A Tutorial on the SWEEP Operator

Abstract

Abstract The importance of the SWEEP operator in statistical computing is not so much that it is an inversion technique, but rather that it is a conceptual tool for understanding the least squares process. The SWEEP operator can be programmed to produce generalized inverses and create, as by-products, such items as the Forward Doolittle matrix, the Cholesky decomposition matrix, the Hermite canonical form matrix, the determinant of the original matrix, Type I sums of squares, the error sum of squares, a solution to the normal equations, and the general form of estimable functions. First, this tutorial describes the use of Gauss-Jordan elimination for least squares and continues with a description of a completely generalized sweep operator that computes and stores (X′X)−, (X′X)− X′X, (X′X)− X′Y, and Y′Y — Y′X(X′X)− X′Y, all in the space of a single upper triangular matrix.