Dynamic Programming Approaches to Optimizing the Blocking Strategy for Basic Matrix Decompositions

Chapter

Abstract

In this chapter, we survey several approaches to optimizing the blocking strategy for basic matrix decompositions, such as LU, Cholesky, and QR. Conventional blocking strategies such as fixed-size blocking and recursive blocking are widely used to optimize the performance of these decompositions. However, these strategies have only a small number of parameters such as the block size or the level of recursion and are not sufficiently flexible to exploit the performance of modern high-performance architectures. As such, several attempts have been made to define a much larger class of strategies and to choose the best strategy among them according to the target machine and the matrix size. The number of candidate strategies is usually exponential in the size of the matrix. However, with the use of dynamic programming, the cost of optimization can be reduced to a realistic level. As representatives of such approaches, we survey variable-size blocking, generalized recursive blocking, and the combination of variable-size blocking and the TSQR algorithm. Directions for future research are also discussed.

Notes

Acknowledgments

We would like to express our sincere gratitude to the anonymous reviewer, whose comments helped us to greatly improve the quality of this chapter. We are also grateful to the members of the Auto-Tuning Research Group for engaging in valuable discussions and to Prof. Shao-Liang Zhang, Prof. Tomohiro Sogabe, and other members of the Zhang laboratory for their continuous support. This study is supported in part by the Ministry of Education, Science, Sports, and Culture through a Grant-in-Aid for Scientific Research on Priority Areas, “i-explosion” (No. 21013014), a Grant-in-Aid for Scientific Research (B) (No. 21300013), a Grant-in-Aid for Scientific Research (C) (No. 21560065), and a Grant-in-Aid for Scientific Research (A) (No. 20246027).

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Copyright information

© Springer New York 2011

Authors and Affiliations

  1. 1.Kobe UniversityKobeJapan

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