Massive memory buys little speed for complete, in-core sparse Cholesky factorization on some scalar computers

Abstract

This note calls into question a claim one sometimes hears about the time it takes to compute a complete sparse Cholesky factorization (after a suitable symbolic factorization phase and without using auxiliary memory). The claim is that loop-free code or code that uses a list with one or more addresses or integers for each arithmetic operation runs considerably faster than code with more modest memory requirements, e.g., memory proportional to the number of nonzeros in the Cholesky factorization. (Loop-free code is a sequence of instructions each of which is executed at most once during the relevant calculation.) On some scalar machines that were commonly used when this paper was first written (e.g., various VAX and Sun-3 computers), one can often come within a factor or two of the fastest possible floating-point operation rate with a scheme that stores one integer per nonzero in the Cholesky factor.