XMR swMATH ID: 7169 Software Authors: Willems, Paul R.; Lang, Bruno Description: A framework for the MR3 algorithm: theory and implementation This paper provides a streamlined and modular presentation of the MR3 algorithm for computing selected eigenpairs of symmetric tridiagonal matrices, thus disentangling the principles driving MR3 and the (recursive) “core” algorithm from the specific (e.g., twisted) decompositions used to represent the matrices at different recursion depths and from the (dqds) transformations converting between them. Our approach allows a modular full proof for the correctness of the MR3 algorithm. This proof is based on five requirements concerning the interplay between the core algorithm and its subcomponents. These requirements can also guide in implementing the algorithm, because they expose quantities that can and should be monitored at runtime. Our new implementation XMR, which is based on the above analysis, is described and compared to xSTEMR from Lapack 3.2.2. Numerical experiments comparing the robustness and performance of both implementations are given. Homepage: http://www-ai.math.uni-wuppertal.de/SciComp/preprints/SC1102.pdf Keywords: symmetric tridiagonal matrix; eigensystem; MRRR algorithm; theory and implementation Related Software: LAPACK; OpenBLAS; EISPACK; ScaLAPACK; Algorithm 977; Matlab; SLATE; BDSVDX; ESSL; DAGuE; SBR Toolbox; MKL; LINPACK; CUDA; BLAS; ATLAS; GitHub; libflame; Algorithm 880 Cited in: 3 Documents Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year A framework for the \(\text{MR}^3\) algorithm: theory and implementation. Zbl 1266.65061Willems, Paul R.; Lang, Bruno 2013 all top 5 Cited by 12 Authors 1 Dongarra, Jack J. 1 Gates, Mark 1 Haidar, Azzam 1 Kurzak, Jakub 1 Lang, Bruno 1 Luszczek, Piotr 1 Quintana-Ortí, Gregorio 1 Tomov, Stanimire Z. 1 van de Geijn, Robert Alexander 1 Van Zee, Field Garritt 1 Willems, Paul R. 1 Yamazaki, Ichitaro Cited in 3 Serials 1 ACM Transactions on Mathematical Software 1 SIAM Review 1 SIAM Journal on Scientific Computing Cited in 2 Fields 3 Numerical analysis (65-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) Citations by Year