A generalization of the implicit LU algorithm to an arbitrary initial matrix

Abstract

The implicit LU algorithm of the (basic) ABS class corresponds to the parameter choices H 1=I, z i =w i =e i . The algorithm can be considered as the ABS version of the classic LU factorization algorithm. In this paper we consider the generalization where the initial matrix H1 is arbitrary except for a certain condition. We prove that every algorithm in the ABS class is equivalent, in the sense of generating the same set of search directions, to a generalized implicit LU algorithm, with suitable initial matrix, that can be interpreted as a right preconditioning matrix. We discuss some consequences of this result, including a straightforward derivation of Bienayme's (1853) classical result on the equivalence of the Gram–Schmidt orthogonalization procedure with Gaussian elimination on the normal equations.