Nearly Optimal Computations with Structured Matrices

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 most popular classes, that is, Toeplitz, Hankel, Cauchy and Van-der-monde 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 \cite{kirrinnis-joc-1998}, except for rational interpolation, which we supply now. All known Boolean cost estimates 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 representation of our tasks and algorithms in terms of both structured matrices and 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.Comment: (2014-04-10

Nearly optimal computations with structured matrices

International audienceWe 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. We supply them now as well as the Boolean complexity estimates for the important problems of multiplication of transposed Vandermonde matrix and its inverse by a vector. All known Boolean cost estimates from [10] for such 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, as well as the transposed Vandermonde matrices. It is known that the solution of Toeplitz, Hankel, Cauchy, Vandermonde, and transposed Vandermonde linear systems of equations is generally prone to numerical stability problems, and numerical problems arise even for multiplication of Cauchy, Vandermonde, and transposed Vandermonde matrices by a vector. Thus our FFT-based results on the Boolean complexity of these important computations could be quite interesting because our estimates are reasonable even for more general classes of structured matrices, showing rather moderate growth of the complexity as the input size increases

Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but typically nonreal roots are much more numerous than the real ones. We dramatically accelerate the known algorithms in this case by exploiting the correlation between the computations with matrices and polynomials, extending the techniques of the matrix sign iteration, and exploiting the structure of the companion matrix of the input polynomial. We extend some of the proposed techniques to the approximation of the real eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm

New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS

Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. Two of them are nearly optimal. The first one, proposed in 1995, relies on recursive factorization of a polynomial, is quite involved, and has never been implemented. The second one, proposed in 2016, relies on subdivision iterations, was implemented in 2018, and promises to be practically competitive, although user's current choice for univariate polynomial root-finding is the package MPSolve, proposed in 2000, revised in 2014, and based on Ehrlich's functional iterations. By proposing and incorporating some novel techniques we significantly accelerate both subdivision and Ehrlich's iterations. Moreover our acceleration of the known subdivision root-finders is dramatic in the case of sparse input polynomials. Our techniques can be of some independent interest for the design and analysis of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table

Accelerated Approximation of the Complex Roots of a Univariate Polynomial (Extended Abstract)

International audienceHighly efficient and even nearly optimal algorithms have been developed for the classical problem of univariate polynomial root-finding (see, e.g., \cite{P95}, \cite{P02}, \cite{MNP13}, and the bibliography therein), but this is still an area of active research. By combining some powerful techniques developed in this area we devise new nearly optimal algorithms, whose substantial merit is their simplicity, important for the implementation