{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T06:12:01Z","timestamp":1764655921280},"reference-count":0,"publisher":"University of Zielona G\u00f3ra, Poland","issue":"4","license":[{"start":{"date-parts":[[2015,12,1]],"date-time":"2015-12-01T00:00:00Z","timestamp":1448928000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The Lanczos algorithm is among the most frequently used iterative techniques for computing a few dominant eigenvalues of a large sparse non-symmetric matrix. At the same time, it serves as a building block within biconjugate gradient (BiCG) and quasi-minimal residual (QMR) methods for solving large sparse non-symmetric systems of linear equations. It is well known that, when implemented on distributed-memory computers with a huge number of processes, the synchronization time spent on computing dot products increasingly limits the parallel scalability. Therefore, we propose synchronization-reducing variants of the Lanczos, as well as BiCG and QMR methods, in an attempt to mitigate these negative performance effects. These so-called <jats:italic>s<\/jats:italic>-step algorithms are based on grouping dot products for joint execution and replacing time-consuming matrix operations by efficient vector recurrences. The purpose of this paper is to provide a rigorous derivation of the recurrences for the <jats:italic>s<\/jats:italic>-step Lanczos algorithm, introduce <jats:italic>s<\/jats:italic>-step BiCG and QMR variants, and compare the parallel performance of these new <jats:italic>s<\/jats:italic>-step versions with previous algorithms.<\/jats:p>","DOI":"10.1515\/amcs-2015-0055","type":"journal-article","created":{"date-parts":[[2016,1,29]],"date-time":"2016-01-29T12:12:01Z","timestamp":1454069521000},"page":"769-785","source":"Crossref","is-referenced-by-count":5,"title":["The Non\u2013Symmetric <i>s<\/i>\u2013Step Lanczos Algorithm: Derivation of Efficient Recurrences and Synchronization\u2013Reducing Variants of BiCG and QMR"],"prefix":"10.61822","volume":"25","author":[{"given":"Stefan","family":"Feuerriegel","sequence":"first","affiliation":[{"name":"Chair for Information Systems Research, University of Freiburg, Platz der Alten Synagoge, 79098 Freiburg, Germany"}]},{"given":"H. Martin","family":"B\u00fccker","sequence":"additional","affiliation":[{"name":"Chair for Advanced Computing, Friedrich Schiller University Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany"}]}],"member":"37438","published-online":{"date-parts":[[2015,12,30]]},"container-title":["International Journal of Applied Mathematics and Computer Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/amcs\/25\/4\/article-p769.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/amcs-2015-0055","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,15]],"date-time":"2024-05-15T22:56:51Z","timestamp":1715813811000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/amcs-2015-0055"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,1]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,12,30]]},"published-print":{"date-parts":[[2015,12,1]]}},"alternative-id":["10.1515\/amcs-2015-0055"],"URL":"https:\/\/doi.org\/10.1515\/amcs-2015-0055","relation":{},"ISSN":["2083-8492"],"issn-type":[{"value":"2083-8492","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,12,1]]}}}