{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:47:52Z","timestamp":1760240872672,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,10,16]],"date-time":"2019-10-16T00:00:00Z","timestamp":1571184000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Nature Science Foundation of China","award":["11701225, 1370147, 11501085 and 61402082"],"award-info":[{"award-number":["11701225, 1370147, 11501085 and 61402082"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["JBK1809003 and ZYGX2016J138"],"award-info":[{"award-number":["JBK1809003 and ZYGX2016J138"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20170173"],"award-info":[{"award-number":["BK20170173"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.<\/jats:p>","DOI":"10.3390\/sym11101302","type":"journal-article","created":{"date-parts":[[2019,10,17]],"date-time":"2019-10-17T04:46:06Z","timestamp":1571287566000},"page":"1302","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9125-9530","authenticated-orcid":false,"given":"Hong-Xiu","family":"Zhong","sequence":"first","affiliation":[{"name":"School of Science, Jiangnan University, Wuxi 214122, Jiangsu, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7895-2050","authenticated-orcid":false,"given":"Xian-Ming","family":"Gu","sequence":"additional","affiliation":[{"name":"School of Economic Mathematics\/Institute of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, Sichuan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shao-Liang","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Karas\u00f6zen, B., Manguo\u01e7lu, M., Tezer-Sezgin, M., G\u00f6ktepe, S., and U\u01e7ur, \u00d6. (2016). Block variants of the COCG and COCR methods for solving complex symmetric linear systems with multiple right-hand sides. Numerical Mathematics and Advanced Applications ENUMATH 2015, Springer International Publishing. Lecture Notes in Computational Science and Engineering 112.","DOI":"10.1007\/978-3-319-39929-4"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/j.cam.2005.07.032","article-title":"A COCR method for solving complex symmetric linear systems","volume":"199","author":"Sogabe","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_3","unstructured":"Siddiqi, A.H., Duff, I.S., and Christensen, O. (2006). Block Krylov space methods for linear systems with multiple right-hand sides: An introduction. 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