Summary: MINIMUM DEVIATION, QUASI-LU FACTORIZATION OF
M.I. BUENO AND C.R. JOHNSON
Abstract. Not all matrices enjoy the existence of an LU factorization. For those that do not,
a number of "repairs" are possible. For nonsingular matrices we offer here a permutation-free repair
in which the matrix is factored ~L ~U, with ~L and ~U collectively as near as possible to lower and upper
triangular (in a natural sense defined herein). Such factorization is not generally unique in any
sense. In the process, we investigate further the structure of matrices without LU factorization and
permutations that produce an LU factorization.
Key words. LU factorization, LPU factorization, quasi-LU factorization, sparsity pattern.
AMS subject classifications. 15A21, 15A23, 05A05, 05B25.
1. Introduction. Factorization of an m-by-m matrix A into a lower triangular
matrix L and an upper triangular matrix U ("LU factorization") is important for a
variety of computational, theoretical and applied reasons. Although the LU factoriza-
tion is well known for its applications to the solution of linear systems of equations,
there are many other applications of this factorization. Consider for instance the use
of the LU factorization to compute the singular values of bidiagonal matrices  or
to prove determinantal inequalities . In any case, the LU factorization is an im-
portant mathematical concept by itself, and a generalization of this idea is studied
here. Unfortunately, the LU factorization does not always exist. Characterizations