Nullity Invariance for Pivot and the Interlace Polynomial

We show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given (square) matrix is described by the symmetric difference operator (for sets). We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof.Comment: small revision of Section 8 w.r.t. v2, 14 pages, 6 figure

Counting Unique-Sink Orientations

Unique-sink orientations (USOs) are an abstract class of orientations of the n-cube graph. We consider some classes of USOs that are of interest in connection with the linear complementarity problem. We summarise old and show new lower and upper bounds on the sizes of some such classes. Furthermore, we provide a characterisation of K-matrices in terms of their corresponding USOs.Comment: 13 pages; v2: proof of main theorem expanded, plus various other corrections. Now 16 pages; v3: minor correction

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

The Interlace Polynomial

In this paper, we survey results regarding the interlace polynomial of a graph, connections to such graph polynomials as the Martin and Tutte polynomials, and generalizations to the realms of isotropic systems and delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials, edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL

Practical improvements to class group and regulator computation of real quadratic fields

We present improvements to the index-calculus algorithm for the computation of the ideal class group and regulator of a real quadratic field. Our improvements consist of applying the double large prime strategy, an improved structured Gaussian elimination strategy, and the use of Bernstein's batch smoothness algorithm. We achieve a significant speed-up and are able to compute the ideal class group structure and the regulator corresponding to a number field with a 110-decimal digit discriminant

Incomplete LU Preconditioner Based on Max-Plus Approximation of LU Factorization

We present a new method for the a priori approximation of the orders of magnitude of the entries in the LU factors of a complex or real matrix A. This approximation is used to determine the positions of the largest entries in the LU factors of A, and these positions are used as the sparsity pattern for an incomplete LU factorization preconditioner. Our method uses max-plus algebra and is based solely on the moduli of the entries of A. We also present techniques for predicting which permutation matrices will be chosen by Gaussian elimination with partial pivoting. We exploit the strong connection between the field of Puiseux series and the max-plus semiring to prove properties of the max-plus LU factors. Experiments with a set of test matrices from the University of Florida Sparse Matrix Collection show that our max-plus LU preconditioners outperform traditional level of fill methods and have similar performance to those preconditioners computed with more expensive threshold-based methods