Polinomis i coeficients de reflexió

Els polinomis es solen representar o bé pels seus coeficients o bé pels seus zeros. Les dues representacions estan lligades per les fórmules de Cardano-Viète que expressen els coeficients com a funcions simètriques elementals dels zeros. La recursió descendent de Levinson defineix els coeficients de reflexió d'un polinomi. En aquest article es veu com es poden caracteritzar els polinomis en termes dels seus coeficients de reflexió, es donen resultats sobre polinomis autoreversos, que juguen un paper singular en aquesta representació, i es donen fórmules homòlogues a les de Cardano-Viète que relacionen zeros amb coeficients de reflexió. També es caracteritzen els polinomis de tipus Kakeya en termes de coeficients de reflexió, cosa que permet donar una demostració alternativa del teorema d'Eneström-Kakeya sobre la localització de zeros d'un polinomi. Molts d'aquests desenvolupaments estan relacionats amb teoria de control i anàlisi de senyals. En aquest context, els texts clàssics de localització de zeros són recursius. Hi ha casos singulars en els quals el procés recursiu queda aturat i s'ha de recórrer a tècniques de pertorbació per continuar-los. Aquestes tècniques sempre funcionen però no estan en general ben fonamentades. Aquí es prova que els polinomis no singulars són densos, amb la norma L2, al disc unitat, cosa que dóna base matemàtica a les tècniques de pertorbació.Polynomials can be represented by their coefficients or by their zeros. The link between these two representations is the Cardan–Vi`ete’s formulas that allow expressing coefficients as elementary symmetric functions in the zeros. Backward Levinson’s recursion defines reflection coefficients of a polynomial. These coefficients can be used to characterize polynomials. A complete classification of the set of all polynomials is obtained and two theorems on self-inversive polynomials are given. As a consequence of Levinson recursion, a counterpart of Cardan-Vi`ete’s formulas is presented. They express polynomial coefficients in terms of its reflection coefficients. Backward Levinson’s recursion for polynomials is used again to obtain the characterization of polynomials of Kakeya type by their reflection coefficients. This result leads to an alternative proof of Enestr¨om and Kakeya theorems on the location of zeros of polynomials. Most of these developments are related to control and signal analysis. In this framework, classical tests for locating zeroes of polynomials are recursive. There are singular cases in which such recursive tests are stopped and perturbation techniques should be applied to proceed. Perturbation techniques, although always successful, are not proven to be well-founded. The non-singular polynomials are proven to be dense in the set of all polynomials with respect to the L2-norm in the unit circle thus giving a mathematical foundation to perturbation techniques

New generalization of discrete Montgomery identity with applications

In this paper, a discrete version of the well-known Montgomery's identity is generalized, and a refinement of an inequality derived by B.G. Pachpatte in 2007 is presented. Finally, the results obtained are applied for expanding a complex multinomial formula in dfferent way of the classical expansion.Peer ReviewedPostprint (published version

A technique to composite a modified Newton's method for solving nonlinear equations

Nova tècnica que permet construir mètodes iteratius d'ordre alt.A zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to compose a given iterative method with a modified Newton’s method that introduces just one evaluation of the function. To carry out this procedure some classical methods with different orders of convergence are used to obtain root-finders with higher efficiency index.Preprin

Order regularity for Birkhoff interpolation with lacunary polynomials

In this short paper we present sufficient conditions for the order regularity problem in Birkhoff interpolation with lacunary polynomials. These conditions are a generalization of the Atkinson-Sharma theorem.Peer ReviewedPostprint (published version

A Technique to Composite a Modified Newton's Method for Solving Nonlinear Equations

A zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to compose a given iterative method with a modified Newton's method that introduces just one evaluation of the function. To carry out this procedure some classical methods with different orders of convergence are used to obtain root-finders with higher efficiency index

Zero and coefficient inequalities for hyperbolic polynomials

In this paper using classical inequalities and Cardan-Viète formulae some inequalities involving zeroes and coefficients of hyperbolic polynomials are given. Furthermore, considering real polynomials whose zeros lie in Re(z)>0, the previous results have been extended and new inequalities are obtained.Peer ReviewedPostprint (published version

Annuli for the zeros of a polynomial

In this paper, ring shaped regions containing all the zeros of a polynomial with complex coeffcients involving binomial coeffcients and Fibonacci-Pell numbers are given. Furthermore, bounds for strictly positive polynomials involving their derivatives are also presented. Finally, using MAPLE, some examples illustrating the bounds proposed are computed and compared with other existing explicit bounds for the zeros.Postprint (published version

On computational order of convergence of some multi-precision solvers of nonlinear systems of equations

Report d'un treball de recerca on es presenten noves tècniques de càlcul de l'ordre de convergència amb una aritmètica adaptativa.In this paper the local order of convergence used in iterative methods to solve nonlinear systems of equations is revisited, where shorter alternative analytic proofs of the order based on developments of multilineal functions are shown. Most important, an adaptive multi-precision arithmetics is used hereof, where in each step the length of the mantissa is defined independently of the knowledge of the root. Furthermore, generalizations of the one dimensional case to m-dimensions of three approximations of computational order of convergence are defined. Examples illustrating the previous results are given.Preprin

Modeling of Real Bistables in VHDL

A complete VHDL model of bistables including their metastable operation is presented. An RS-NAND latch has been modelled as a basic structure, orienting its implementation towards its inclusion in a cell library. Two applications are included: description of a more complex latch (D-type) and description of a circuit containing three latches where metastable signals are propagated. Simulation results show that the presented niodel provides very realistic information about the device behavior, which until now had to be obtained through electric simulation

New CMOS VLSI Linear Self-Timed Architectures

The implementation of digital signal processor circuits via self-timed techniques is currently a valid altemative to solve some problems encountered in synchronous VLSI circuits. However; a main difference between synchronous and asynchronous circuits is the hardware resources needed to implement asynchronous circuits. This communication presents four less-costly alternatives to a previously reported linear selftimed architecture, and their application in the design of FIFO memories. Furthermore, the integration and characterization in the laboratory of prototypes of these FIFOs are presented