Recognition and ordering algorithms concerning global inheritance in LU factorisations
Date
1988
Authors
Slater, Terence Arthur
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Abstract
The concept of forward lower restricted orderings of digraphs arises naturally from the study of inherited entries in LU factorizations of matrices. If a matrix A has a unique left unit LU factorization A = W then an entry u, 1 of U with J ~ i is said to be inherited from A if u, 1 = a, 1 for reasons related to the combinational structure of A and not because of the specific values of the nonzero matrix elements It can be proved that the entire strict upper triangular part of A is inherited if and only if the digraph D of the matrix has the property that for all, and J with, < J there is no path of length greater than one from , to J in D such that all intermediate nodes of the path are m { 1 2 , -1} Any digraph on the node set { 1 2 n } having this property is said to be forward lower restricted (FLR) ordered.
We state and prove theorems which characterize the FLR ordered trees and the maximal FLR ordered digraphs Polynomial time algorithms for deciding if a given ordered digraph is FLR ordered and for finding an FLR ordering of an arbitrary digraph are presented The latter algorithm also detects that a digraph is not FLR orderable In addition, a more efficient algorithm for finding FLR orderings of digraphs containing a spanning tree is given some results concerning an attempt to characterize FLR orderable digraphs in terms of their cycle structure are included.
All of the results of this thesis also extend to inheritance of entries in the matrix L of the LU factorization and to inheritance in UL factorizations.