Preliminaries

Abstract

In recent years, the price of computer hardware has dropped drastically. Powerful, small size computers are now easily purchased by many groups, including statistics departments. The resulting ease with which data may be gathered and manipulated has led to a corresponding increase in the size of the average statistical problem. To analyze data from such problems, the statistician must have access to algorithms that are sufficiently economical of space that they will run on these small machines. One area of analysis where there is a need for algorithms that are economical of space is in the fitting of linear models. In many instances the number of parameters may be quite large, as for example when data are blocked by location, subject, or time period. Computer implementations of standard algorithms such as QR decompositions (Lawson and Hanson 1974) or the symmetric sweep method (Goodnight 1979) require O(r²) storage to fit a model having r parameters, since they must store a triangular structure of side r. When r is large, it may not be possible to retain both the data and the triangular structures in the computer’s memory at the same time. In such a situation the data or the triangular structure (or both) could be placed on a secondary storage medium such as magnetic disk, and retrieved when necessary. This may slow down fitting considerably. For problems that are not too large, there is an alternative. An algorithm having only O(r) storage requirements may not need to use secondary storage, and consequently may run faster.