Towards an Efficient Tile Matrix Inversion of Symmetric Positive Definite Matrices on Multicore Architectures

TL;DR

This paper explores efficient parallel algorithms for inverting symmetric positive definite matrices using tile matrix techniques on multicore architectures, demonstrating the need for advanced compiler optimizations to enhance scalability.

Contribution

It introduces a tile-based approach for matrix inversion on multicore systems and discusses compiler techniques to improve parallelism beyond existing methods.

Findings

Tile algorithms can be effectively applied to matrix inversion.

Compiler optimizations significantly improve parallel performance.

Preliminary results show promising scalability improvements.

Abstract

The algorithms in the current sequential numerical linear algebra libraries (e.g. LAPACK) do not parallelize well on multicore architectures. A new family of algorithms, the tile algorithms, has recently been introduced. Previous research has shown that it is possible to write efficient and scalable tile algorithms for performing a Cholesky factorization, a (pseudo) LU factorization, and a QR factorization. In this extended abstract, we attack the problem of the computation of the inverse of a symmetric positive definite matrix. We observe that, using a dynamic task scheduler, it is relatively painless to translate existing LAPACK code to obtain a ready-to-be-executed tile algorithm. However we demonstrate that non trivial compiler techniques (array renaming, loop reversal and pipelining) need then to be applied to further increase the parallelism of our application. We present preliminary experimental results.