When solving large linear systems of equations arising from the discretization of elliptic boundary value problems, a combination of iterative methods and preconditioners based on incomplete LU factorizations is frequently used. Given a model problem with variable coefficients, we investigate a class of incomplete LU factorizations depending on a relaxation parameter. We show that the associate...

Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then the catenary degree of S (denoted c(S)) and the tame degree of S (denoted t(S)) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we describe met...

Matrices with off-diagonal decay appear in a variety of fields in mathematics and in numerous applications, such as signal processing, statistics, communications engineering, condensed matter physics, and quantum chemistry. Numerical algorithms dealing with such matrices often take advantage (implicitly or explicitly) of the empirical observation that this off-diagonal decay property seems to b...

The null-space method for solving saddle point systems of equations has long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified the equivalence of the null-space method and matrix factorizations. In this report, we review these findings, highlight links between them, and bring t...

We prove the following two variants of the defect theorem. Let X be a nite set of words over a nite alphabet. Then if a nonperiodic bi-in nite word w has two X-factorizations, then the combinatorial rank of X is at most card(X) 1, i.e. there exists a set F such that X F+ with card(F ) < card(X). Further, if card(X) = 2 and a bi-in nite word possesses two X-factorizations which are not shift-equ...

This paper presents an e cient implementation of incomplete LU (ILU) factorizations that are derived from the Crout version of Gaussian elimination (GE). At step k of the elimination, the k-th row of U and the k-th column of L are computed using previously computed rows of U and columns of L. The data structure and implementation borrow from already known techniques used in developing both spar...

Let NA be the monoid generated byA = {a1, . . . ,an} ⊆ Z.We introduce the homogeneous catenary degree of NA as the smallest N ∈ N with the following property: for each a ∈ NA and any two factorizations u,v of a, there exists factorizations u = w1, . . . ,wt = v of a such that, for every k, d(wk,wk+1) ≤ N, where d is the usual distance between factorizations, and the length of wk, |wk|, is less ...

We consider the problem of counting transitive factorizations of permutations; that is, we study tuples (σr , . . . , σ1) of permutations on {1, . . . , n} such that (1) the product σr · · · σ1 is equal to a given target permutation π , and (2) the group generated by the factors σi acts transitively on {1, . . . , n}. This problem is widely known as the Hurwitz Enumeration Problem, since an enc...

Time-Varying Lossless Systems and the Inversion of Large Structured Matrices In the inversion of large matrices, direct methods might give undesired ‘unstable’ results. Valuable insight into the mechanism of this effect is obtained by viewing the matrix as the input-output operator of a time-varying system, which allows to translate ‘unstable’ into ‘anticausal’ but bounded inverses. Inner-outer...

In 1979, two constructions for making partitionable graphs were introduced in (by Chv2 atal et al. (Ann. Discrete Math. 21 (1984) 197)). The graphs produced by the second construction are called CGPW graphs. A near-factorization (A; B) of a %nite group is roughly speaking a non-trivial factorization of G minus one element into two subsets A and B. Every CGPW graph with n vertices turns out to b...