Aggregations of Elementary Transformations

The techniques of aggregating transformations have been used in the development of blocked algorithms, in particular, the compact representations of aggregated Householder transformation have been wellknown. We show in this paper that the aggregation techniques can be integrated in a framework for aggregating general elementary transformations. The framework consists of three basic compact forms, as extensions of the three compact forms for aggregated Householder transformations, and two componentwise representations of the compact forms. The componentwise representations, introduced in this paper, are a central link to various computational procedures (known and new) for the compact forms, and provide a stepping stone for numerical analysis of blocked algorithms. We show also the connection between the computation of the compact forms and the inverse of general nonsingular triangular matrices.