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Research Project","award":["60973015","61170311","11126103","11201055","11171054","2011JY0002","20120185120026"],"award-info":[{"award-number":["60973015","61170311","11126103","11201055","11171054","2011JY0002","20120185120026"]}]},{"name":"Doctoral Program of Higher Education","award":["60973015","61170311","11126103","11201055","11171054","2011JY0002","20120185120026"],"award-info":[{"award-number":["60973015","61170311","11126103","11201055","11171054","2011JY0002","20120185120026"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>The Biconjugate<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:math>-Orthogonal Residual (BiCOR) method carried out in finite precision arithmetic by means of the biconjugate<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:math>-orthonormalization procedure may possibly tend to suffer from two sources of numerical instability, known as two kinds of breakdowns, similarly to those of the Biconjugate Gradient (BCG) method. This paper naturally exploits the composite step strategy employed in the development of the composite step BCG (CSBCG) method into the BiCOR method to cure one of the breakdowns called as pivot breakdown. Analogously to the CSBCG method, the resulting interesting variant, with only a minor modification to the usual implementation of the BiCOR method, is able to avoid near pivot breakdowns and compute all the well-defined BiCOR iterates stably on the assumption that the underlying biconjugate<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:math>-orthonormalization procedure does not break down. Another benefit acquired is that it seems to be a viable algorithm providing some further practically desired smoothing of the convergence history of the norm of the residuals, which is justified by numerical experiments. In addition, the exhibited method inherits the promising advantages of the empirically observed stability and fast convergence rate of the BiCOR method over the BCG method so that it outperforms the CSBCG method to some extent.<\/jats:p>","DOI":"10.1155\/2013\/408167","type":"journal-article","created":{"date-parts":[[2013,2,20]],"date-time":"2013-02-20T22:36:18Z","timestamp":1361399778000},"page":"1-16","source":"Crossref","is-referenced-by-count":1,"title":["Exploiting the Composite Step Strategy to the Biconjugate<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:math>-Orthogonal Residual Method for Non-Hermitian Linear Systems"],"prefix":"10.1155","volume":"2013","author":[{"given":"Yan-Fei","family":"Jing","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ting-Zhu","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bruno","family":"Carpentieri","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Computing Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong","family":"Duan","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","first-page":"1155","volume":"43","year":"2005","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1002\/nla.499"},{"key":"3","first-page":"22","volume":"2","year":"2000","journal-title":"Computer Science and Engineering"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2010.12.009"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970937"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718003"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2009.05.022"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.2528\/PIER09101901"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2010.07.034"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1137\/100794031"},{"key":"11","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BFb0080116","volume-title":"Conjugate gradient methods for indefinite systems","volume":"506","year":"1976"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1137\/0613056"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1137\/0613037"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479890188803"},{"volume-title":"Block Krylov space methods for linear systems withmultiple right-hand sides: an introduction","year":"2006","key":"16"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1137\/0117118"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385699"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1007\/BF02141258"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385726"},{"volume-title":"The unsymmetric Lanczos algorithms and their relations to Pade approximation, continued fraction and the QD algorithm","year":"1990","key":"21"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.2307\/2007796"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1137\/0613036"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1007\/BF02142326"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385846"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(96)00217-9"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050439"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.1137\/0914009"},{"key":"30","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492900002737"},{"key":"33","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2008.05.018"},{"first-page":"309","volume-title":"A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations","year":"1976","key":"31"},{"key":"32","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144504445376"},{"issue":"23","key":"34","volume":"97","year":"1997","journal-title":"NA Digest"},{"key":"35","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(98)00117-2"},{"key":"36","doi-asserted-by":"publisher","DOI":"10.1007\/BF03041466"},{"key":"37","doi-asserted-by":"publisher","DOI":"10.1137\/040603917"},{"key":"38","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7176"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/408167.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/408167.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/408167.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,7]],"date-time":"2020-05-07T08:11:40Z","timestamp":1588839100000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2013\/408167\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":36,"alternative-id":["408167","408167"],"URL":"https:\/\/doi.org\/10.1155\/2013\/408167","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2013]]}}}