An LU factorization algorithm for parallel supercomputers with memory hierarchies

Abstract

A newly designed parallel algorithm for solving LU factorization of huge dense matrices was developed for parallel vector supercomputers with a hierarchy of memory layers (i.e., local memories, shared memory, semiconductor extended storage and magnetic disk) . The algorithm is based on Gaussian elimination and optimizes data transfers among memory layers by recursively using a block partitioning method. Using four memory layers, an LU factorization for a 32,768 x 32,768 dense matrix was calculated in 10 hours and 40 minutes on the HPP-LHS supercomputer system developed under the MITI (the Ministry of International Trade and Industry) Supercomputer Project.Required memory capacity for the gigantic matrix is 8 GB, and the whole matrix data area was allocated to magnetic disk for this calculation. The execution speed with 4 processors was 2.8 times faster than that with 1 processor, even using a magnetic disk, and the algorithm was proved to be effective.