{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:47:28Z","timestamp":1767340048462,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,8,29]],"date-time":"2019-08-29T00:00:00Z","timestamp":1567036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["NSFC-11601081","NSFC-11671105"],"award-info":[{"award-number":["NSFC-11601081","NSFC-11671105"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the research fund for distinguished young scholars of Fujian Agriculture and Forestry University","award":["No. xjq201727"],"award-info":[{"award-number":["No. xjq201727"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In the linear response eigenvalue problem arising from quantum chemistry and physics, one needs to compute several positive eigenvalues together with the corresponding eigenvectors. For such a task, in this paper, we present a FEAST algorithm based on complex contour integration for the linear response eigenvalue problem. By simply dividing the spectrum into a collection of disjoint regions, the algorithm is able to parallelize the process of solving the linear response eigenvalue problem. The associated convergence results are established to reveal the accuracy of the approximated eigenspace. Numerical examples are presented to demonstrate the effectiveness of our proposed algorithm.<\/jats:p>","DOI":"10.3390\/a12090181","type":"journal-article","created":{"date-parts":[[2019,8,29]],"date-time":"2019-08-29T11:26:22Z","timestamp":1567077982000},"page":"181","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A FEAST Algorithm for the Linear Response Eigenvalue Problem"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1533-1534","authenticated-orcid":false,"given":"Zhongming","family":"Teng","sequence":"first","affiliation":[{"name":"College of Computer and Information Science, Fujian Agriculture and Forestry University, Fuzhou 350002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Linzhang","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/060651653","article-title":"Numerical methods for electronic structure calculations of materials","volume":"52","author":"Saad","year":"2010","journal-title":"SIAM Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1137\/16M1102641","article-title":"A structure preserving Lanczos algorithm for computing the optical absorption spectrum","volume":"39","author":"Shao","year":"2018","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1137\/110838960","article-title":"Minimization principle for linear response eigenvalue problem, I: Theory","volume":"33","author":"Bai","year":"2012","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.laa.2017.01.005","article-title":"A symmetric structure-preserving \u0393QR algorithm for linear response eigenvalue problems","volume":"520","author":"Li","year":"2017","journal-title":"Linear Algebra Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/j.laa.2016.08.023","article-title":"Error bounds for approximate deflating subspaces for linear response eigenvalue problems","volume":"528","author":"Wang","year":"2017","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1137\/110838972","article-title":"Minimization principle for linear response eigenvalue problem, II: Computation","volume":"34","author":"Bai","year":"2013","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"12001","DOI":"10.1209\/0295-5075\/78\/12001","article-title":"Reduction of the RPA eigenvalue problem and a generalized Cholesky decomposition for real-symmetric matrices","volume":"78","author":"Papakonstantinou","year":"2007","journal-title":"Europhys. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.cam.2013.01.003","article-title":"Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem","volume":"247","author":"Teng","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_9","first-page":"505","article-title":"A block Lanczos method for the linear response eigenvalue problem","volume":"46","author":"Teng","year":"2017","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1007\/s10444-016-9455-2","article-title":"A block Chebyshev-Davidson method for linear response eigenvalue problems","volume":"42","author":"Teng","year":"2016","journal-title":"Adv. Comput. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"164109","DOI":"10.1063\/1.3494540","article-title":"Ab initio calculations of optical absorpation spectra: solution of the Bethe-Salpeter equation within density matrix perturbation theory","volume":"133","author":"Rocca","year":"2010","journal-title":"J. Chem. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.laa.2015.09.036","article-title":"Structure preserving parallel algorithms for solving the Bethe-Salpeter eigenvalue problem","volume":"488","author":"Shao","year":"2016","journal-title":"Linear Algebra Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.cpc.2017.07.017","article-title":"Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory","volume":"221","author":"Vecharynski","year":"2017","journal-title":"Comput. Phys. Commun."},{"key":"ref_14","first-page":"153","article-title":"Weighted Golub-Kahan-Lanczos bidiagonalization algorithms","volume":"47","author":"Zhong","year":"2017","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhong, H.X., Teng, Z., and Chen, G. (2019). Weighted block Golub-Kahan-Lanczos algorithms for linear response eigenvalue problem. Mathematics, 7.","DOI":"10.3390\/math7010053"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11425-016-0297-1","article-title":"Linear response eigenvalue problem solved by extended locally optimal preconditioned conjugate gradient methods","volume":"59","author":"Bai","year":"2016","journal-title":"Sci. China Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"115112","DOI":"10.1103\/PhysRevB.79.115112","article-title":"Density-matrix-based algorithm for solving eigenvalue problems","volume":"79","author":"Polizzi","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1137\/13090866X","article-title":"FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection","volume":"35","author":"Tang","year":"2014","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"S772","DOI":"10.1137\/15M1026572","article-title":"FEAST eigensolver for non-Hermitian problems","volume":"38","author":"Kestyn","year":"2016","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.jocs.2018.05.006","article-title":"FEAST eigensolver for nonlinear eigenvalue problems","volume":"27","author":"Gavin","year":"2018","journal-title":"J. Comput. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2012.11.014","article-title":"Dissecting the FEAST algorithm for generalized eigenproblems","volume":"244","author":"Galgon","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_22","first-page":"A2100","article-title":"Zolotarev quadrature rules and load balancing for the FEAST eigensolver","volume":"37","author":"Guttel","year":"2015","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1268","DOI":"10.1137\/16M1086601","article-title":"A fast contour-integral eigensolver for non-Hermitian matrices","volume":"38","author":"Ye","year":"2017","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"e2092","DOI":"10.1002\/nla.2092","article-title":"A FEAST algorithm with oblique projection for generalized eigenvalue problems","volume":"24","author":"Yin","year":"2017","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s00211-014-0681-6","article-title":"Convergence of the block Lanczos method for eigenvalue clusters","volume":"131","author":"Li","year":"2015","journal-title":"Numer. Math."},{"key":"ref_26","unstructured":"Stein, E.M., and Shakarchi, R. (2010). Complex Analysis, Princeton University Press."},{"key":"ref_27","first-page":"127","article-title":"Parallel stochastic estimation method of eigenvalue distribution","volume":"2","author":"Futamura","year":"2010","journal-title":"J. SIAM Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1002\/nla.2048","article-title":"Efficient estimation of eigenvalue counts in an interval","volume":"23","author":"Napoli","year":"2016","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Saad, Y. (2011). Numerical Methods for Large Eigenvalue Problems: Revised Version, SIAM.","DOI":"10.1137\/1.9781611970739"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.laa.2019.04.036","article-title":"A harmonic FEAST algorithm for non-Hermitian generalized eigenvalue problems","volume":"578","author":"Yin","year":"2019","journal-title":"Linear Algebra Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.cam.2019.01.016","article-title":"On the non-Hermitian FEAST algorithms with oblique projection for eigenvalue problems","volume":"355","author":"Yin","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Saad, Y. (2003). Iterative Methods for Sparse Linear Systems, SIAM.","DOI":"10.1137\/1.9780898718003"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"856","DOI":"10.1137\/0907058","article-title":"GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems","volume":"7","author":"Saad","year":"1986","journal-title":"SIAM J. Sci. 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