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Register a new userLU factorization with panel rank revealing pivoting and its communication avoiding version
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Amal Khabou
Publication date
2012
fields
Informatics Engineering
and research's language is
English
Authors
Amal Khabou
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We present the LU decomposition with panel rank revealing pivoting (LU_PRRP), an LU factorization algorithm based on strong rank revealing QR panel factorization. LU_PRRP is more stable than Gaussian elimination with partial pivoting (GEPP). Our extensive numerical experiments show that the new factorization scheme is as numerically stable as GEPP in practice, but it is more resistant to pathological cases and easily solves the Wilkinson matrix and the Foster matrix. We also present CALU_PRRP, a communication avoiding version of LU_PRRP that minimizes communication. CALU_PRRP is based on tournament pivoting, with the selection of the pivots at each step of the tournament being performed via strong rank revealing QR factorization. CALU_PRRP is more stable than CALU, the communication avoiding version of GEPP. CALU_PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail.
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We present a novel recursive algorithm for reducing a symmetric matrix to a triangular factorization which reveals the rank profile matrix. That is, the algorithm computes a factorization $mathbf{P}^Tmathbf{A}mathbf{P} = mathbf{L}mathbf{D}mathbf{L}^T$ where $mathbf{P}$ is a permutation matrix, $mathbf{L}$ is lower triangular with a unit diagonal and $mathbf{D}$ is symmetric block diagonal with $1{times}1$ and $2{times}2$ antidiagonal blocks. The novel algorithm requires $O(n^2r^{omega-2})$ arithmetic operations. Furthermore, experimental results demonstrate that our algorithm can even be slightly more than twice as fast as the state of the art unsymmetric Gaussian elimination in most cases, that is it achieves approximately the same computational speed. By adapting the pivoting strategy developed in the unsymmetric case, we show how to recover the rank profile matrix from the permutation matrix and the support of the block-diagonal matrix. There is an obstruction in characteristic $2$ for revealing the rank profile matrix which requires to relax the shape of the block diagonal by allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient. This relaxed decomposition can then be transformed into a standard $mathbf{P}mathbf{L}mathbf{D}mathbf{L}^Tmathbf{P}^T$ decomposition at a negligible cost.
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