New Generalized Data Structures for Matrices Lead to a Variety of High Performance Dense Linear Algebra Algorithms

Abstract

This paper is a condensation and continuation of [9]. We present a novel way to produce dense linear algebra factorization algorithms. The current state-of-the-art (SOA) dense linear algebra algorithms have a performance inefficiency and hence they give sub-optimal performance for most of Lapack’s factorizations. We show that standard Fortran and C two dimensional arrays are the main reason for the inefficiency. For the other standard format ( packed one dimensional arrays for symmetric and/or triangular matrices ) the situation is much worse. We introduce RFP (Rectangular Full Packed) format which represent a packed array as a full array. This means that performance of Lapack’s packed format routines becomes equal to or better than their full array counterparts. Returning to full format, we also show how to correct these performance inefficiencies by using new data structures (NDS) along with so-called kernel routines. The NDS generalize the current storage layouts for both standard layouts. We use the Algorithms and Architecture approach to justify why our new methods gives higher efficiency. The simplest forms of the new factorization algorithms are a direct generalization of the commonly used LINPACK algorithms. All programming for our NDS can be accomplished in standard Fortran, through the use of three- and four-dimensional arrays. Thus, no new compiler support is necessary. Combining RFP format with square blocking or just using SBP (Square Block Packed) format we are led to new high performance ways to produce ScaLapack type algorithms.