LU factorization with panel rank revealing pivoting and its communication avoiding version

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), with a theoretical upper bound of the growth factor of (1+ τb) n b , where b is the size of the panel used during the block factorization, τ is a parameter of the strong rank revealing QR factorization, and n is the number of columns of the matrix. For example, if the size of the panel is b = 64, and τ = 2, then (1+2b) = (1.079) 2n−1, where 2n−1 is the upper bound of the growth factor of 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. The LU PRRP factorization does only O(nb) additional floating point operations compared to GEPP. 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, with a theoretical upper bound of the growth factor of (1 + τb) n b (H+1)−1, where b is the size of the panel used during the factorization, τ is a parameter of the strong rank revealing QR factorization, n is the number of columns of the matrix, and H is the height of the reduction tree used during tournament pivoting. The upper bound of the growth factor of CALU is 2n(H+1)−1. CALU PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail. Key-words: LU factorization, numerical stability, communication avoiding, strong rank revealing QR factorization LU PRRP and CALU PRRP 3