Accurate and Efficient Expression Evaluation and Linear Algebra

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use the so-called Traditional Model (TM). We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high accuracy algorithm or a proof that none exists. When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as $x+y+z$, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case. Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl

Enumeration of Matchings: Problems and Progress

This document is built around a list of thirty-two problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and on-line literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems. (Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley), Mathematical Science Research Institute publication #37, Cambridge University Press, 199

The Power of Bidiagonal Matrices

Bidiagonal matrices are widespread in numerical linear algebra, not least because of their use in the standard algorithm for computing the singular value decomposition and their appearance as LU factors of tridiagonal matrices. We show that bidiagonal matrices have a number of interesting properties that make them powerful tools in a variety of problems, especially when they are multiplied together. We show that the inverse of a product of bidiagonal matrices is insensitive to small componentwise relative perturbations in the factors if the factors or their inverses are nonnegative. We derive componentwise rounding error bounds for the solution of a linear system $Ax = b$, where $A$ or $A^{-1}$ is a product $B_1 B_2\dots B_k$ of bidiagonal matrices, showing that strong results are obtained when the $B_i$ are nonnegative or have a checkerboard sign pattern. We show that given the \fact\ of an $n\times n$ totally nonnegative matrix $A$ into the product of bidiagonal matrices, $\|A^{-1}\|_{\infty}$ can be computed in $O(n^2)$ flops and that in floating-point arithmetic the computed result has small relative error, no matter how large $\|A^{-1}\|_{\infty}$ is. We also show how factorizations involving bidiagonal matrices of some special matrices, such as the Frank matrix and the Kac--Murdock--Szeg\"o matrix, yield simple proofs of the total nonnegativity and other properties of these matrices