Linear algebra provides the foundational setting for the study of multivariable mathematics which in turn is the bedrock upon which most modern theories of mathematical physics rest including classical mechanics (rigid body mechanics), continuum mechanics (the mechanics of deformable material bodies), relativistic mechanics, quantum mechanics, etc. At the heart of linear algebra is the notion o...

In this chapter we discuss numerical software for linear algebra problems on parallel computers. We focus on some of the most common numerical operations: linear system solving and eigenvalue computations. Numerical operations such as linear system solving and eigenvalue calculations can be applied to two di erent kinds of matrix: dense and sparse. In dense systems, essentially every matrix ele...

at the current iterate x ∈ Rn. In practice, we find that the bulk of computational resources (memory, time) consumed by these algorithms is spent constructing and solving these linear systems, and not in the higher-level algorithmic details. That is not to say that the higher-level details do not impact performance; for example, algorithmic improvements could result in a reduction in the number...

Theoret ica l B a c k g r o u n d The procedure bisect is designed to replace the procedures tridibi I and 2 given in ES]. All three procedures are based essentially on the following theorem. Given a symmetric tridiagonal matrix with diagonal elements q to cn and off-diagonal elements b, to b, (with b 1 set equal to zero for completeness) and b i 4=0, then for any number 2 let the sequence Po (...

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This article describes the context, design, and recent development of the LAPACK for clusters (LFC) project. It has been developed in the framework of Self-Adapting Numerical Software (SANS) since we believe such an approach can deliver the convenience and ease of use of existing sequential environments bundled with the power and versatility of highly tuned parallel codes that execute on cluste...