Piecewise polynomial approximation based on taylor series with efficient realization using Farrow structure

Over the last decades the finite element method, which was introduced by engineers in the 1960s, has become the perhaps most important numerical method for partial differential equations, particularly for equations of elliptic and parabolic types. This method is based on the variational form of the boundary value problem and approximates the exact solution by a piecewise polynomial function. It is more easily adapted to the geometry of the underlying domain than the finite difference method, and for symmetric positive definite elliptic problems it reduces to a finite linear system with a symmetric positive definite matrix. We first introduce this method in Sect. 5.1 for the case of a two-point boundary value problem and show a number of error estimates. In Sect. 5.2 we then formulate the method for a two-dimensional model problem. Here the piecewise polynomial approximations are defined on triangulations of the spatial domain, and in the following Sect. 5.3 we study such approximation in more detail. In Sect. 5.4 we show basic error estimates for the finite element method for the model problem, using piecewise linear approximating functions. All error bounds derived up to this point contain a norm of the unknown exact solution and are therefore often referred to as a priori error estimates. In Sect. 5.5 we show a so-called a posteriori error estimate in which the error bound is expressed in terms of the data of the problem and the computed solution. In Sect. 5.6 we analyze the effect of numerical integration, which is often used when the finite element equations are assembled in a computer program. In Sect. 5.7 we briefly describe a so-called mixed finite element method.KeywordsFinite Element MethodElliptic EquationQuadrature FormulaInterpolation ErrorPosteriori Error EstimateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Solving parametric piecewise polynomial systems

This paper presents an event-driven simulation methodology for injection-locked oscillators. The proposed method adopts a phase-domain macromodel based on a perturbation projection vector (PPV), which expresses the oscillator's phase response using a nonlinear, time-varying differential equation. By expressing the PPV using a piecewise polynomial function, the PPV model can be simulated in an event-driven fashion on a digital logic simulator like SystemVerilog. While this piecewise polynomial approximation introduces a trade-off between the simulation accuracy and speed, our results show that the simulation accuracy quickly reaches its optimal value as the piecewise interval number increases even with first-order polynomials. The experimental results with an LC oscillator, ring oscillator, and burst-mode CDR models operating with both periodic and aperiodic inputs demonstrate that the proposed models achieve speeds up to 2 ~ 30× that of transistor-level Spectre simulations with the same accuracy.

Let\(TA(f) = \int\limits_a^b {\frac{1}{2}(P_a^n (x) + P_b^n (x))dx} \) and let\(TA(f) = \int\limits_a^b {P_{(a + b)/2}^{n + 1} (x)dx} \), where Pcn denoles the Taylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of the Trapezoidal rule and the midpoint rule, respectively, and are each exact for all polynomials of degree ≤n+1. We let L(f)=αTM(f)+(1−α)TA(f), where\(\alpha = \frac{{2^{n + 1} (n + 1)}}{{2^{n + 1} (n + 1) + 1}}\), to obtain a numerical integration rule L which is exact for all polynomials of degree≤n+3 (see Theorem 1). The case n=0 is just the classical Simpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacing P(a+b)2/n+1(x) by the Hermite cubic interpolant at a and b, we obtain some known formulae by a different approach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by taking piecewise polynomials of odd degree, each piece being the Taylor polynomial of f at a and b, respectively. Of course all of our formulae can be compounded over subintervals of [a,b].

In smooth support vector machine (SSVM), the plus function must be approximated by some smooth function, and the approximate error will affect the classification ability. This paper studies the smooth approximation to the plus function by piecewise polynomials. First, some standard piecewise polynomial smooth approximation problems are formulated. Then, the existence and uniqueness of solution for these problems are proved and the analytic solutions are achieved. The comparison between the results in this paper and the previous ones shows that the piecewise polynomial functions in this paper achieve better approximation to the plus function

This paper studies the design and implementation of FIR and IIR VDFs, whose frequency characteristics can be controlled continuously by some control or tuning parameters. A least squares (LS) approach is proposed to design FIR VDFs with the impulse response expressed as a linear combination of basis functions. By choosing the basis functions as piecewise polynomials, VDFs with larger tuning range than the ordinary polynomial based approach result. These VDFs can be efficiently implemented using the Farrow structure. Making use of the FIR VDF, an eigensystem realization algorithm (ERA)-based model reduction technique is proposed to obtain a stable IIR VDF with lower system order. It does not suffer from the undesirable transient response during parameter tuning. For frequency selective VDF, about 40% of the multiplications can be saved using the IIR VDF. The implementation of the proposed FIR VDF using the SOPOT coefficients and the multiplier block technique is also studied. Results show that about two-thirds of the additions in implementing the SOPOT coefficients can be saved using the multiplier block.

This contribution presents an idea of polynomial-based filters with segments of different lengths which can be effectively used for sampling rate conversion. For these filters the impulse response is characterized by the following: in each interval of length T/sub n/ the impulse response is expressed as polynomial of low order M, there are all together N intervals, and the impulse response is symmetric around its middle point. In the case of interpolation (decimation), the length of every polynomial segment can be arbitrary integer multiple of input (output) sampling period. In this paper, the effective implementation structure for these filters is derived. The implementation structure is based on the known Farrow structure which is commonly used for implementation of various types of polynomial-based filters. It is also shown that the number of fixed coefficients in the novel structure is the same as in the case of modified and prolonged modified Farrow structure. The novel structure offers tradeoff between filtering requirements and system delay, keeping the number of multipliers unchanged, as it is shown through examples.

The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F -spaces N 0 p , n {\mathbf {N}}_0^{p,n} (introduced for this purpose in [2]). We obtain two o -results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in N 0 p , n {\mathbf {N}}_0^{p,n} we define a derivative D n , σ f {D^{n,\sigma }}f in L σ , σ = ( n + p − 1 ) − 1 {L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}} , which agrees with D n f {D^n}f for smooth f , and prove several properties of D n , σ {D^{n,\sigma }} .

Model order reduction (MOR) of nonlinear systems draws great attention in the past several decades. This paper presents a new MOR method in time domain for nonlinear dynamical systems. The new algorithm is based on the combination of the Taylor series expansion for the state variable x(t) and the trajectory piecewise polynomial technique. Firstly, the nonlinear system is approximated by a piecewise polynomial representation. Then, based on the Taylor series coefficients of x(t), we formulate the projection matrix V for the piecewise polynomial system and the compact model of the piecewise polynomial system is obtained in the following. Besides, error estimation and stability analysis are also presented in this paper. Finally, two nonlinear systems are tested to verify the effectiveness of the algorithm.

We design a new, fast algorithm for agnostically learning univariate probability distributions whose densities are well-approximated by piecewise polynomial functions. Let ƒ be the density function of an arbitrary univariate distribution, and suppose that ƒ is OPT-close in Li- distance to an unknown piecewise polynomial function with t interval pieces and degree d. For any γ > 0, our algorithm draws n = Õγ(t(d + 1)/ ∊2) samples from ƒ, runs in time Õ(n), and with probability at least 9/10 outputs an Ογ (t)-piecewise degree-d hypothesis H that is (3 + γ) · OPT + ∊ close to f. Our approximation factor almost matches the best known information-theoretic (but computationally inefficient) upper bound of 3.Our general algorithm yields (nearly) sample- optimal and nearly-linear time estimators for a wide range of structured distribution families over both continuous and discrete domains in a unified way. For most of our applications, these are the first sample-optimal and nearly-linear time estimators in the literature. As a consequence, our work resolves the sample and computational complexities of a broad class of inference tasks via a single “meta-algorithm”. Moreover, we demonstrate that our algorithm performs very well in experiments.Our algorithm consists of three levels: (i) At the top level, we employ an iterative greedy algorithm for finding a good partition of the real line into the pieces of a piecewise polynomial. (ii) For each piece, we show that the sub-problem of finding a good polynomial fit on the current interval can be solved efficiently with a separation oracle method. (iii) We reduce the task of finding a separating hyperplane to a combinatorial problem and design a nearly-linear algorithm for this problem. Combining these three procedures gives a density estimation algorithm with the claimed guarantees.

This paper studies the design and multiplier-less realization of variable fractional delay matched filters (VFD-MFs), which provide matching filtering and variable fractional delay of the filter output. It offers greater flexibility and lower delay in symbol-timing adjustment than directly cascading a match filter with a fractional delayer. The design of VFD-MFs, which can be viewed as a variable digital filter (VDF) design problem subject to the matched filtering condition, is formulated as a second order cone programming (SOCP) problem with least square design criteria. The proposed VFD-MFs can be efficiently implemented using the Farrow structure. By employing sum-of-power-of-two (SOPOT) coefficients and the multiplier block (MB) technique, very efficient multiplier-less realization of the VFD-MF with low hardware complexity is obtained. A design example is given to demonstrate the effectiveness of the proposed approach.

Taylor series, a fundamental concept in mathematics, expresses a function as an infinite sum of terms which is calculated from the values of its derivatives at a single point. Its applications span various fields, notably physics and finance, where it simplifies complex functions for analysis and problem-solving. This study employs a qualitative approach, reviewing existing literature on Taylor series applications. It analyzes case studies in physics, such as approximating motion equations, and in finance, like option pricing models. The method includes mathematical derivations and simulations to demonstrate efficacy. Findings indicate that Taylor series significantly enhance computational efficiency and accuracy. In physics, it aids in predicting particle behavior under varying forces. In finance, it provides valuable insights into the pricing of derivatives and risk assessment. The Taylor series proves to be an indispensable tool in both physics and finance, facilitating complex calculations and improving analytical capabilities. Its versatility underscores the importance of mathematical foundations in real-world applications.

The monograph published by E.Ya. Remez in 1957 addressed numerical methods with Chebyshev approximations. Particularly, the problem of the best uniform approximation of a function that is convex on an interval with continuous piecewise linear functions with free nodes was considered. In 1975, A.M. Vershik, V.N. Malozemov, and A.B. Pevnyi developed a general approach for constructing the best piecewise polynomial approximations with free nodes. The notion of partition with equal deviations was introduced, and it was found that such partition exists and generates the best piecewise polynomial approximation. In addition, a numerical method for constructing a partition with equal deviations was proposed. This paper gives three examples to describe the general approach to solving the problem of the best piecewise linear approximation with free nodes. In the case of an arbitrary continuous function, its best piecewise linear approximation in general is not continuous. It is continuous when approximating strictly convex and strictly concave functions.

SynopsisIf Ω is a bounded domain in Rn satisfying certain conditions, Ωk denotes its intersection with a k-dimensional hyperplane, 1 ≦ k ≦ n, it is shown that the embedding of the Sobolev space Ws,p(Ω), s>0, into Lq(Ωk) is of type lm if for q<p<∞. The same result is obtained for the space of Bessel potentials Ls,p(Ω). Piecewise polynomial and Fourier approximations of functions and interpolation theorems areused.

The paper is concerned with some characteristics of approximation by piecewise polynomial functions with nonfixed knots in various integral metrics Lp(a,b), 0 <p ≤∞ .