1,228 research outputs found
Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees
We present a practical implementation based on Newton's method to find all
roots of several families of complex polynomials of degrees exceeding one
billion () so that the observed complexity to find all roots is between
and (measuring complexity in terms of number of
Newton iterations or computing time). All computations were performed
successfully on standard desktop computers built between 2007 and 2012.Comment: 24 pages, 19 figures. Update in v2 incorporates progress on
polynomials of even higher degrees (greater than 1 billion
A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems
We improve the local generic position method for isolating the real roots of
a zero-dimensional bivariate polynomial system with two polynomials and extend
the method to general zero-dimensional polynomial systems. The method mainly
involves resultant computation and real root isolation of univariate polynomial
equations. The roots of the system have a linear univariate representation. The
complexity of the method is for the bivariate case, where
, resp., is an upper bound on the degree, resp., the
maximal coefficient bitsize of the input polynomials. The algorithm is
certified with probability 1 in the multivariate case. The implementation shows
that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure
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
Descartes' Rule of Signs for Polynomial Systems supported on Circuits
We give a multivariate version of Descartes' rule of signs to bound the
number of positive real roots of a system of polynomial equations in n
variables with n+2 monomials, in terms of the sign variation of a sequence
associated both to the exponent vectors and the given coefficients. We show
that our bound is sharp and is related to the signature of the circuit.Comment: 25 pages, 3 figure
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