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

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    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 (10910^9) so that the observed complexity to find all roots is between O(dlnd)O(d\ln d) and O(dln3d)O(d\ln^3 d) (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

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    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 O~B(N10)\tilde{O}_B(N^{10}) for the bivariate case, where N=max(d,τ)N=\max(d,\tau), dd resp., τ\tau 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

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    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

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    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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