*Jack Dongarra and Piotr Luszczek*

The increasing availability of advanced-architecture computers, at affordable costs, has had a significant effect on all spheres of scientific computation. In this chapter, we'll show the need for designers of computing algorithms to make expeditious and substantial adaptations to algorithms, in reaction to architecture changes, by closely examining one simple but important algorithm in mathematical software: Gaussian elimination for the solution of linear systems of equations.

At the application level, science has to be captured in mathematical models, which in turn are expressed algorithmically and ultimately encoded as software. At the software level, there is a continuous tension between performance and portability on the one hand, and understandability of the underlying code. We'll examine these issues and look at trade-offs that have been made over time. Linear algebra—in particular, the solution of linear systems of equations—lies at the heart of most calculations in scientific computing. This chapter focuses on some of the recent developments in linear algebra software designed to exploit advanced-architecture computers over the decades.

There are two broad classes of algorithms: those for dense matrices and those for sparse matrices. A matrix is called
*sparse* if it contains a substantial number of zero elements. For sparse matrices, radical savings in space ...

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