Abstract
The algorithm of Multiple Relatively Robust Representations (MRRR or MR3) computes k eigenvalues and eigenvectors of a symmetric tridiagonal matrix in O(nk) arithmetic operations. Large problems can be effectively tackled with existing distributed-memory parallel implementations of MRRR; small and medium size problems can instead make use of LAPACK’s routine xSTEMR. However, xSTEMR is optimized for single-core CPUs, and does not take advantage of today’s multi-core and future many-core architectures. In this paper we discuss some of the issues and trade-offs arising in the design of MR3–SMP, an algorithm for multi-core CPUs and SMP systems. Experiments on application matrices indicate that MR3–SMP is both faster and obtains better speedups than all the tridiagonal eigensolvers included in LAPACK and Intel’s Math Kernel Library (MKL).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Wilkinson, J.: The Calculation of the Eigenvectors of Codiagonal Matrices. Comp. J. 1(2), 90–96 (1958)
Francis, J.: The QR Transform - A Unitary Analogue to the LR Transformation, Part I and II. The Comp. J. 4 (1961/1962)
Kublanovskaya, V.: On some Algorithms for the Solution of the Complete Eigenvalue Problem. Zh. Vych. Mat. 1, 555–572 (1961)
Cuppen, J.: A Divide and Conquer Method for the Symmetric Tridiagonal Eigenproblem. Numer. Math. 36, 177–195 (1981)
Gu, M., Eisenstat, S.C.: A Divide-and-Conquer Algorithm for the Symmetric Tridiagonal Eigenproblem. SIAM J. Matrix Anal. Appl. 16(1), 172–191 (1995)
Dhillon, I., Parlett, B.: Multiple Representations to Compute Orthogonal Eigenvectors of Symmetric Tridiagonal Matrices. Linear Algebra Appl. 387, 1–28 (2004)
Anderson, E., Bai, Z., Bishof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. SIAM (1999)
Demmel, J., Marques, O., Parlett, B., Vömel, C.: Performance and Accuracy of LAPACK’s Symmetric Tridiagonal Eigensolvers. SIAM J. Sci. Comp. 30, 1508–1526 (2008)
Dongarra, J., Du Cruz, J., Duff, I., Hammarling, S.: A Set of Level 3 Basic Linear Algebra Subprograms. ACM Trans. Math. Software 16, 1–17 (1990)
Demmel, J., Dhillon, I., Ren, H.: On the Correctness of some Bisection-like Parallel Eigenvalue Algorithms in Floating Point Arithmetic. Electron. Trans. Numer. Anal. 3, 116–149 (1995)
Parlett, B., Dhillon, I.: Relatively Robust Representations of Symmetric Tridiagonals. Linear Algebra Appl. 309, 121–151 (1999)
Dhillon, I., Parlett, B.: Orthogonal Eigenvectors and Relative Gaps. SIAM J. Matrix Anal. Appl. 25, 858–899 (2004)
Parlett, B., Marques, O.: An Implementation of the DQDS Algorithm (Positive Case). Linear Algebra Appl. 309, 217–259 (1999)
Parlett, B.: The Symmetric Eigenvalue Problem. Prentice-Hall (1980)
Bientinesi, P., Dhillon, I., van de Geijn, R.: A Parallel Eigensolver for Dense Symmetric Matrices Based on Multiple Relatively Robust Representations. SIAM J. Sci. Comp. 21, 43–66 (2005)
Vömel, C.: ScaLAPACK’s MRRR Algorithm. ACM Trans. on Math. Software 37(1), 1:1–1:35 (2010)
Dhillon, I., Parlett, B., Vömel, C.: The Design and Implementation of the MRRR Algorithm. ACM Trans. on Mathem. Software 32, 533–560 (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Petschow, M., Bientinesi, P. (2012). The Algorithm of Multiple Relatively Robust Representations for Multi-core Processors. 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_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-28151-8_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28150-1
Online ISBN: 978-3-642-28151-8
eBook Packages: Computer ScienceComputer Science (R0)