1,807 research outputs found
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
Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial
To appearInternational audienceThe known algorithms approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time. They are, however, quite involved and require a high precision of computing when the degree of the input polynomial is large, which causes numerical stability problems. We observe that these difficulties do not appear at the initial stages of the algorithms, and in our present paper we extend one of these stages, analyze it, and avoid the cited problems, still achieving the solution within a nearly optimal complexity estimates, provided that some mild initial isolation of the roots of the input polynomial has been ensured. The resulting algorithms promise to be of some practical value for root-finding and can be extended to the problem of polynomial factorization, which is of interest on its own right. We conclude with outlining such an extension, which enables us to cover the cases of isolated multiple roots and root clusters
Simple and Nearly Optimal Polynomial Root-finding by Means of Root Radii Approximation
We propose a new simple but nearly optimal algorithm for the approximation of
all sufficiently well isolated complex roots and root clusters of a univariate
polynomial. Quite typically the known root-finders at first compute some crude
but reasonably good approximations to well-conditioned roots (that is, those
isolated from the other roots) and then refine the approximations very fast, by
using Boolean time which is nearly optimal, up to a polylogarithmic factor. By
combining and extending some old root-finding techniques, the geometry of the
complex plane, and randomized parametrization, we accelerate the initial stage
of obtaining crude to all well-conditioned simple and multiple roots as well as
isolated root clusters. Our algorithm performs this stage at a Boolean cost
dominated by the nearly optimal cost of subsequent refinement of these
approximations, which we can perform concurrently, with minimum processor
communication and synchronization. Our techniques are quite simple and
elementary; their power and application range may increase in their combination
with the known efficient root-finding methods.Comment: 12 pages, 1 figur
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection
Given a zero-dimensional polynomial system consisting of n integer
polynomials in n variables, we propose a certified and complete method to
compute all complex solutions of the system as well as a corresponding
separating linear form l with coefficients of small bit size. For computing l,
we need to project the solutions into one dimension along O(n) distinct
directions but no further algebraic manipulations. The solutions are then
directly reconstructed from the considered projections. The first step is
deterministic, whereas the second step uses randomization, thus being
Las-Vegas.
The theoretical analysis of our approach shows that the overall cost for the
two problems considered above is dominated by the cost of carrying out the
projections. We also give bounds on the bit complexity of our algorithms that
are exclusively stated in terms of the number of variables, the total degree
and the bitsize of the input polynomials
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
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