems whose matrices admit the described block structures.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A classification of symmetric double saddle-point systems into block-arrow and block-tridiagonal matrix forms is presented, including invertibility conditions, spectral properties, and block preconditioners.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Symmetric double saddle-point systems divide into block-arrow and block-tridiagonal matrix forms that control their invertibility, spectra, and preconditioners.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e696e0bd602f0aefcaf4b1dc8b512830a2c12fc47664ff57d95092cb3c96d6ed"},"source":{"id":"2605.14157","kind":"arxiv","version":1},"verdict":{"id":"babc917f-d273-4c1f-b880-0d29f1104304","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:52:54.804514Z","strongest_claim":"At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms.","one_line_summary":"A classification of symmetric double saddle-point systems into block-arrow and block-tridiagonal matrix forms is presented, including invertibility conditions, spectral properties, and block preconditioners.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The systems under consideration are symmetric double saddle-point systems whose matrices admit the described block structures.","pith_extraction_headline":"Symmetric double saddle-point systems divide into block-arrow and block-tridiagonal matrix forms that control their invertibility, spectra, and preconditioners."},"references":{"count":86,"sample":[{"doi":"","year":2021,"title":"James H. Adler, Thomas R. Benson, Eric C. Cyr, Patrick E. Farrell, Scott P. MacLachlan, and Ray S. Tuminaro. Monolithic multigrid meth- ods for magnetohydrodynamics.SIAM Journal on Scientific Computin","work_id":"3430b725-83d6-4d64-9900-b021864df9fe","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Safique Ahmad and Pinki Khatun","work_id":"718bd984-c237-40a8-8e32-cea11e15667f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Safique Ahmad and Pinki Khatun","work_id":"68125046-f111-4625-b5f0-ad685b9efecc","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Ali Beik and M","work_id":"ae6663ab-b47e-4149-a311-92ba63e76083","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Antonietti, Jacopo De Ponti, Luca Formaggia, and Anna Scotti","work_id":"3a85891a-fedb-48be-9157-1fcf904998d7","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":86,"snapshot_sha256":"1b0ae7ee4bd71538732448ed7bbe972be777d9e8064f77de3ec39b32a8b45fd1","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"276b63fd0f60f4171ea86f367984e14c1726c7b454c199c0e2f80dea77c60c0b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}