Linear algebra on high performance computers

This is a survey of some work recently done at Argonne National Laboratory in an attempt to discover ways to construct numerical software for high performance computers. The numerical algorithms discussed are taken from several areas of numerical linear algebra. We discuss certain architectural features of advanced computer architectures that will affect the design of algorithms. The technique of restructuring algorithms in terms of certain modules is reviewed. This technique has proved very successful in obtaining a high level of transportability without severe loss of performance on a wide variety of both vector and parallel computers. The module technique is demonstrably effective for dense linear algebra problems. However, in the case of sparse and structured problems it may be difficult to identify general modules that will be as effective. New algorithms have been devised for certain problems in this category. We present examples in three important areas: banded systems, sparse QR - factorization, and symmetric eigenvalue problems.