Taylor Series

This paper presents new, fast hardware for computing the exponential function, sine, and cosine. The main new idea is to use low-precision arithmetic components to approximate high precision computations, and then to correct very quickly the approximation error periodically so that the effect is to get high precision computation at near low-precision speed. The algorithm used in the paper is a nontrivial modification of the well-known CORDIC algorithm, and might be applicable to the computation of other functions than the ...

When a transformer is energized, magnetizing inrush current arises due to short-term overfluxing in the transformer core. The quantity of inrush current may be as high as ten times of transformer rated current which causes maloperation of differential protection system. This inflow current comprises decaying DC and superior second harmonic components which may cause unwanted effects like poor power quality and minimized mean lifespan of transformer. At the moment of transformer energizing if fault transpires, the protection system crash to distinguish between inrush and fault current. This paper provides a method for ejection of DC component and discrimination between inrush and fault condition to intend power transformer protection. For elimination of DC component, a compensating signal is generated with help of algorithm based on Taylor series expansion of inrush current. As second harmonic component is supreme in inrush current, discrimination is carried out utilising harmonic restraint. Based on ratio of second harmonic component to fundamental component algorithm is evolved to discriminate inrush from fault current, which is validated on a case study developed in Matlab/Simulink.

This paper intends to introduce the Taylor series for multi-variable real functions. More than a demostration of the teorema, it shows how to expose the series in a compact notation. Generalization of the jacobean of any order of a function with multiple dependence is defined. For this we use the differential operator Nabla with multiple tensor products. Also is established a special multiplication of the derivatives with the displacement of independent variables. This multiplicaction is identified as the contraction of indexes. At the end there is a new proof of the Taylor's Theorem for vectorial and tensorial functions.  Also it is included the multi-index notation version of the series.

The study of mixed convection evaporation of an inclined wet flat plate was carried out. As the resolution of partial differential equations occupies an important place in the world of research, our study can serve as a reference. We have proposed two methods for understanding and solving the physical phenomena interfering in mixed convection. The first being the semi-analytical resolution allowed us to analyse and to study the preponderance of natural convection and then forced convection. The other is the numerical resolution by the finite difference implicit method which displays results similar to the first, just by varying the Richardson number. The influence of the inclination is also presented in each case. Mass and heat transfers correspond to forced convection and natural convection, respectively. The combination of the two gives rise to mixed convection. In our case, we chose air but the equations can be used for other fluids for heat exchange with a solid surface.

"from my book "functions and infinite series" www.mpantes.gr"---------------


the author      attempted  to expound the most useful facts about Taylor series of real and complex variables, as about Fourier transforms and Fourier series  in a mathematical narrative, in a semi-popular design. The book is not intended  for mathematics specialists, a reader is assumed whose mathematical  grounding  in other respects goes no further than the traditional second year University course in mathematics.

Every s×s matrix A yields a composition map acting on polynomials on R s . Specifically, for every polynomial p we define the mapping C A by the formula (C A p)(x):=p(Ax), x∈R s . For each nonnegative integer n, homogeneous polynomials of degree n form an invariant subspace for C A . We let A (n) be the matrix representation of C A relative to the monomial basis and call A (n) a binomial matrix. This paper studies the asymptotic behavior of A (n) as n→∞. The special case of 2×2 matrices A with the property that A 2=I corresponds to discrete Taylor series and motivated our original interest in binomial matrices.

It has been developed a method of arbitrary degree based on Taylor series for multi-variable functions. The method is proposed for solving a system of homogeneous equations f(x)=0 in ℝᴺ. Methods of degree one and two are revised and a strategy for degree n is proposed, supported on the best estimation criterion. The multi-variable polynomials are solved sequentially from degree one of Newton-Raphson to degree n.

The paper demonstrates the advanced simulation methodology based on differentiation of the discretized finite element (FE) equations to parametric simulation of micro-electro-mechanical-systems (MEMS). The idea of the approach is to compute not only the governing system matrices but also high order derivatives (HOD) with regard to design parameters by means of automatic differentiation (AD). As result, Taylor vectors of the model response can be expanded in the vicinity of the initial position with regard to dimensional and physical parameters. The objective of this presentation is to demonstrate the viability of HOD methods to parametric simulation of MEMS in the static, modal, frequency response domains on the basis of the structural analysis and macromodeling.

Los métodos numéricos son modelos estudiados en matemáticas como una herramienta efectiva a la hora de resolver de problemas, dado que de forma aproximada algunos de estos son imposibles de solucionar analíticamente. Este libro pretende ser el módulo introductorio al estudio de los métodos numéricos y se espera que a lo largo del estudio de este el lector potencie el proceso matemático de aproximar. Como acompañamiento del texto el estudiante podrá acceder a la aplicación móvil MeNuMericos, la cual le permitirá facilitar algunas cuentas y graficar algunas situaciones. Se espera que el lector pueda cuantificar el error, solucionar ecuaciones de variable real y ecuaciones diferenciales, aproximar áreas bajo la curva, derivar sobre un conjunto de datos discreto, interpolar conjuntos de puntos y calcular regresiones.

Two formulations of a nonlinear model predictive control scheme based on the second-order Volterra series model are presented. The first formulation determines the control action using successive substitution, and the second method directly solves a fourth-order nonlinear ...

Nowadays, there are many new methods for slope stability analysis; including probabilistic methods assessing geotechnical uncertainties to develop safety factors. In this paper, a reliability index analysis for the Sungun copper mine slope stability is evaluated based on three methods of uncertainties consisting Taylor series method, Rosenblueth point estimate method and Monte-Carlo simulation method. Sungun copper mine will be one of the Iran’s biggest mines with final pit’s height of 700 meters. For this study two of its main slopes were assessed, one dipping to the NE (030) and the other to the SE (140). Probability density function of cohesion and angle of friction for the slopes were developed using limit equilibrium methods. These shear strengths were then used to determine the probability density function of safety factor and reliability index using the probabilistic methods. Results of the probabilistic analysis indicate that with ascending values of the uncertainties the reliability index decreases. Furthermore, it was determined that with the Monte Carlo simulation the seed number used has little effect on the reliability index of the safety factor especially with seed numbers in excess of 1200. Variations in the overall reliability index of safety factor were observed between the two slopes and this difference is explained by the differences in complexities of the geology within the cross-section.

This paper addresses the problem of initial synchronization of pseudo-noise code. A new code acquisition technique for spread-spectrum communication systems using band-limited chip waveforms is presented. Unlike conventional power detector based on testing the estimated maximum of the ambiguity function, the devised detector exploits a fast parabolic interpolation, running on three estimated ambiguity samples in the neighborhood of the coarse estimate. Performance analysis is carried out in comparison with conventional detector. Mathematical expressions for the probability of false alarm and detection are derived. They are numerically evaluated, under operating settings, by a reduced Tayloriquests expansion up to the second order. The theoretical results, substantiated by computer simulations, have evidenced that the devised method is well suited for asynchronous spread-spectrum communications. In particular, the acquisition performance depends on the actual offset between the received and the reference code waveforms, which are randomly distributed (in chip-asynchronous systems) within one sampling period. In fact, the parabolic interpolation technique outperforms the conventional detector for a wide range of code offsets because it is able to self-synchronize the testing variable around the true offset.

Continuous-discrete filtering (CDF) arises in many real-world problems such as ballistic projectile tracking, ballistic missile tracking, bearing-only tracking in 2D, angle-only tracking in 3D, and satellite orbit determination. We develop CDF algorithms using the extended Kalman filter (EKF), unscented Kalman filter (UKF), and particle filter (PF) with applications to the angle-only tracking in 3D. The modified spherical coordinates are used to represent the target state. Monte Carlo simulations are performed to compare the performance and computational complexity of the proposed filtering algorithms. Our results show that the CDF algorithms based on the EKF and UKF have the best state estimation accuracy and nearly the same computational cost.