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

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 d...