Solutions of linear dynamic systems by generalized orthogonal polynomials

Generalized orthogonal polynomials that represent all types of orthogonal polynomial are introduced in this paper. Using the idea of orthogonal polynomial functions that can be expressed by power series, and vice versa, the operational matrix for integration of a generalized orthogonal polynomial is first derived and then applied to solve the equations of linear dynamic systems. The characteristics of each kind of orthogonal polynomial in relation to solving linear dynamic systems is demonstrated. The computational strategy for finding the expansion coefficients of the state variables is very simple, straightforward and easy. The operational matrix is simpler than those of conventional orthogonal polynomials. Hence the expansion coefficients are more easily calculated from the proposed recursive formula when compared with those obtained from conventional orthogonal polynomial approximations.     

A general sampling expansion

The Shannon sampling theorem has been extended by Kramer to include the general kernel K(t,x) instead of the exponential kernel and also to be orthogonal in a finite interval. Jerri obtained the sampling theorem using Laguerre polynomials for the kernel. In this note we obtain the sampling expansion for a Hermite polynomial and parabolic functions which are orthogonal in the infinite interval.     

Solution of a scaled system via generalized orthogonal polynomials

Generalized orthogonal polynomials which include all types of orthogonal polynomial are introduced first. Using the idea of orthogonal polynomials that can be expressed by a Taylor power series and vice versa, the operational matrix for the integration of the generalized orthogonal polynomials is first derived. A stretched operational matrix of diagonal form is also derived. Both the operational matrix for the integration and the stretched operational matrix of generalized orthogonal polynomials are applied to solve functional differential equations. The characteristics of each kind of orthogonal polynomial in solving the scaled system is demonstrated. The computational strategy for finding the expansion coefficients of the state variables is very simple, straightforward and easy. The inversion of only one matrix, which has the same dimension as the state variables, is required. The expansion coefficients of the state variables are obtained by the proposed recursive formula. Much computer time is thus saved and computational results are obtained that are very accurate compared with previous methods.     

The Fourth-order Difference Equation Satisfied by the Associated Orthogonal Polynomials of the Δ -Laguerre–Hahn Class

Starting from the D_ω-Riccati difference equation satisfied by the Stieltjes function of a linear functional, we work out an algorithm which enables us to write the general fourth-order difference equation satisfied by the associated of any integer order of orthogonal polynomials of the Δ -Laguerre–Hahn class. Moreover, in classical situations (Meixner, Charlier, Krawtchouk and Hahn), we give these difference equations explicitly; and from the Hahn difference equation, by limit processes we recover the difference equations satisfied by the associated of the classical discrete orthogonal polynomials and the differential equations satisfied by the associated of the classical continuous orthogonal polynomials.     

Fehlerabschätzungen für gewisse Näherungskoeffizienten

Some simple error estimations for the approximate coefficients in the expansion of a real function by orthonormal polynomials are given. First we replace the function by an interpolation polynomial of degreen according to anyn+1 interpolation nodes, and in the second case we choose the nodes as the roots of some orthonormal polynomials of degreen+1.     

The computation of fourth virial coefficients for pairwise-additive spherically symmetric interaction potentials

A method is given for computing the fourth virial coefficient D(T) for a pairwise additive spherically symmetric interaction potential. Taking one of the four interacting particles as origin and using the appropriate co-ordinate transformations in the usual way the ninefold integrals defining D(T) are reduced to sixfold integrals which are then formally reduced to triple integrals by expanding out those Ursell-Mayer functions in the integrals not involving the origin particle as infinite series in Legendre polynomials P_S(cos?), where ? is the angle between the radius vectors of the interacting particles. The coefficients in these expansions are integrals of highly oscillatory functions, especially for large s, and are evaluated using the Chebyshev polynomial expansion for the Ursell-Mayer functions, thus making explicit use of the oscillatory behaviour of P_S(cos?). The triple integrals are evaluated using a nonproduct integration formula of the seventh degree employed earlier in the computation of the thirdh virial coefficient. The values of D(T) computed by the present method have the same qualitative behaviour as the literature values but appear to be more accurate, particularly at lower temperatures.     

Analysis and parameter estimation of bilinear systems via generalized orthogonal polynomials

An effective method of using generalized orthogonal polynomials (GOP) for analysing and identifying the parameters of a process whose behaviour can be modelled by a bilinear equation is presented. The integration operational matrix and the operational matrix for the product of tⁱ with the GOP vector are derived. These two kinds of operational matrices of the GOP are related to any type of individual orthogonal polynomial. By expanding the state and control functions into a series of GOP, the bilinear equation can be converted into a set of linear algebraic equations. The expansion coefficients of state variables are solved from these linear algebraic equations. The unknown parameters are evaluated by using the least squares method in conjunction with the individual orthogonal polynomial expansion. Two examples are given to illustrate the validity of the method. Very satisfactory results are obtained.     

The number of registers required for evaluating arithmetic expressions

We study the number of registers required for evaluating arithmetic expressions. This parameter of binary trees appears in various computer science problems as well as in numerous natural sciences applications where it is known as the Strahler number.We give several enumeration results describing the distribution of the number of registers for trees of size n. The average number of registers has the asymptotic expansion log₄n + D(log₄n) + 0(1); here, function D is periodic of period one, and its Fourier expansion can be explicitly determined in terms of Riemann's zeta function and Euler's gamma function.     

Computations with quasiseparable polynomials and matrices

In this paper, we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and univariate polynomials. Quasiseparable matrices generalize two classical matrix classes, Jacobi (tridiagonal) matrices and unitary Hessenberg matrices that are known to correspond to real orthogonal polynomials and Szegö polynomials, respectively. The latter two polynomial families arise in a wide variety of applications, and their short recurrence relations are the basis for a number of efficient algorithms. For historical reasons, algorithm development is more advanced for real orthogonal polynomials. Recent variations of these algorithms tend to be valid only for the Szegö polynomials; they are analogues and not generalizations of the original algorithms.     

Study of D-decompositions by the methods of computational real-valued algebraic geometry

New methods to study the D-decomposition with the use of the computational realvalued algebraic geometry were proposed. The number of domains of D-decomposition for the polynomial parametric families of polynomials and matrices was estimated. This technique which requires construction of the Gr?bner bases and cylindrical decomposition sometimes proves to be more precise than the traditional technique. The symbolic calculation system Maple v.14 and, in particular, its package RegularChains are used.     

Pseudospectral methods based on nonclassical orthogonal polynomials for solving nonlinear variational problems

Two direct pseudospectral methods based on nonclassical orthogonal polynomials are proposed for solving finite-horizon and infinite-horizon variational problems. In the proposed finite-horizon and infinite-horizon methods, the rate variables are approximated by the Nth degree weighted interpolant, using nonclassical Gauss-Lobatto and Gauss points, respectively. Exponential Freud type weights are introduced for both of nonclassical orthogonal polynomials and weighted interpolation. It is shown that the absolute error in weighted interpolation is dependent on the selected weight, and the weight function can be tuned to improve the quality of the approximation. In the finite-horizon scheme, the functional is approximated based on Gauss-Lobatto quadrature rule, thereby reducing the problem to a nonlinear programming one. For infinite-horizon problems, an strictly monotonic transformation is used to map the infinite domain onto a finite interval. We transcribe the transformed problem to a nonlinear programming using Gauss quadrature rule. Numerical examples demonstrate the accuracy of the proposed methods.     

Polytope-based computation of polynomial ranges

Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate a polytope defined as the convex hull of the coefficients with respect to the tensorial Bernstein basis, and we formulate several polytopes based on the Bernstein polynomials of the domain. These Bernstein polytopes can be defined by a polynomial number of halfspaces. We give the number of vertices, the number of hyperfaces, and the volume of each polytope for n=1,2,3,4, and we compare the computed range widths for random n-variate polynomials for n?10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.     

Numerical modelling of trace element fractionation during diffusion controlled crystallization

A FORTRAN IV computer program is presented and described which models the fractionation of trace elements during simple diffusion controlled crystallization of magmas. Two mathematical techniques are used: Crank—Nicolson finite difference and Lanczos tau polynomial methods, because it was determined that there were regions in which either one or the other was unsuitable. The regions of applicability of the respective methods are identified. The program can be used in several manners: (1) It can model diffusion controlled crystallization in which the melt is initially homogeneous in composition, with K (partition coefficient), D (diffusion coefficient), and R (rate of crystal growth) specified. Any of these variables may be changed during crystallization; (2) It can model a situation where the melt has compositional heterogeneity (specified by user), with K, R and D also specified. These variables may be changed during crystallization; (3) If the solid profile is specified, as well as K, the program can be made to calculate best-fit values for R/D ratio. Output from the program compares favourably with actual analytical data from the Bushveld Complex, South Africa. Although the geological basis for the model probably is conceptually simplistic, the model provides a basis for comparison with natural data, and thus can assist in obtaining greater insight into the processes involved in magmatic crystallization.     

Stability regions in the parameter space: D-decomposition revisited

The challenging problem in linear control theory is to describe the total set of parameters (controller coefficients or plant characteristics) which provide stability of a system. For the case of one complex or two real parameters and SISO system (with a characteristic polynomial depending linearly on these parameters) the problem can be solved graphically by use of the so-called D-decomposition. Our goal is to extend the technique and to link it with general M-Δ framework. In this way we investigate the geometry of D-decomposition for polynomials and estimate the number of root invariant regions. Several examples verify that these estimates are tight. We also extend D-decomposition for the matrix case, i.e. for MIMO systems. For instance, we partition real axis or complex plane of the parameter k into regions with invariant number of stable eigenvalues of the matrix A+kB. Similar technique can be applied to double-input double-output systems with two parameters.     

<Emphasis Type="Italic">D</Emphasis>-decomposition technique state-of-the-art

It is a survey of recent extensions and new applications for the classical D-decomposition technique. We investigate the structure of the parameter space decomposition into root invariant regions for single-input single-output systems linear depending on the parameters. The D-decomposition for uncertain polynomials is considered as well as the problem of describing all stabilizing controllers of the certain structure (for instance, PID-controllers) that satisfy given H _∞-criterion. It is shown that the D-decomposition technique can be naturally linked with M-Δ framework (a general scheme for analysis of uncertain systems) and it is applicable for describing feasible sets for linear matrix inequalities. The problem of robust synthesis for linear systems can be also treated via D-decomposition technique.     

Solution of variational problems via general orthogonal polynomials

The general orthogonal polynomials approximation is employed to solve variational problems. The operational matrix of integration is applied to reduce an integral equation to an algebraic equation with expansion coefficients. A simple and straightforward algorithm is then developed to calculate the expansion coefficients of the general orthogonal polynomials. The proposed method is general and various classical orthogonal polynomial approximations of the same problem can be obtained as a special case of the derived results.     

基于正交完备U-系统的图形分类与识别方法

为了探索有效的图形分类与识别的新方法,引进一类正交完备的分段k次多项式系统(简称U-系统).U-系统是一类属于L²[0,1]的正交完备分段k次多项式系统.该系统下的U级数展开式具有良好的平方逼近及一致逼近性质.基于U-系统理论,提出了U描述子的概念,给出了U描述子的性质并在理论上予以证明.为了更好地对图形分类与识别,对U描述子进行了归一化,同时在理论上证明了归一化U描述子具有旋转、平移、尺度大小等不变的性质.实验表明,归一化的U描述子能够高效、准确地对图形进行分类与识别,与Fourier描述子相比,具有更好的识 别率.     

Structure-preserving numerical algorithm for solving discrete-time LMI and DARS

We present a numerical algorithm to solve a discrete-time linear matrix inequality (LMI) and discrete-time algebraic Riccati system (DARS). With a given system (A,B,C,D) we associate a para-hermitian matrix pencil. Then we transform it by an orthogonal transformation matrix into a block-triangular para-hermitian form. Under either of the two assumptions (1) matrix pair (A,B) is controllable or (2) matrix pair (A,B) is reachable and (A,B,C,D) is a left invertible system, we extract the solution of LMI and DARS by the entries of the orthogonal transformation matrix.     

Programs for the approximation of real and imaginary single- and multi-valued functions by means of Hermite-Padé-approximants

We present programs for the calculation and evaluation of special type Hermite-Padé-approximations. They allow the user to either numerically approximate multi-valued functions represented by a formal series expansion or to compute explicit approximants for them. The approximation scheme is based on Hermite-Padé polynomials and includes both Padé and algebraic approximants as limiting cases. The algorithm for the computation of the Hermite-Padé polynomials is based on a set of recursive equations which were derived from a generalization of continued fractions. The approximations retain their validity even on the cuts of the complex Riemann surface which allows for example the calculation of resonances in quantum mechanical problems. The programs also allow for the construction of multi-series approximations which can be more powerful than most summation methods.

Program summary

Title of program: hp.srCatalogue identifier: ADSOProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSOProgram obtainable from: CPC Program Library, Queen's University Belfast, Northern IrelandLicensing provisions: Persons requesting the program must sign the standard CPC non-profit use licenseComputer: Sun Ultra 10Installation: Computing Center, University of Regensburg, GermanyOperating System: Sun Solaris 7.0Program language used: MapleV.5Distribution format: tar gzip fileMemory required to execute with typical data: 32 MB; the program itself needs only about 20 kBNumber of bits in a word: 32No. of processors used: 1Has the code been vectorized?: noNo. of bytes in distributed program, including test data etc.: 38194No. of lines in distributed program, including test data, etc.: 4258Nature of physical problem: Many physical and chemical quantum systems lead to the problem of evaluating a function for which only a limited series expansion is known. These functions can be numerically approximated by summation methods even if the corresponding series is only asymptotic. With the help of Hermite-Padé-approximants many different approximation schemes can be realized. Padé and algebraic approximants are just well-known examples. Hermite-Padé-approximants combine the advantages of highly accurate numerical results with the additional advantage of being able to sum complex multi-valued functions.Method of solution: Special type Hermite-Padé polynomials are calculated for a set of divergent series. These polynomials are then used to implicitly define approximants for one of the functions of this set. This approximant can be numerically evaluated at any point of the Riemann surface of this function. For an approximation order not greater than 3 the approximants can alternatively be expressed in closed form and then be used to approximate the desired function on its complete Riemann surface.Restriction on the complexity of the problem: In principle, the algorithm is only limited by the available memory and speed of the underlying computer system. Furthermore the achievable accuracy of the approximation only depends on the number of known series coefficients of the function to be approximated assuming of course that these coefficients are known with enough accuracy.Typical running time: 10 minutes with parameters comparable to the testrunsUnusual features of the program: none