George Poole

A Survey on M-Matrices

This is an expository-research paper which attempts to unify many of the more recent results on nonsingular and singular M-matrices (in the sense of Ostrowski and Schneider). Insight and direction are given to additional research on this very useful class of matrices and some new results are presented.

A Geometric Analysis of Gaussian Elimination. II

Abstract In Part I of this work, we began a discussion of the numeric consequences of hyperplane orientation in Gaussian elimination. We continue this discussion by introducing the concept of back-substitution-phase error multipliers . These error multipliers help to explain many of the previously unproven or poorly understood observations concerning Gaussian elimination in a finite-precision environment. A new pivoting strategy designed to control both sweepout phase roundoff error and back-substitution-phase instability is also presented. This new strategy, called rooks pivoting , is only slightly more expensive than partial pivoting yet produces results comparable to those produced by complete pivoting.

Computing nonnegative rank factorizations

Abstract The existence of nonnegative generalized inverses in terms of nonnegative rank factorizations is considered. An algorithm is presented which computes a nonnegative rank factorization of a nonnegative matrix when a nonnegative 1-inverse exists.

The Rook's pivoting strategy

Abstract Based on the geometric analysis of Gaussian elimination (GE) found in Neal and Poole (Linear Algebra Appl. 173 (1992) 239–264) and Poole and Neal (Linear Algebra Appl. 149 (1991) 249–272; 162–164 (1992) 309–324), a new pivoting strategy, Rook’s pivoting (RP), was introduced in Neal and Poole (Linear Algebra Appl. 173 (1992) 239–264) which encourages stability in the back-substitution phase of GE while controlling the growth of round-off error during the sweep-out. In fact, Foster (J. Comput. Appl. Math. 86 (1997) 177–194) has previously shown that RP, as with complete pivoting, cannot have exponential growth error. Empirical evidence presented in Neal and Poole (Linear Algebra Appl. 173 (1992) 239–264) showed that RP produces computed solutions with consistently greater accuracy than partial pivoting. That is, Rooks pivoting is, on average, more accurate than partial pivoting, with comparable costs. Moreover, the overhead to implement Rooks pivoting in a scalar or serial environment is only about three times the overhead to implement partial pivoting. The theoretical proof establishing this fact is presented here, and is empirically confirmed in this paper and supported in Foster (J. Comput. Appl. Math. 86 (1997) 177–194).

Generalized

Recently, two distinct directions have been taken in an attempt to generalize the definition of an M-matrix. Even for nonsingular matrices, these two generalizations are not equivalent. The role of these and other classes of recently defined matrices is indicated showing their usefulness in various applications.

M

Abstract Following Berman and Plemmons [5], Werner [17], Poole and Barker [13], and others, we investigate some generalizations of matrix monotonicity and consider their relation to MP matrices. We prove some observations on almost monotone, MP, and group monotone matrices. However, of much interest to us is the problem whether a symmetric positive semidefinite matrix is MP if and only if it is monotone on its range. To consider this question we obtain several results on the structure of range monotone matrices.

-matrices and applications

One of Leo Mosers geometry problems is referred to as the Worm Problem [10]: “What is the (convex) region of smallest area which will accommodate (or cover) every planar arc of length 1?” For example, it is easy to show that the circular disk with diameter 1 will cover every planar arc of length 1. The area of the disk is approximately 0.78539. Here we show that a solution to the Worm Problem of Moser is a region with area less than 0.27524.

More on generalizations of matrix monotonicity

Abstract If A is a nonsingular M -matrix, the elements of the sequence { A − k } all have the same zero pattern. Using the Drazin inverse, we show that a similar zero pattern invariance property holds for a class of matrices which is larger than the generalized M -matrices.

The worm problem of Leo Moser

Abstract For the linear system Ax = b , the ordered pair ( D, F ) of nonsingular diagonal matrices determine a scaling of the system through the two equations D ( AF ) y = Db , y = F −1 x . When scaling is implemented along with partial pivoting (PP) to solve Ax = b by Gaussian elimination (GE), it is well known that certain ordered pairs ( D, F ) produce better computed solutions than those obtained in the absence of scaling, while others produce worse solutions. The two most common explanations of this fact are (1) ( D, F ) modifies (magnifies or reduces) the classical condition number of A , and (2) ( D, F ) modifies the magnitudes of the elements of A . In case (2), if a scaling yields entries of approximately the same magnitude, it is called an equilibration. Here, the underlying hyperplane geometry of both the sweepout phase and the back-substitution phase of GE is used to achieve a new level of understanding. We present what we believe to be a better explanation of how scaling or equilibration influences PP in the selection of pivot equations, a process critical to both phases of GE.

Nonnegative matrices with power invariant zero patterns

In this paper, a class of Hessenberg matrices is presented for adoption as test matrices. The Moore-Penrose inverse and the Drazin inverse for each member of this class are determined explicitly.