Truncation approximants to probabilistic evolution of ordinary differential equations under initial conditions via bidiagonal evolution matrices: simple case

In this work, we investigate the probabilistic evolution approach (PEA) to ordinary differential equations whose evolution matrices are composed of only two diagonals under certain initial value impositions. We have been able to develop analytic expressions for truncation approximants which can be generated by using finite left uppermost square blocks in the denumerable infinite number of PEA equations and their infinite limits. What we have revealed is the fact that the truncation approximants converge for initial value parameter, values residing at most in a disk centered at the expansion point and excluding the nearest zero(es). The numerical implementations validate this formation.     

Routh approximations for reducing order of linear, time-invariant systems

A new method of approximating the transfer function of a high-order linear system by one of lower order is proposed. Called the "Routh approximation method" because it is based on an expansion that uses the Routh table of the original transfer function, the method has a number of useful properties: if the original transfer function is stable, then all approximants are stable; the sequence of approximants converge monotonically to the original in terms of "impulse response" energy; the approximants are partial Padé approximants in the sense that the firstkcoefficients of the power series expansions of thekth-order approximant and of the original are equal; the poles and zeros of the approximants move toward the poles and zeros of the original as the order of the approximation is increased. A numerical example is given for the calculation of the Routh approximants of a fourth-order transfer function and for illustration of some of the properties.     

行星轮点轨迹的图形分类与应用研究

行星轮上点的轨迹图形存在规则与无规则之分,各有对应的应用价值。通过建立行星轮上点的轨迹方程,令一个坐标在一个位置上的一至三阶导数同时等于零,得到了图形分类的基本关系,提出了规则正多边形、规则正多角形、多圈回归螺旋形与环状相邻缠绕形的设计条件,为设计规则图形、在一个行程单极限位置或双极限位置作高阶停歇的机构、多个行程单极限位置或双极限位置作高阶停歇的机构提供了理论基础。     

Generalized Homogeneous Multivariate Matrix PadÉ-Type Approximants and PadÉ Approximants

Generalized homogeneous multivariate matrix Pade-type approximants (GHMPTA) and Pade approximants are studied in ways similar to those of Brezinski and Kida in the scalar cases. By choosing an arbitrary monic bivariate scalar polynomial from the triangular form as the generating one of the approximant, we discuss their several typical important properties and study the connection between generalized homogeneous bivariate matrix Pade-type approximants and Pade approximants. The arguments given in detail in two variables can be extended directly to the case of d variables (d ges 2).     

On the Convergence of Global Rational Approximants for Stochastic Discrete Event Systems

Difficulties often arise in analyzing stochastic discrete event systems due to the so-called curse of dimensionality. A typical example is the computation of some integer-parameterized functions, where the integer parameter represents the system size or dimension. Rational approximation approach has been introduced to tackle this type of computational complexity. The underline idea is to develop rational approximants with increasing orders which converge to the values of the systems. Various examples demonstrated the effectiveness of the approach. In this paper we investigate the convergence and convergence rates of the rational approximants. First, a convergence rate of order O(1/ ) is obtained for the so-called Type-1 rational approximant sequence. Secondly, we establish conditions under which the sequence of [n/n] Type-2 rational approximants has a convergence rate of order .     

On the approximation of the transport equation by the method of weighted residuals

A general class of approximate models for the transport equation is derived by the Weighted Residual Method using polynomial trial and test spaces. For two, particularly relevant, of such models a closed form of the approximate solution is given in the Laplace-transform domain. A strict connection is established between the approximate transfer function and the Padé approximants to exp(-sx). Furthermore it is shown that Galerkin model minimizes in the considered class a measure of the initial state approximation error. Finally an application to the solution of systems of first order hyperbolic equations is reported.     

A critical review of asymptotic numerical methods

Summary Various sorts of asymptotic-numerical methods have been propsed in the literature: the reduced basis technique, direct computation of series or the use of Padé approximants. The efficiency of the method may also depend on the chosen path parameter, on the order of truncature and on alternative parameters. In this paper, we compare the three classes of asymptotic-numerical method, with a view to define the “best” numerical strategy.     

A Numerically Efficient Dissipation-Preserving Implicit Method for a Nonlinear Multidimensional Fractional Wave Equation

The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has been proved to be an effective tool for solving large scattered data interpolation problems. However, in order to achieve a good accuracy, the question about how many points we have to consider on each local subdomain, i.e. how large can be the local data sets, needs to be answered. Moreover, it is well-known that also the shape parameter affects the accuracy of the local RBF approximants and, as a consequence, of the PU interpolant. Thus here, both the shape parameter used to fit the local problems and the size of the associated linear systems are supposed to vary among the subdomains. They are selected by minimizing an a priori error estimate. As evident from extensive numerical experiments and applications provided in the paper, the proposed method turns out to be extremely accurate also when data with non-homogeneous density are considered.     

A simple and direct method of reducing order of linear systems using routh approximations in the frequency domain

A simple and direct method of approximating higher order systems by lower order ones based on Routh approximants without using reciprocal transformations is proposed. New algorithms for constructing reduced order transfer functions are presented. A method of approximating unstable systems with poles in RHP is also proposed resulting in unique transfer functions.     

Recursive formulation of the matrix Padé approximation in packed storage

The Extended Euclidean algorithm for matrix Padé approximants is applied to compute matrix Padé approximants when the coefficient matrices of the input matrix polynomial are triangular. The procedure given by Bjarne S. Anderson et al. for packing a triangular matrix in recursive packed storage is applied to pack a sequence of lower triangular matrices of a matrix polynomial in recursive packed storage. This recursive packed storage for a matrix polynomial is applied to compute matrix Padé approximants of the matrix polynomial using the Matrix Padé Extended Euclidean algorithm in packed form. The CPU time and memory comparison, in computing the matrix Padé approximants of a matrix polynomial, between the packed case and the non-packed case are described in detail.     

Approximation of multivariable linear systems with impulse response and autocorrelation sequences

This paper considers the construction of approximants of multi-input-multi-output, discrete-time linear systems from the finite data of the impulse response and autocorrelation sequences. In the approximation of a multivariable linear system, it is common practice to use a finite portion of its impulse response sequence. This is formally equivalent to the Padé approximation technique, which may produce unstable approximants, even though the original system is stable. Mullis and Roberts proposed a new method, which yields stable approximants, in connection with approximation of digital filters. This is, however, restricted to the single-input-single-output case. This paper extends their method to the multi-input-multi-output case and shows a fast recursive algorithm to construct stable approximants of linear systems.     

Design of non-rational MIMO feedback systems with inputs and outputs satisfying certain bounding conditions by using rational approximants

When a feedback system has components described by non-rational transfer functions, a standard practice in designing such a system is to replace the non-rational functions with rational approximants and then carry out the design with the approximants by means of a method that copes with rational systems. In order to ensure that the design carried out with the approximants still provides satisfactory results for the original system, a criterion of approximation should be explicitly taken into account in the design formulation. This paper derives such a criterion for multi-input multi-output (MIMO) feedback systems whose design objective is to ensure that the absolute values of every error and every controller output components always stay within prescribed bounds whenever the inputs satisfy certain bounding conditions. The obtained criterion generalizes a known result which was derived for single-input single-output (SISO) systems; furthermore, for a given rational approximant matrix, it is expressed as a set of inequalities that can be solved in practice. Finally, a controller for a binary distillation column is designed by using the criterion in conjunction with the method of inequalities. The numerical results clearly demonstrate that the usefulness of the criterion in obtaining a design solution for the original system.     

Padé error estimates for the logarithm of a matrix

The error of Padé approximations to the logarithm of a matrix and related hypergeometric functions is analysed. By obtaining an exact error expansion with positive coefficients, it is shown that the error in the matrix approximation at X is always less than the scalar approximation error at x, when ∥X∥ < x. A more detailed analysis, involving the interlacing properties of the zeros of the Padé denominator polynomials, shows that for a given order of approximation, the diagonal Padé approximants are the most accurate. Similarly, knowing that the denominator zeros must lie in the interval (1,∞) leads to a simple upper bound on the condition number of the matrix denominator polynomial, which is a crucial indicator of how accurately the matrix Padé approximants can be evaluated numerically. In this respect the Padé approximants to the logarithm are very well conditioned for ∥X∥ < 0·25. This latter condition can be ensured by using the ‘inverse scaling and squaring’ procedure for evaluating the logarithm.     

Construction of the characteristic polynomial of a matrix

A connection between Padé approximants and characteristic polynomial of a matrix with complex, in general, elements is established. A system of homogeneous, first-order ordinary differential equations is associated with the given matrix and after Laplace transform the individual transformed components are shown to be special Padé approximants.     

Geometry and Convergence Analysis of Algorithms for Registration of 3D Shapes

The computation of a rigid body transformation which optimally aligns a set of measurement points with a surface and related registration problems are studied from the viewpoint of geometry and optimization. We provide a convergence analysis for widely used registration algorithms such as ICP, using either closest points (Besl and McKay, 1992) or tangent planes at closest points (Chen and Medioni, 1991) and for a recently developed approach based on quadratic approximants of the squared distance function (Pottmann et al., 2004). ICP based on closest points exhibits local linear convergence only. Its counterpart which minimizes squared distances to the tangent planes at closest points is a Gauss–Newton iteration; it achieves local quadratic convergence for a zero residual problem and—if enhanced by regularization and step size control—comes close to quadratic convergence in many realistic scenarios. Quadratically convergent algorithms are based on the approach in (Pottmann et al., 2004). The theoretical results are supported by a number of experiments; there, we also compare the algorithms with respect to global convergence behavior, stability and running time.     

A fourth order multiderivative method for linear second order boundary value problems

Fourth order methods are developed and analysed for the numerical solution of linear second order boundary value problems.

The methods are developed by replacing the exponential terms in a three-point recurrence relation by Padé approximants.

The derivations of second order and sixth order methods from the recurrence relation are outlined briefly.

One method is tested on two problems from the literature, one of which is mildly nonlinear.     

A recursive formulation of collocation in terms of canonical polynomials

In this paper we discuss two related but analytically different techniques: the collocation method and Ortiz's recursive formulation of the Tau Method. Specifically, we show that it is possible to simulate with the Tau Method collocation approximants for any desired degree. We give a representation for collocation approximants in terms ofshifted canonical polynomials, which are introduced here. We show that in the linear case computing a collocation approximant of orderN by this new approach requiresO(N) arithmetic operations while obtaining the same approximant by the direct approach involvesO(N ³). Furthermore, our technique leads to a recursive formulation of collocation. We discuss separately the linear and nonlinear cases and propose a more efficienteconomized approach for the latter.     

Feature selection for best mean square approximation of class densities

A criterion for feature selection is proposed which is based on mean square approximation of class density functions. It is shown that for the widest possible class of approximants, the criterion reduces to Devijer's Bayesian distance. For linear approximants the criterion is equivalent to well known generalized Fisher criteria.     

Padé-approximants in operator theory for the solution of nonlinear differential and integral equations

The Padé approximation problem in operator theory and its solution will be repeated briefly together with some properties of the Padé approximants. Effective methods for the solution of differential and integral equations are of great practical importance and there is a vast literature devoted to this subject.

Here a few methods, resulting from the use of Padé approximants in operator theory, are introduced and illustrated by means of some typical examples (initial value problems, boundary value problems, partial differential equations, nonlinear integral equations).

Among the new methods is an iterative scheme which we will call Halley's method and which proves to be very useful in the neighbourhood of singularities.

Well known methods such as Newton's and Chebyshev's method prove to be special cases of the class of iterative procedures.     

一般非线性系统的零动态、正则标准形与局部反馈渐近镇定

研究了一般非线性系统的零动态算法,得到了应用广泛的正则标准形,只需对系统 做较弱的正则性假设,即平衡点附近某些分布是非奇异的.同时讨论了该标准形在局部反馈 渐近镇定中的应用,该结果适用于解决局部反馈渐近镇定的临界问题.仿射非线性系统的结 果成为该文结果的特例.