In this section, we first discuss the uses of dense LU factorization in several fields. We next develop a block-partitioned version of the k, or right-looking, variant of the LU factorization algorithm. In subsequent sections, the parallelization of this algorithm is described in detail in order to highlight the issues and considerations that must be taken into account in developing an efficient, scalable, and transportable dense linear algebra library for MIMD, distributed memory, concurrent computers.
Dense matrix computations, such as LU factorization, have important applications, as discussed in a recent survey by Edelman [28]. A major source of large dense linear systems is the solution of problems by the boundary element method. In this method integral equations defined on the boundary of a region of interest are used to compute some final desired quantity in three-dimensional space. The dense linear systems generated are commonly solved using LU factorization. Electromagnetic scattering studies make use of the boundary element method, which is usually referred to as the method of moments in this context [34, 49]. This approach is used in the design of aircraft with small radar cross-sections, and in the design of satellite antennae. Boundary element methods are also used in the study of fluid flows, and here the variant of the boundary element method used is called the panel method [36, 37].