## IRAM

 swMATH ID: 8795 Software Authors: Sorensen, Danny C. Description: Implicit application of polynomial filters in a k-step Arnoldi method. The author describes and analyses a new implementation of the Arnoldi method for computing a few eigenvalues and the corresponding eigenvectors of a large general square matrix (which reduces to the Lanczos method in the symmetric case). Using a truncated variant of the implicitly shifted QR-iteration, the author applies a polynomial filter to the Arnoldi (Lanczos) vector on each iteration. This approach generalizes explicit restart methods. Advantages of the method are discussed and some preliminary computational results using parallel and vector computers are given Homepage: http://epubs.siam.org/doi/abs/10.1137/0613025 Keywords: Arnoldi method; eigenvalues; eigenvectors; Lanczos method; implicitly shifted $$QR$$-iteration; polynomial filter; explicit restart methods; parallel and vector computers Related Software: eigs; ARPACK; JDQZ; JDQR; TRLan; Harwell-Boeing sparse matrix collection; Matlab; lobpcg.m; LAPACK; PROPACK; SLEPc; SparseMatrix; MatrixMarket; Eigtool; PRIMME; Algorithm 432; mctoolbox; EIGIFP; a-TRLan; FEAST Cited in: 205 Publications
all top 5

### Cited by 307 Authors

 10 Jia, Zhongxiao 9 Wu, Gang 8 Meerbergen, Karl 7 Morgan, Ronald B. 6 Baglama, James 6 Lin, Wen-Wei 6 Reichel, Lothar 6 Sorensen, Danny C. 5 Faßbender, Heike 5 Zhou, Yunkai 4 Abdel-Aziz, Mohammedi R. 4 Benner, Peter 4 Calvetti, Daniela 4 Dax, Achiya 4 Miao, Cun-Qiang 4 Vandebril, Raf 3 Aishima, Kensuke 3 Bhuruth, Muddun 3 Camps, Daan 3 Dookhitram, Kumar 3 Embree, Mark 3 Gu, Chuanqing 3 Katsouleas, Georgios 3 Maroulas, John 3 Niu, Datian 3 Watkins, David S. 2 Antoulas, Athanasios C. 2 Bella, Tom 2 Boojhawon, Ravindra 2 Brandts, Jan H. 2 Bujanović, Zvonimir 2 Bultheel, Adhemar François 2 Canning, Andrew 2 Chen, Guizhi 2 Chen, Zhibing 2 Daube, Olivier 2 De Samblanx, Gorik 2 Gugercin, Serkan 2 Hansen, Per Christian 2 Hochstenbach, Michiel E. 2 Hou, Thomas Yizhao 2 Huang, Tsungming 2 Huang, Weiqiang 2 Kressner, Daniel 2 Le Quéré, Patrick 2 Li, Yiming 2 Lu, Linzhang 2 Mehrmann, Volker 2 Picucci, Jennifer 2 Plestenjak, Bor 2 Polizzi, Eric 2 Román, Jose E. 2 Rommes, Joost 2 Rostami, Minghao W. 2 Ruhe, Axel 2 Saad, Yousef 2 Sadkane, Miloud 2 Shieh, Shih-Feng 2 Simoncini, Valeria 2 Stoll, Martin 2 Van der Vorst, Henk Albertus 2 Van Dooren, Paul M. 2 Vömel, Christof 2 Wang, Lin-Wang 2 Yuan, Xuegang 1 Afanasiev, Konstantin 1 Ainsworth, G. O. jun. 1 Amodei, Luca 1 Andriano, C. 1 Apel, Thomas 1 Astudillo, Reinaldo 1 Bagheri, Shervin 1 Bai, Zhaojun 1 Ballard, Grey M. 1 Bekas, Constantine 1 Booten, Albert G. L. 1 Bosch, Jessica 1 Brogaard Riis, Nicolai André 1 Bucci, Michele Alessandro 1 Buchot, Jean-Marie 1 Bunse-Gerstner, Angelika 1 Caboussat, Alexandre 1 Calo, Victor Manuel 1 Campos, Carmen 1 Carden, Russell L. 1 Carson, Erin Claire 1 Chang, Shu-Ming 1 Chang, Wei-Jen 1 Cherubini, Stefania 1 Choi, Young-Geun 1 Chou, So-Hsiang 1 Cooke, S. J. 1 Cox, Steven J. 1 Cullum, Jane K. 1 da Silva, Ricardo Reis 1 D’Almeida, Filomena D. 1 Darnell, Dean 1 Dartora, C. A. 1 David, Roden J. A. 1 Demmel, James Weldon ...and 207 more Authors
all top 5

### Cited in 50 Serials

 30 Linear Algebra and its Applications 26 Journal of Computational and Applied Mathematics 12 Journal of Computational Physics 12 Applied Mathematics and Computation 12 BIT 12 Numerical Algorithms 10 SIAM Journal on Scientific Computing 8 Mathematical and Computer Modelling 8 Numerical Linear Algebra with Applications 7 Computer Methods in Applied Mechanics and Engineering 5 Applied Numerical Mathematics 5 SIAM Journal on Matrix Analysis and Applications 4 Computers & Mathematics with Applications 4 Mathematics of Computation 4 Numerische Mathematik 4 Journal of Scientific Computing 2 International Journal for Numerical Methods in Fluids 2 Journal of Fluid Mechanics 2 Systems & Control Letters 2 Japan Journal of Industrial and Applied Mathematics 2 Multiscale Modeling & Simulation 2 East Asian Journal on Applied Mathematics 1 Computers and Fluids 1 Physics Reports 1 Calcolo 1 Journal of Multivariate Analysis 1 Mathematics and Computers in Simulation 1 Mitteilungen der Gesellschaft für Angewandte Mathematik und Mechanik 1 Computer Aided Geometric Design 1 Physica D 1 Computational Mechanics 1 Applied Mathematics Letters 1 Journal of Parallel and Distributed Computing 1 Applications of Mathematics 1 Mathematical Programming. Series A. Series B 1 Archive of Applied Mechanics 1 ETNA. Electronic Transactions on Numerical Analysis 1 ELA. The Electronic Journal of Linear Algebra 1 Mathematical Problems in Engineering 1 European Journal of Control 1 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 1 Archives of Computational Methods in Engineering 1 SIAM Journal on Applied Dynamical Systems 1 Acta Numerica 1 GAMM-Mitteilungen 1 Communications in Computational Physics 1 Science China. Mathematics 1 Afrika Matematika 1 International Journal of Applied and Computational Mathematics 1 Results in Applied Mathematics
all top 5

### Cited in 28 Fields

 173 Numerical analysis (65-XX) 40 Linear and multilinear algebra; matrix theory (15-XX) 14 Systems theory; control (93-XX) 12 Partial differential equations (35-XX) 12 Fluid mechanics (76-XX) 9 Mechanics of deformable solids (74-XX) 7 Statistical mechanics, structure of matter (82-XX) 5 Optics, electromagnetic theory (78-XX) 5 Quantum theory (81-XX) 4 Computer science (68-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Operations research, mathematical programming (90-XX) 2 History and biography (01-XX) 2 Ordinary differential equations (34-XX) 2 Operator theory (47-XX) 2 Mechanics of particles and systems (70-XX) 1 Combinatorics (05-XX) 1 Functions of a complex variable (30-XX) 1 Special functions (33-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Approximations and expansions (41-XX) 1 Integral equations (45-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Probability theory and stochastic processes (60-XX) 1 Statistics (62-XX) 1 Geophysics (86-XX) 1 Biology and other natural sciences (92-XX) 1 Information and communication theory, circuits (94-XX)