1,190 research outputs found
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 , 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 , where or is a product of bidiagonal
matrices, showing that strong results are obtained when the are
nonnegative or have a checkerboard sign pattern. We show that given the \fact\
of an totally nonnegative matrix into the product of bidiagonal
matrices, can be computed in flops and that in
floating-point arithmetic the computed result has small relative error, no
matter how large 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
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