{"paper":{"title":"The Shifting Technique for Computing the Extreme Solutions of $X + A^\\top X^{-1} A = Q$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Chun-Yueh Chiang, Matthew M. Lin","submitted_at":"2013-02-27T12:56:33Z","abstract_excerpt":"We propose a new way for speeding up the search of the maximal solution $X_+$ of $X + A^\\top X^{-1} A = Q$. It is known that the speed of convergence of traditional approaches for solving this problem depends highly on the spectral radius $\\rho(X_+^{-1}A)$. If $\\rho(X_+^{-1}A)$ is close to one or equal to one, the iterations of traditional approaches converges very slowly or does not converge. Our goal is to come up with a shifting tactic to remove the singularities embedded in $\\rho(X_+^{-1}A)$. Finally, an example is used to demonstrate the capacity of our method."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.6752","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}