I/O-Optimal Algorithms for Symmetric Linear Algebra Kernels

Abstract

In this paper, we consider two fundamental symmetric kernels in linear algebra: the Cholesky factorization and the symmetric rankupdate (SYRK), with the classical three nested loops algorithms for these kernels. In addition, we consider a machine model with a fast memory of size and an unbounded slow memory. In this model, all computations must be performed on operands in fast memory, and the goal is to minimize the amount of communication between slow and fast memories. As the set of computations is fixed by the choice of the algorithm, only the ordering of the computations (the schedule) directly influences the volume of communications.We prove lower bounds of 1 3 √ 2 3 CCS CONCEPTS• Theory of computation → Shared memory algorithms; Communication complexity; • Mathematics of computing → Solvers.