@conference {1303, title = {Matrix Powers Kernels for Thick-Restart Lanczos with Explicit External Deflation}, booktitle = {International Parallel and Distributed Processing Symposium (IPDPS)}, year = {2019}, month = {2019-05}, publisher = {IEEE}, organization = {IEEE}, address = {Rio de Janeiro, Brazil}, abstract = {Some scientific and engineering applications need to compute a large number of eigenpairs of a large Hermitian matrix. Though the Lanczos method is effective for computing a few eigenvalues, it can be expensive for computing a large number of eigenpairs (e.g., in terms of computation and communication). To improve the performance of the method, in this paper, we study an s-step variant of thick-restart Lanczos (TRLan) combined with an explicit external deflation (EED). The s-step method generates a set of s basis vectors at a time and reduces the communication costs of generating the basis vectors. We then design a specialized matrix powers kernel (MPK) that reduces both the communication and computational costs by taking advantage of the special properties of the deflation matrix. We conducted numerical experiments of the new TRLan eigensolver using synthetic matrices and matrices from electronic structure calculations. The performance results on the Cori supercomputer at the National Energy Research Scientific Computing Center (NERSC) demonstrate the potential of the specialized MPK to significantly reduce the execution time of the TRLan eigensolver. The speedups of up to 3.1{\texttimes} and 5.3{\texttimes} were obtained in our sequential and parallel runs, respectively.}, author = {Zhaojun Bai and Jack Dongarra and Ding Lu and Ichitaro Yamazaki} } @inbook {747, title = {LAPACK}, booktitle = {Handbook of Linear Algebra}, year = {2013}, publisher = {CRC Press}, organization = {CRC Press}, edition = {Second}, address = {Boca Raton, FL}, abstract = {With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters.}, isbn = {9781466507289}, author = {Zhaojun Bai and James Demmel and Jack Dongarra and Julien Langou and Jenny Wang} } @article {icl:50, title = {LAPACK Users{\textquoteright} Guide, 3rd ed.}, journal = {Philadelphia: Society for Industrial and Applied Mathematics}, year = {1999}, month = {1999-01}, author = {Ed Anderson and Zhaojun Bai and Christian Bischof and Susan Blackford and James Demmel and Jack Dongarra and Jeremy Du Croz and Anne Greenbaum and Sven Hammarling and Alan McKenney and Danny Sorensen} }