Abstract
The QR algorithm computes the Schur form of a matrix and is by far the most popular approach for solving dense nonsymmetric eigenvalue problems. Multishift and aggressive early deflation (AED) techniques have led to significantly more efficient sequential implementations of the QR algorithm during the last decade. More recently, these techniques have been incorporated in a novel parallel QR algorithm on hybrid distributed memory HPC systems. While leading to significant performance improvements, it has turned out that AED may become a computational bottleneck as the number of processors increases. In this paper, we discuss a two-level approach for performing AED in a parallel environment, where the lower level consists of a novel combination of AED with the pipelined QR algorithm implemented in the ScaLAPACK routine PDLAHQR. Numerical experiments demonstrate that this new implementation further improves the performance of the parallel QR algorithm.
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References
Adlerborn, B., Kågström, B., Kressner, D.: Parallel Variants of the Multishift QZ Algorithm with Advanced Deflation Techniques. In: Kågström, B., Elmroth, E., Dongarra, J., Waśniewski, J. (eds.) PARA 2006. LNCS, vol. 4699, pp. 117–126. Springer, Heidelberg (2007)
Anderson, E., Bai, Z., Bischof, C.H., Blackford, S., Demmel, J.W., Dongarra, J.J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.C.: LAPACK User’s Guide, 3rd edn. SIAM, Philadelphia (1999)
Bai, Z., Demmel, J.W.: On a Block Implementation of Hessenberg Multishift QR Iteration. Intl. J. of High Speed Comput. 1, 97–112 (1989)
Bai, Z., Demmel, J.W.: On Swapping Diagonal Blocks in Real Schur Form. Linear Algebra Appl. 186, 73–95 (1993)
Blackford, L.S., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J.W., Dhillon, I., Dongarra, J.J., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., Whaley, R.C.: ScaLAPACK Users’ Guide. SIAM, Philadelphia (1997)
Braman, K., Byers, R., Mathias, R.: The Multishift QR Algorithm. Part I: Maintaining Well-focused Shifts and Level 3 Performance. SIAM J. Matrix Anal. Appl. 23(4), 929–947 (2002)
Braman, K., Byers, R., Mathias, R.: The Multishift QR Algorithm. Part II: Aggressive Early Deflation. SIAM J. Matrix Anal. Appl. 23(4), 948–973 (2002)
Byers, R.: LAPACK 3.1 xHSEQR: Tuning and Implementation Notes on the Small Bulge Multi-shift QR Algorithm with Aggressive Early Deflation. LAPACK Working Note 187 (2007)
Golub, G., Uhlig, F.: The QR Algorithm: 50 Years Later Its Genesis by John Francis and Vera Kublanovskaya and Subsequent Developments. IMA J. Numer. Anal. 29(3), 467–485 (2009)
Granat, R., Kågström, B., Kressner, D.: A Novel Parallel QR Algorithm for Hybrid Distributed Memory HPC Systems. SIAM J. Sci. Comput. 32(4), 2345–2378 (2010) (An earlier version appeared as LAPACK Working Note 216)
Granat, R., Kågström, B., Kressner, D.: Parallel Eigenvalue Reordering in Real Schur Forms. Concurrency and Computat.: Pract. Exper. 21(9), 1225–1250 (2009)
GOTO-BLAS – High-performance BLAS by Kazushige Goto, http://www.tacc.utexas.edu/tacc-projects/#blas
Henry, G., van de Geijn, R.: Parallelizing the QR Algorithm for the Nonsymmetric Algebraic Eigenvalue Problem: Myths and Reality. SIAM J. Sci. Comput. 17, 870–883 (1997)
Henry, G., Watkins, D.S., Dongarra, J.J.: A Parallel Implementation of the Nonsymmetric QR Algorithm for Distributed Memory Architectures. SIAM J. Sci. Comput. 24(1), 284–311 (2002)
Kressner, D.: Numerical Methods for General and Structured Eigenvalue Problems. LNCSE, vol. 46. Springer, Heidelberg (2005)
Kressner, D.: The Effect of Aggressive Early Deflation on the Convergence of the QR Algorithm. SIAM J. Matrix Anal. Appl. 30(2), 805–821 (2008)
Lang, B.: Effiziente Orthogonaltransformationen bei der Eigen- und Singulärwertzerlegung. Habilitationsschrift (1997)
Watkins, D.S.: The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods. SIAM, Philadelphia (2007)
Watkins, D.S.: Francis’s Algorithm. Amer. Math. Monthly (2010) (to appear)
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Kågström, B., Kressner, D., Shao, M. (2012). On Aggressive Early Deflation in Parallel Variants of the QR Algorithm. In: Jónasson, K. (eds) Applied Parallel and Scientific Computing. PARA 2010. Lecture Notes in Computer Science, vol 7133. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28151-8_1
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DOI: https://doi.org/10.1007/978-3-642-28151-8_1
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