3,539 research outputs found

    A family of root-finding methods with accelerated convergence

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    AbstractA parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given

    Laguerre-like methods for the simultaneous approximation of polynomial multiple zeros

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    Two new methods of the fourth order for the simultaneous determination of multiple zeros of a polynomial are proposed. The presented methods are based on the fixed point relation of Laguerre's type and realized in ordinary complex arithmetic as well as circular complex interval arithmetic. The derived iterative formulas are suitable for the construction of modified methods with improved convergence rate with negligible additional operations. Very fast convergence of the considered methods is illustrated by two numerical examples

    Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)

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    We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.Comment: 102 pages, 31 figure

    Some classical multiple orthogonal polynomials

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    Recently there has been a renewed interest in an extension of the notion of orthogonal polynomials known as multiple orthogonal polynomials. This notion comes from simultaneous rational approximation (Hermite-Pade approximation) of a system of several functions. We describe seven families of multiple orthogonal polynomials which have he same flavor as the very classical orthogonal polynomials of Jacobi, Laguerre and Hermite. We also mention some open research problems and some applications

    On two models of orthogonal polynomials and their applications

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    37 pages, no figures.-- MSC2000 codes: 33C45, 42C05.This contribution deals with some models of orthogonal polynomials as well as their applications in several areas of mathematics. Some new trends in the theory of orthogonal polynomials are summarized. In particular, we emphasize on two kinds of orthogonality, i.e., the standard orthogonality in the unit circle and a non standard one, which is called multi-orthogonality. Both have attracted the interest of researchers during the past ten years.This work has been supported by Dirección General de Investigación (MCyT) of Spain under grant BFM2000-0206-C04-01 and INTAS project 2000-272. J. Arvesú was partially supported by the Dirección General de Investigación (Comunidad Autónoma de Madrid).Publicad

    How smooth are particle trajectories in a Λ\LambdaCDM Universe?

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    It is shown here that in a flat, cold dark matter (CDM) dominated Universe with positive cosmological constant (Λ\Lambda), modelled in terms of a Newtonian and collisionless fluid, particle trajectories are analytical in time (representable by a convergent Taylor series) until at least a finite time after decoupling. The time variable used for this statement is the cosmic scale factor, i.e., the "aa-time", and not the cosmic time. For this, a Lagrangian-coordinates formulation of the Euler-Poisson equations is employed, originally used by Cauchy for 3-D incompressible flow. Temporal analyticity for Λ\LambdaCDM is found to be a consequence of novel explicit all-order recursion relations for the aa-time Taylor coefficients of the Lagrangian displacement field, from which we derive the convergence of the aa-time Taylor series. A lower bound for the aa-time where analyticity is guaranteed and shell-crossing is ruled out is obtained, whose value depends only on Λ\Lambda and on the initial spatial smoothness of the density field. The largest time interval is achieved when Λ\Lambda vanishes, i.e., for an Einstein-de Sitter universe. Analyticity holds also if, instead of the aa-time, one uses the linear structure growth DD-time, but no simple recursion relations are then obtained. The analyticity result also holds when a curvature term is included in the Friedmann equation for the background, but inclusion of a radiation term arising from the primordial era spoils analyticity.Comment: 16 pages, 4 figures, published in MNRAS, this paper introduces a convergent formulation of Lagrangian perturbation theory for LCD
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