M-matrices satisfy Newton's inequalities

Newton's inequalities $c_n^2 \ge c_{n-1}c_{n+1}$ are shown to hold for the normalized coefficients $c_n$ of the characteristic polynomial of any $M$- or inverse $M$-matrix. They are derived by establishing first an auxiliary set of inequalities also valid for both of these classes. They are also used to derive some new necessary conditions on the eigenvalues of nonnegative matrices.Comment: 6 page

Applications of the duality method to generalizations of the Jordan canonical form

The Jordan normal form for a matrix over an arbitrary field and the canonical form for a pair of matrices under contragredient equivalence are derived using Ptak's duality method.Comment: 6 page

Communication-optimal Parallel and Sequential Cholesky Decomposition

Numerical algorithms have two kinds of costs: arithmetic and communication, by which we mean either moving data between levels of a memory hierarchy (in the sequential case) or over a network connecting processors (in the parallel case). Communication costs often dominate arithmetic costs, so it is of interest to design algorithms minimizing communication. In this paper we first extend known lower bounds on the communication cost (both for bandwidth and for latency) of conventional (O(n^3)) matrix multiplication to Cholesky factorization, which is used for solving dense symmetric positive definite linear systems. Second, we compare the costs of various Cholesky decomposition implementations to these lower bounds and identify the algorithms and data structures that attain them. In the sequential case, we consider both the two-level and hierarchical memory models. Combined with prior results in [13, 14, 15], this gives a set of communication-optimal algorithms for O(n^3) implementations of the three basic factorizations of dense linear algebra: LU with pivoting, QR and Cholesky. But it goes beyond this prior work on sequential LU by optimizing communication for any number of levels of memory hierarchy.Comment: 29 pages, 2 tables, 6 figure

Toward accurate polynomial evaluation in rounded arithmetic

Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here ``accurately'' means with relative error less than 1, i.e., with some correct leading digits. The answer depends on the model of rounded arithmetic: We assume that for any arithmetic operator $op(a,b)$, for example $a+b$ or $a \cdot b$, its computed value is $op(a,b) \cdot (1 + \delta)$, where $| \delta |$ is bounded by some constant $\epsilon$ where $0 < \epsilon \ll 1$, but $\delta$ is otherwise arbitrary. This model is the traditional one used to analyze the accuracy of floating point algorithms.Our ultimate goal is to establish a decision procedure that, for any $p$ and $\cal D$, either exhibits an accurate algorithm or proves that none exists. In contrast to the case where numbers are stored and manipulated as finite bit strings (e.g., as floating point numbers or rational numbers) we show that some polynomials $p$ are impossible to evaluate accurately. The existence of an accurate algorithm will depend not just on $p$ and $\cal D$, but on which arithmetic operators and which constants are are available and whether branching is permitted. Toward this goal, we present necessary conditions on $p$ for it to be accurately evaluable on open real or complex domains ${\cal D}$. We also give sufficient conditions, and describe progress toward a complete decision procedure. We do present a complete decision procedure for homogeneous polynomials $p$ with integer coefficients, {\cal D} = \C^n, and using only the arithmetic operations $+$, $-$ and $\cdot$.Comment: 54 pages, 6 figures; refereed version; to appear in Foundations of Computational Mathematics: Santander 2005, Cambridge University Press, March 200