{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:03:35Z","timestamp":1750219415188,"version":"3.41.0"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,8,10]],"date-time":"2024-08-10T00:00:00Z","timestamp":1723248000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,8,10]],"date-time":"2024-08-10T00:00:00Z","timestamp":1723248000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271401"],"award-info":[{"award-number":["12271401"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s11075-024-01907-8","type":"journal-article","created":{"date-parts":[[2024,8,10]],"date-time":"2024-08-10T07:02:32Z","timestamp":1723273352000},"page":"1081-1110","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the block Kaczmarz-Tanabe methods with relaxation parameters for solving linear systems"],"prefix":"10.1007","volume":"99","author":[{"given":"Chuan-gang","family":"Kang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,8,10]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Baumeister, J., Kaltenbacher, B., Leit$$\\tilde{a}$$o, A.: On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations. Inverse Probl. Imag. 4(3), 335\u2013350 (2010)","key":"1907_CR1","DOI":"10.3934\/ipi.2010.4.335"},{"doi-asserted-by":"crossref","unstructured":"Wei, K.: Solving systems of phaseless equations via Kaczmarz methods: a proof of concept study. Inverse Probl. 31, 125008 (2015)","key":"1907_CR2","DOI":"10.1088\/0266-5611\/31\/12\/125008"},{"key":"1907_CR3","first-page":"355","volume":"35","author":"S Kaczmarz","year":"1937","unstructured":"Kaczmarz, S.: Angen\u00e4herte aufl\u00f6sung von systemen linearer gleichungen. Bulletin de Academie Polonaise des Sciences et Lettres 35, 355\u2013357 (1937)","journal-title":"Bulletin de Academie Polonaise des Sciences et Lettres"},{"key":"1907_CR4","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1016\/0022-5193(70)90109-8","volume":"29","author":"R Gordon","year":"1970","unstructured":"Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstrction techniques (ART) for three dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29, 471\u2013481 (1970)","journal-title":"J. Theor. Biol."},{"key":"1907_CR5","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1007\/BF01436376","volume":"17","author":"K Tanabe","year":"1971","unstructured":"Tanabe, K.: Projection method for solving a singular system of linear equations and its applications. Numer. Math 17, 203\u2013214 (1971)","journal-title":"Numer. Math"},{"doi-asserted-by":"crossref","unstructured":"Natterer, F.: The mathematics of computerized tomography. SIAM (2001)","key":"1907_CR6","DOI":"10.1137\/1.9780898719284"},{"key":"1907_CR7","doi-asserted-by":"publisher","first-page":"600","DOI":"10.1109\/42.241889","volume":"12","author":"GT Herman","year":"1993","unstructured":"Herman, G.T., Meyer, L.B.: Algebraic reconstruction techniques can be made computationally efficient. IEEE Trans. Med Imaging 12, 600\u2013609 (1993)","journal-title":"IEEE Trans. Med Imaging"},{"doi-asserted-by":"crossref","unstructured":"Jiao, Y.L., Jin, B.T., Lu, X.L.: Preasymptotic convergence of randomized Kaczmarz method. Inverse Probl. 33 (2017)","key":"1907_CR8","DOI":"10.1088\/1361-6420\/aa8e82"},{"key":"1907_CR9","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/s10543-020-00824-1","volume":"61","author":"JD Moorman","year":"2021","unstructured":"Moorman, J.D., Thomas, K.T., Molitor, D., Needell, D.: Randomized Kaczmarz with averaging. BIT 61, 337\u2013359 (2021)","journal-title":"BIT"},{"key":"1907_CR10","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1007\/s00041-008-9030-4","volume":"15","author":"T Strohmer","year":"2009","unstructured":"Strohmer, T., Vershynin, R.: A randomized Kaczmarz algorithm with exponential convergence. J. Fourier Anal. Appl. 15, 262\u2013278 (2009)","journal-title":"J. Fourier Anal. Appl."},{"doi-asserted-by":"crossref","unstructured":"Tan, Y.S., Vershynin, R.: Phase retrieval via randomized Kaczmarz: theoretical guarantees. Information & Inference (2018)","key":"1907_CR11","DOI":"10.1093\/imaiai\/iay005"},{"doi-asserted-by":"crossref","unstructured":"Ma, A., Molitor, D.: Randomized Kaczmarz for tensor linear systems. arXiv:2006.01246v1 (2020)","key":"1907_CR12","DOI":"10.1007\/s10543-021-00877-w"},{"doi-asserted-by":"crossref","unstructured":"Eggermont, P.P.B., Herman, G.T., Lent, A.: Iterative algorithms for large partitioned linear systems, with applications to image reconstruction. Linear Algebra Appl. 40(OCT), 37\u201367 (1981)","key":"1907_CR13","DOI":"10.1016\/0024-3795(81)90139-7"},{"issue":"1","key":"1907_CR14","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/BF01396307","volume":"41","author":"Y Censor","year":"1983","unstructured":"Censor, Y., Eggermont, P.B.B., Dan, G.: Strong underrelaxation in Kaczmarz\u2019s method for inconsistent systems. Numer. Math. 41(1), 83\u201392 (1983)","journal-title":"Numer. Math."},{"key":"1907_CR15","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/j.laa.2012.12.022","volume":"441","author":"D Needell","year":"2014","unstructured":"Needell, D., Tropp, J.A.: Paved with good intentions:analysis of a randomized block Kaczmarz method. Linear Algebra Appl. 441, 199\u2013221 (2014)","journal-title":"Linear Algebra Appl."},{"key":"1907_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11075-017-0425-7","volume":"79","author":"C Popa","year":"2018","unstructured":"Popa, C.: Convergence rates for Kaczmarz-type algorithms. Numer. Algor. 79, 1\u201317 (2018)","journal-title":"Numer. Algor."},{"doi-asserted-by":"crossref","unstructured":"Kang, C.: Convergence rates of the Kaczmarz-Tanabe method for linear system. J. Comput. Appl. Math. 394, 113577 (2021)","key":"1907_CR17","DOI":"10.1016\/j.cam.2021.113577"},{"doi-asserted-by":"crossref","unstructured":"Kang, C.: The standard forms and convergence theory of the Kaczmarz-Tanabe type methods for solving linear systems. J. Comput. Appl. Math. 434, 115333 (2023)","key":"1907_CR18","DOI":"10.1016\/j.cam.2023.115333"},{"doi-asserted-by":"crossref","unstructured":"Kang, C.: The standard form and convergence theory of the relaxation Kaczmarz-Tanabe method for solving linear systems. arXiv:2305.12105v1 (2023)","key":"1907_CR19","DOI":"10.1016\/j.cam.2023.115333"},{"unstructured":"Elman, H.C.: Iterative methods for large, sparse, nonsymmetric systems of linear equations, vol. 229. Yale University (1982)","key":"1907_CR20"},{"key":"1907_CR21","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1016\/0167-8191(89)90114-2","volume":"9","author":"C Kamath","year":"1988","unstructured":"Kamath, C., Sameh, A.: A projection method for solving nonsymmetric linear systems on multiprocessors. Parallel Comput. 9, 291\u2013312 (1988)","journal-title":"Parallel Comput."},{"issue":"3","key":"1907_CR22","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.cmpb.2013.01.003","volume":"110","author":"J Klukowska","year":"2013","unstructured":"Klukowska, J., Davidi, R., Herman, G.T.: SNARK09 - A software package for reconstruction of 2d images from 1d projections. Comput. Methods Prog. Biomed. 110(3), 424\u2013440 (2013)","journal-title":"Comput. Methods Prog. Biomed."},{"key":"1907_CR23","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s11075-017-0430-x","volume":"79","author":"PC Hansen","year":"2018","unstructured":"Hansen, P.C., Jorgensen, J.S.: AIR Tools II: algebraic iterative reconstruction method, improved implementation. Numer. Algor. 79, 107\u2013137 (2018)","journal-title":"Numer. Algor."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01907-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01907-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01907-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T08:43:25Z","timestamp":1750149805000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01907-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,10]]},"references-count":23,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["1907"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01907-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2024,8,10]]},"assertion":[{"value":"28 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 August 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not Applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}