5,056 research outputs found

    Over-constrained Weierstrass iteration and the nearest consistent system

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    We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method is the Gauss-Newton iteration to find the nearest system which has at least kk common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of our method to the optimization problem formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate case we generalize the expressions of Karmarkar and Lakshman, and give explicitly several iteration functions to compute the optimum. The arithmetic complexity of the iterations is detailed

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

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    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

    Optimization via Chebyshev Polynomials

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    This paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices (CPSDMs). In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one- and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy.Comment: 26 pages, 6 figures, 2 table

    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

    On the degree and half degree principle for symmetric polynomials

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    In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte. It says that a symmetric real polynomial FF of degree dd in nn variables is positive on Rn\R^n (on R0n\R^{n}_{\geq 0}) if and only if it is so on the subset of points with at most max{d/2,2}\max\{\lfloor d/2\rfloor,2\} distinct components. We deduce Timofte's original statement as a corollary of a slightly more general statement on symmetric optimization problems. The idea we are using to prove this statement is to relate it to a linear optimization problem in the orbit space. The fact that for the case of the symmetric group SnS_n this can be viewed as a question on normalized univariate real polynomials with only real roots allows us to conclude the theorems in a very elementary way. We hope that the methods presented here will make it possible to derive similar statements also in the case of other groups.Comment: (v2) revision based on suggestions by refere
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