Optimal and nearly optimal algorithms for approximating polynomial zeros

Optimal and Nearly Optimal Algorithms for Approximating Polynomial Zeros

K e y w o r d s C o m p l e x polynomial zeros, Approximation, Polynomial factorization, Parallel algorithms, Computational complexity, Sequential algorithms. 1. I N T R O D U C T I O N 1.1. T h e Subject , Some History, and a S u m m a r y of Our Resul ts The problem of solving a polynomial equation p ( x ) = 0 substantially motivated the development of mathematics throughout the centuries. As...

Nearly Optimal Algorithms for Canonical Matrix Forms

A Las Vegas type probabilistic algorithm is presented for nding the Frobenius canonical form of an n n matrix T over any eld K. The algorithm requires O~(MM(n)) = MM(n) (logn) O(1) operations in K, where O(MM(n)) operations in K are suucient to multiply two n n matrices over K. This nearly matches the lower bound of (MM(n)) operations in K for this problem, and improves on the O(n 4) operations...

Nearly Optimal Algorithms for Univariate Polynomial Factorization and Rootfinding II: Computing a Basic Annulus for Splitting

Optimal Algorithms for Parallel Polynomial Evaluation

Approximating optimal point configurations for multivariate polynomial interpolation

Efficient and effective algorithms are designed to compute the coordinates of nearly optimal points for multivariate polynomial interpolation on a general geometry. “Nearly optimal” refers to the property that the set of points has a Lebesgue constant near to the minimal Lebesgue constant with respect to multivariate polynomial interpolation on a finite region. The proposed algorithms range fro...