{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T07:41:22Z","timestamp":1694590882074},"reference-count":29,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"crossref","award":["MTM2010-18674"],"award-info":[{"award-number":["MTM2010-18674"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>We propose a parallel preconditioner for the Newton method in the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners starting from an enhanced approximate inverse RFSAI (Bergamaschi and Mart\u00ednez, 2012) and enriched by a BFGS-like update formula is proposed to accelerate the preconditioned conjugate gradient solution of the linearized Newton system to solve<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>A<\/mml:mi><mml:mi mathvariant=\"bold\">u<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>q<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi mathvariant=\"bold\">u<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mi mathvariant=\"bold\">u<\/mml:mi><\/mml:math>,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>q<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi mathvariant=\"bold\">u<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>being the Rayleigh quotient. In a previous work (Bergamaschi and Mart\u00ednez, 2013) the sequence of preconditioned Jacobians is proven to remain close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to 1.5 million unknowns account for the efficiency and the scalability of the proposed low rank update of the RFSAI preconditioner. The overall RFSAI-BFGS preconditioned Newton algorithm has shown comparable efficiencies with a well-established eigenvalue solver on all the test problems.<\/jats:p>","DOI":"10.1155\/2013\/767042","type":"journal-article","created":{"date-parts":[[2013,9,26]],"date-time":"2013-09-26T21:12:02Z","timestamp":1380229922000},"page":"1-10","source":"Crossref","is-referenced-by-count":4,"title":["Parallel RFSAI-BFGS Preconditioners for Large Symmetric Eigenproblems"],"prefix":"10.1155","volume":"2013","author":[{"given":"L.","family":"Bergamaschi","sequence":"first","affiliation":[{"name":"Department of Civil, Environmental, and Architectural Engineering, University of Padua, Via Trieste 63, 35100 Padova, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"Mart\u00ednez","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Padua, Via Trieste 63, 35100 Padova, 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