In electrical engineering, the designer is often presented with the problem of synthesizing a circuit for which the mathematical specifications are unsuitable for physical realization. Hence, the engineer must approximate as well as possible the prescribed network function by another function which is realizable. This paper describes a new approximation method for solving the problem of realizing passive network transfer functions, where the realization is carried out through the use of passive, reciprocal,lumped, linear, and time-invariant elements.

An automatic algorithm for accurate numerical gradient calculations has been developed. The algorithm is based on both finite differences and Chebyshev interpolation approximations. The novelty of the method is an automated tuning of the step size perturbation required for both methods. This automation guaranties the best possible solution using these approaches without the requirement of user inputs. The algorithm treats the functions as a black box, which makes it extremely useful when general and complex problems are considered. This is the case of spacecraft trajectory design problems and complex optimization systems. An efficient procedure for the automatic implementation is presented. Several examples based on an Earth-Moon free return trajectory are presented to validate and demonstrate the accuracy of the method. A state transition matrix (STM) procedure is developed as a reference for the validation of the method. / text

We propose novel methods to utilize orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, long-term orbit propagation. These methods have immediate application to efficient propagation of catalogs of Resident Space Objects (RSOs) and improved accounting for the uncertainty in the ephemeris of these objects. More fundamentally, the methodology promises to be of broad utility in solving initial and two point boundary value problems from a wide class of mathematical representations of problems arising in engineering, optimal control, physical sciences and applied mathematics. We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10^−9ms^−2, globally) are required near the Earths surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with both speed and storage efficiency op- timized using radial adaptation. The second class of problems addressed includes orbit propagation and solution of associated boundary value problems. The successive Chebyshev-Picard path approximation method is shown well-suited to solving these problems with over an order of magnitude speedup relative to known methods. Furthermore, the approach is parallel-structured so that it is suited for parallel implementation and further speedups. Used in conjunction with orthogonal Finite Element Model (FEM) gravity approximations, the Chebyshev-Picard path approximation enables truly revolutionary speedups in orbit propagation without accuracy loss.

Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to tenth-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes, and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 8 uppsatser.

Thesis (Ph. D.)--West Virginia University, 1998. / Title from document title page. Document formatted into pages; contains xi, 116 p. : ill. (some col.) Includes abstract. Includes bibliographical references (p. 94-98).

Orientador: Masayoshi Tsuchida / Banca: José Manoel Balthazar / Banca: Elso Drigo Filho / Resumo: Neste trabalho estuda-se alguns sistemas dinâmicos utilizando um novo método para aproximar a matriz de transição de estados (STM) para sistemas periódicos no tempo. Este método é baseado na transformação de Lyapunov-Floquet (L-F), e utiliza a expansão polinomial de Chebyshev para aproximar o termo periódico. O método iterativo de Picard é usado para aproximar a STM. Os multiplicadores de Floquet, determinados através deste método, permitem construir o diagrama de estabilidade do sistema dinâmico. Esta técnica é aplicada para analisar a estabilidade e os pontos de bifurcação do sistema dinâmico formado por um pêndulo elástico com excitação vertical periódica no suporte. Além dessa aplicação, é analisada também a equação de Mathieu e a estabilidade do sistema dinâmico constituído por partículas carregadas e imersas em um campo magnético perturbado. / Abstract: In this work some dynamic systems are studied using a new method to approach state transition matrix (STM) for time-periodic systems. This method is based on Lyapunov- Floquet transformation (transformation L-F) and uses the Chebyshev polynomial expansion to approach the periodical term. The Picard iterative method is used to approach the STM. The Floquet multipliers determined through this method, allow to draw the stability diagram of the dynamic system. This technique is applied to analyze the stability and bifurcation points of the dynamic system formed by an elastic pendulum with periodic vertical excitation on support. Besides this application, the Mathieu equation is analyzed and also the stability of the dynamical system constituted by charged particle in a perturbed magnetic field is discussed. / Mestre

abstract: Renewable portfolio standards prescribe for penetration of high amounts of re-newable energy sources (RES) that may change the structure of existing power systems. The load growth and changes in power flow caused by RES integration may result in re-quirements of new available transmission capabilities and upgrades of existing transmis-sion paths. Construction difficulties of new transmission lines can become a problem in certain locations. The increase of transmission line thermal ratings by reconductoring using High Temperature Low Sag (HTLS) conductors is a comparatively new technology introduced to transmission expansion. A special design permits HTLS conductors to operate at high temperatures (e.g., 200oC), thereby allowing passage of higher current. The higher temperature capability increases the steady state and emergency thermal ratings of the transmission line. The main disadvantage of HTLS technology is high cost. The high cost may place special emphasis on a thorough analysis of cost to benefit of HTLS technology im-plementation. Increased transmission losses in HTLS conductors due to higher current may be a disadvantage that can reduce the attractiveness of this method. Studies described in this thesis evaluate the expenditures for transmission line re-conductoring using HTLS and the consequent benefits obtained from the potential decrease in operating cost for thermally limited transmission systems. Studies performed consider the load growth and penetration of distributed renewable energy sources according to the renewable portfolio standards for power systems. An evaluation of payback period is suggested to assess the cost to benefit ratio of HTLS upgrades. The thesis also considers the probabilistic nature of transmission upgrades. The well-known Chebyshev inequality is discussed with an application to transmission up-grades. The Chebyshev inequality is proposed to calculate minimum payback period ob-tained from the upgrades of certain transmission lines. The cost to benefit evaluation of HTLS upgrades is performed using a 225 bus equivalent of the 2012 summer peak Arizona portion of the Western Electricity Coordi-nating Council (WECC). / Dissertation/Thesis / M.S. Electrical Engineering 2014

Quand on veut évaluer ou manipuler une fonction mathématique f, il est fréquent de la remplacer par une approximation polynomiale p. On le fait, par exemple, pour implanter des fonctions élémentaires en machine, pour la quadrature ou la résolution d'équations différentielles ordinaires (ODE). De nombreuses méthodes numériques existent pour l'ensemble de ces questions et nous nous proposons de les aborder dans le cadre du calcul rigoureux, au sein duquel on exige des garanties sur la précision des résultats, tant pour l'erreur de méthode que l'erreur d'arrondi.Une approximation polynomiale rigoureuse (RPA) pour une fonction f définie sur un intervalle [a,b], est un couple (P, Delta) formé par un polynôme P et un intervalle Delta, tel que f(x)-P(x) appartienne à Delta pour tout x dans [a,b].Dans ce travail, nous analysons et introduisons plusieurs procédés de calcul de RPAs dans le cas de fonctions univariées. Nous analysons et raffinons une approche existante à base de développements de Taylor.Puis nous les remplaçons par des approximants plus fins, tels que les polynômes minimax, les séries tronquées de Chebyshev ou les interpolants de Chebyshev.Nous présentons aussi plusieurs applications: une relative à l'implantation de fonctions standard dans une bibliothèque mathématique (libm), une portant sur le calcul de développements tronqués en séries de Chebyshev de solutions d'ODE linéaires à coefficients polynômiaux et, enfin, un processus automatique d'évaluation de fonction à précision garantie sur une puce reconfigurable. / For purposes of evaluation and manipulation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floating-point implementations of elementary functions, integration, ordinary differential equations (ODE) solving. For that, a wide range of numerical methods exists. We consider the application of such methods in the context of rigorous computing, where we need guarantees on the accuracy of the result, with respect to both the truncation and rounding errors.A rigorous polynomial approximation (RPA) for a function f defined over an interval [a,b] is a couple (P, Delta) where P is a polynomial and Delta is an interval such that f(x)-P(x) belongs to Delta, for all x in [a,b]. In this work we analyse and bring forth several ways of obtaining RPAs for univariate functions. Firstly, we analyse and refine an existing approach based on Taylor expansions. Secondly, we replace them with better approximations such as minimax approximations, Chebyshev truncated series or interpolation polynomials.Several applications are presented: one from standard functions implementation in mathematical libraries (libm), another regarding the computation of Chebyshev series expansions solutions of linear ODEs with polynomial coefficients, and finally an automatic process for function evaluation with guaranteed accuracy in reconfigurable hardware.

In this thesis we present some fundamental results regarding orthogonal polynomials and biorthogonal polynomials, the latter defined as in the article "Cauchy Biorthogonal Polynomials", authored by Bertola, Gekhtman, and Szmigielski. We show that total positivity of the kernel can be weakened and how this implies that interlacement for biorthogonal polynomials holds in general. A counterexample is provided showing that in general there does not exist a four-term recurrence relation such as the one found for the Cauchy kernel. As a direct consequence we show that biorthogonal polynomial sequences cannot be considered orthogonal polynomial sequences by an appropriate choice of orthogonality measure. Furthermore, we motivate a conjecture stating that the more general form of interlacement that exists for orthogonal polynomials also exists for biorthogonal polynomials. We end with suggesting some further work that could be of interest.