Nearly optimal computations with structured matrices
Victor Y. Pan,Elias P. Tsigaridas +1 more
- pp 21-30
TLDR
In this paper, the complexity of solving nonsingular linear systems of equations with structured matrices has been studied and the authors present a Boolean complexity analysis for the problem of polynomial multiplication and division.Abstract:
We estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic and most popular classes, that is, Toeplitz, Hankel, Cauchy and Vandermonde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in [10], except for rational interpolation, and we supply them now. All known Boolean cost estimates from [10] for these problems rely on using Kronecker product. This implies the d-fold precision increase for the d-th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions. This also enables further extensions of our estimates to cover Trummer's important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes.read more
Citations
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Proceedings ArticleDOI
FAQ: Questions Asked Frequently
TL;DR: InsideOut as mentioned in this paper is a variation of the traditional dynamic programming approach for constraint programming based on variable elimination, which adds a couple of simple twists to basic variable elimination in order to deal with the generality of FAQ, to take full advantage of Grohe and Marx's fractional edge cover framework, and of the analysis of recent worstcase optimal relational join algorithms.
Journal ArticleDOI
Multivariate polynomials, duality, and structured matrices
Bernard Mourrain,Victor Y. Pan +1 more
TL;DR: Simplifying and/or generalizing the known reduction of the multivariate polynomial systems to the matrix eigenproblem, the derivation of the Bezout and Bernshtein bounds on the number of the roots, and the construction of multiplication tables yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving multivariate poaching systems of equations.
Proceedings ArticleDOI
Matrix-vector multiplication in sub-quadratic time: (some preprocessing required)
TL;DR: Any matrix over any finite semiring can be preprocessed in O time, such that all subsequent vector multiplications with A can be performed in O(εlog) time, for all ε > 0.
Journal ArticleDOI
Structured matrices and newton's iteration: unified approach
Victor Y. Pan,Youssef Rami +1 more
TL;DR: In this paper, the convergence rate of Newton's iteration for n × n structured matrices with O(n ) entries of their short generators is studied. But the authors do not consider the problem of controlling the length of the generators, which tends to grow quite rapidly in the iterative process.
References
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Book
The Art of Computer Programming
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