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Minimal-storage high-performance Cholesky factorization via blocking and recursion

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2 Author(s)
Gustavson, F.G. ; IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, USA ; Jonsson, I.

We present a novel practical algorithm for Cholesky factorization when the matrix is stored in packed format by combining blocking and recursion. The algorithm simultaneously obtains Level 3 performance, conserves about half the storage, and avoids the production of Level 3 BLAS for packed format. We use recursive packed format, which was first described by Andersen et al. [1] . Our algorithm uses only DGEMM and Level 3 kernel routines; it first transforms standard packed format to packed recursive lower row format. Our new algorithm outperforms the Level 3 LAPACK routine DPOTRF even when we include the cost of data transformation. (This is true for three IBM platforms—the POWER3, the POWER2, and the PowerPC 604e.) For large matrices, blocking is not required for acceptable Level 3 performance. However, for small matrices the overhead of pure recursion and/or data transformation is too high. We analyze these costs analytically and provide de tailed cost estimates. We show that blocking combined with recursion reduces all overheads to a tiny, acceptable level. However, a new problem of nonlinear addressing arises. We use two-dimensional mappings (tables) or data copying to overcome the high costs of directly computing addresses that are nonlinear functions of i and j.

Note: The Institute of Electrical and Electronics Engineers, Incorporated is distributing this Article with permission of the International Business Machines Corporation (IBM) who is the exclusive owner. The recipient of this Article may not assign, sublicense, lease, rent or otherwise transfer, reproduce, prepare derivative works, publicly display or perform, or distribute the Article.  

Published in:

IBM Journal of Research and Development  (Volume:44 ,  Issue: 6 )