LU Factorization with Panel Rank Revealing Pivoting and Its Communication Avoiding Version

Abstract

We present block LU factorization with panel rank revealing pivoting (block LU_PRRP), a decomposition algorithm based on strong rank revealing QR panel factorization. Block LU_PRRP is more stable than Gaussian elimination with partial pivoting (GEPP), with a theoretical upper bound of the growth factor of $(1+ \tau b)^{(n/ b)-1}$, where $b$ is the size of the panel used during the block factorization, $\tau$ is a parameter of the strong rank revealing QR factorization, $n$ is the number of columns of the matrix, and for simplicity we assume that n is a multiple of b. We also assume throughout the paper that $2\leq b \ll n$. For example, if the size of the panel is $b = 64$, and $\tau = 2$, then $(1+2b)^{(n/b)-1} = (1.079)^{n-64} \ll 2^{n-1}$, where $2^{n-1}$ 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. The block LU_PRRP factorization does only $O(n^2 b)$ additional floating point operations compared to GEPP. We also present block CALU_PRRP, a version of block LU_PRRP that minimizes communication and 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. Block CALU_PRRP is more stable than CALU, the communication avoiding version of GEPP, with a theoretical upper bound of the growth factor of $(1+ \tau b)^{{n\over b}(H+1)-1}$, where $H$ is the height of the reduction tree used during tournament pivoting. The upper bound of the growth factor of CALU is $2^{n(H+1)-1}$. Block CALU_PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail.