{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T00:20:17Z","timestamp":1759191617015},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2017,1,11]],"date-time":"2017-01-11T00:00:00Z","timestamp":1484092800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2017,10]]},"DOI":"10.1007\/s11075-017-0263-7","type":"journal-article","created":{"date-parts":[[2017,1,11]],"date-time":"2017-01-11T03:17:01Z","timestamp":1484104621000},"page":"413-425","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The parameterized upper and lower triangular splitting methods for saddle point problems"],"prefix":"10.1007","volume":"76","author":[{"given":"Jing-Tao","family":"Li","sequence":"first","affiliation":[]},{"given":"Chang-Feng","family":"Ma","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,1,11]]},"reference":[{"key":"263_CR1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492904000212","volume":"14","author":"M Benzi","year":"2005","unstructured":"Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1\u2013137 (2005)","journal-title":"Acta Numer."},{"key":"263_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1236463.1236469","volume":"33","author":"HC Elman","year":"2007","unstructured":"Elman, H.C., Ramage, A., Silvester, D.J.: A Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33, 1\u201318 (2007)","journal-title":"ACM Trans. Math. Softw."},{"key":"263_CR3","doi-asserted-by":"crossref","unstructured":"Ferziger, J.H., Peric, M.: Computational Methods for Fluid Dynamics, vol. 3. Springer, Berlin (1996)","DOI":"10.1007\/978-3-642-97651-3"},{"issue":"5","key":"263_CR4","doi-asserted-by":"crossref","first-page":"1621","DOI":"10.1088\/0266-5611\/20\/5\/018","volume":"20","author":"E Haber","year":"2004","unstructured":"Haber, E., Modersitzki, J.: Numerical methods for volume preserving image registration. Inverse Prob. 20(5), 1621 (2004)","journal-title":"Inverse Prob."},{"key":"263_CR5","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics","author":"F Brezzi","year":"1991","unstructured":"Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics. Springer-Verlag, New York (1991)"},{"key":"263_CR6","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971484","volume-title":"Numerical Methods for Least Squares Problems","author":"A Bj\u00f6rck","year":"1996","unstructured":"Bj\u00f6rck, A.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)"},{"key":"263_CR7","doi-asserted-by":"crossref","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"J Nocedal","year":"1999","unstructured":"Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (1999)"},{"key":"263_CR8","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.cam.2015.04.006","volume":"288","author":"N-M Zhang","year":"2013","unstructured":"Zhang, N.-M., Shen, P.: Constraint preconditioners for solving singular saddle point problems. J. Comput. Appl. Math. 288, 116\u2013125 (2013)","journal-title":"J. Comput. Appl. Math."},{"issue":"4","key":"263_CR9","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1137\/S0895479896297732","volume":"18","author":"R Fletcher","year":"1997","unstructured":"Fletcher, R., Johnson, T.: On the stability of null-space methods for KKT systems. SIAM J. Matrix Anal. Appl. 18(4), 938\u2013958 (1997)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"263_CR10","doi-asserted-by":"crossref","first-page":"2900","DOI":"10.1016\/j.laa.2008.01.018","volume":"428","author":"Z-Z Bai","year":"2008","unstructured":"Bai, Z.-Z., Wang, Z.-Q.: On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl. 428, 2900\u20132932 (2008)","journal-title":"Linear Algebra Appl."},{"key":"263_CR11","first-page":"765","volume":"206","author":"F Chen","year":"2008","unstructured":"Chen, F., Jiang, Y.-L.: A generalization of the inexact parameterized Uzawa methods for saddle point problems. Appl. Math. comput. 206, 765\u2013771 (2008)","journal-title":"Appl. Math. comput."},{"key":"263_CR12","doi-asserted-by":"crossref","unstructured":"Liang, Z.-Z., Zhang, G.-F.: Variants of accelerated parameterized inexact Uzawa method for saddle-point problems. BIT Numer. Math., 1\u201320 (2015)","DOI":"10.1007\/s10543-015-0582-9"},{"key":"263_CR13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00211-005-0643-0","volume":"102","author":"Z-Z Bai","year":"2005","unstructured":"Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: ON generalized successive overrelaxation methods for augmented linear systems. Numer. Math. 102, 1\u201338 (2005)","journal-title":"Numer. Math."},{"key":"263_CR14","first-page":"88","volume":"252","author":"A-L Yang","year":"2015","unstructured":"Yang, A.-L., Li, X., Wu, Y.-J.: On semi-convergence of the Uzawa-HSS method for singular saddle-point problems. Appl. Math. Comput. 252, 88\u201398 (2015)","journal-title":"Appl. Math. Comput."},{"key":"263_CR15","first-page":"89","volume":"221","author":"Z-Z Liang","year":"2013","unstructured":"Liang, Z.-Z., Zhang, G.-F.: On block-diagonally preconditioned accelerated parameterized inexact Uzawa method for singular saddle point problems. Appl. Math. Comput. 221, 89\u2013101 (2013)","journal-title":"Appl. Math. Comput."},{"key":"263_CR16","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s00607-010-0101-4","volume":"89","author":"Z-Z Bai","year":"2010","unstructured":"Bai, Z.-Z.: On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems. Computing 89, 171\u2013197 (2010)","journal-title":"Computing"},{"key":"263_CR17","doi-asserted-by":"crossref","first-page":"2338","DOI":"10.1016\/j.cam.2011.11.020","volume":"236","author":"W Li","year":"2012","unstructured":"Li, W., Liu, Y.-P., Peng, X.-F.: The generalized HSS method for solving singular linear systems. J. Comput. Appl. Math. 236, 2338\u20132353 (2012)","journal-title":"J. Comput. Appl. Math."},{"key":"263_CR18","first-page":"552","volume":"244","author":"X Li","year":"2014","unstructured":"Li, X., Wu, Y.-J., Yang, A.-L., Yuan, J.-Y.: Modified accelerated parameterized inexact Uzawa method for singular and nonsingular saddle point problems. Appl. Math. Comput. 244, 552\u2013560 (2014)","journal-title":"Appl. Math. Comput."},{"key":"263_CR19","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.cam.2013.05.015","volume":"255","author":"N-M Zhang","year":"2014","unstructured":"Zhang, N.-M., Lu, T.T, Wei, Y.-M.: Semi-convergence analysis of Uzawa methods for singular saddle point problems. J. Comput. Appl. Math. 255, 334\u2013345 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"263_CR20","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1016\/j.camwa.2013.12.006","volume":"67","author":"H-T Fan","year":"2014","unstructured":"Fan, H.-T., Zheng, B.: A preconditioned GLHSS iteration method for non-Hermitian singular saddle point problems. Comput. Math. Appl. 67, 614\u2013626 (2014)","journal-title":"Comput. Math. Appl."},{"issue":"6","key":"263_CR21","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.1137\/0731085","volume":"31","author":"HC Elman","year":"1994","unstructured":"Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31(6), 1645\u20131661 (1994)","journal-title":"SIAM J. Numer. Anal."},{"key":"263_CR22","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1023\/B:BITN.0000039428.54019.15","volume":"44","author":"L Wang","year":"2004","unstructured":"Wang, L., Bai, Z.-Z.: Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts. BIT Numer. Math. 44, 363\u2013386 (2004)","journal-title":"BIT Numer. Math."},{"issue":"6","key":"263_CR23","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1002\/nla.626","volume":"16","author":"Z-Z Bai","year":"2009","unstructured":"Bai, Z.-Z.: Optimal paremeters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl. 16(6), 447\u2013479 (2009)","journal-title":"Numer. Linear Algebra Appl."},{"key":"263_CR24","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1002\/nla.772","volume":"18","author":"Y Cao","year":"2011","unstructured":"Cao, Y., Jiang, M.-Q., Zheng, Y.-L: A splitting preconditioner for saddle point problems. Numer. Linear Algebra Appl. 18, 875\u2013895 (2011)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"2","key":"263_CR25","first-page":"240","volume":"26","author":"J-F Yin","year":"2008","unstructured":"Yin, J.-F., Bai, Z.-Z.: The restrictively preconditioned conjugate gradient methods on normal residual for block two-by-two linear systems. J. Comput. Math. 26(2), 240\u2013249 (2008)","journal-title":"J. Comput. Math."},{"key":"263_CR26","volume-title":"Studies in Nonlinear Programming","author":"K Arrow","year":"1958","unstructured":"Arrow, K., Hurwicz, L., Uzawa, H.: Studies in Nonlinear Programming. Stanford University Press, Stanford (1958)"},{"key":"263_CR27","volume-title":"Iterative Solution for Large Linear Systems","author":"DM Yang","year":"1971","unstructured":"Yang, D.M.: Iterative Solution for Large Linear Systems. Academic Press, New York (1971)"},{"key":"263_CR28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00211-004-0521-1","volume":"98","author":"Z-Z Bai","year":"2004","unstructured":"Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1\u201332 (2004)","journal-title":"Numer. Math."},{"key":"263_CR29","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.cam.2015.11.026","volume":"298","author":"Q-Q Zheng","year":"2016","unstructured":"Zheng, Q.-Q., Ma, C.-F.: A class of triangular splitting methods for saddle point problems. J. Comput. Apll. Math. 298, 13\u201323 (2016)","journal-title":"J. Comput. Apll. Math."},{"key":"263_CR30","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/j.cam.2008.10.006","volume":"228","author":"S-L Wu","year":"2009","unstructured":"Wu, S.-L., Huang, T.-Z., Zhao, X.-L.: A modified SSOR iterative method for augmented systems. J. Comput. Appl. Math. 228, 424\u2013433 (2009)","journal-title":"J. Comput. Appl. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-017-0263-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-017-0263-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-017-0263-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,17]],"date-time":"2019-09-17T11:05:39Z","timestamp":1568718339000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-017-0263-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,11]]},"references-count":30,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,10]]}},"alternative-id":["263"],"URL":"https:\/\/doi.org\/10.1007\/s11075-017-0263-7","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1,11]]}}}