On the algebraic fitting of experimental thermodynamic data with special regard to employing of orthonormal polynomials

Following the Weierstrass approximation theorem the thermodynamic excess functions are representable with arbitrary high accuracy by means of polynomials of sufficient high degrees in the mole fraction x. So, algebraic fitting of experimental thermodynamic excess data can be based upon mathematical polynomial expressions without any loss of generality. With respect to the necessary scattering of experimental results, algebraic evaluation of those data can only be solved by employing the calculus of observations. The least square method is the only principle of fitting with full justification by statistical mathematics, and which can be applied directly for algebraic fitting of experimental data by means of a computer. The general linear problem of fitting is solved explicitly (i) by means of Gauss method of elimination, and (ii) by employing the property of “orthonormality” of polynomials. In the latter case the explicit form of the “orthonormal” polynomials depends strongly on the number of experimental data which has to be fitted. A convenient procedure is presented to generate polynomials which are orthonormal with respect to an actual set of experimental data. Computer-programs in PORTRAN-language are enclosed 1) to employ Gauss method of elimination, and 2) to generate discrete orthonormal polynomials.     

Optimal Algorithm for Algebraic Factoring

This paper presents on optimized method for factoring multivariate polynomials over algebraic extension fields defined by an irreducible ascending set. The basic idea is to convert multivariate polynomials to univariate polynomials and algebraic extension fields to algebraic number fields by suitable integer substituteions.Then factorize the univariate polynomials over the algebraic number fields.Finally,construct mulativariate factors of the original polynomial by Hensel lemma and TRUEFACTOR test.Some examples with timing are included.     

Algebraic approximation of optimal cyclic control problems

Algebraic polynomials are used to approximate the state trajectory, the control and the functions in differential equations and in the criterion for the constrained optimal cyclic control problem. Several discretized problems using algebraic polynomials are proposed to approximate the basic control problem. Sufficient conditions for the convergence of solutions of approximating problems to the optimal solution of the basic problem are given and the convergence rate is estimated. The application to a class of industrial optimal cyclic control problems is discussed.     

Application of general discrete orthogonal polynomials to optimal-control systems

The shift-transformation matrix of general discrete orthogonal polynomials is introduced. General discrete orthogonal polynomials are adopted to obtain the modified discrete Euler-Lagrange equations. Then general discrete orthogonal polynomials are applied to simplify the discrete Euler-Lagrange equations into a set of linear algebraic ones for the approximation of state and control variables of digital systems. An example is included to demonstrate the simplicity and applicability of the method. Also, a comparison of the results obtained via several classical discrete orthogonal polynomials for the same problem is given.     

2‐D ALGEBRAIC TEST FOR ROBUST STABILITY OF QUASIPOLYNOMIALS WITH INTERVAL PARAMETERS

Determining the robust stability of interval quasipolynomials leads to a NP problem: an enormous number of testing edge polynomials. This paper develops an efficient approach to reducing the number of testing edge polynomials. This paper solves the stability test problem of interval quasipolynomials by transforming interval quasipolynomials into two‐dimensional (2‐D) interval polynomials. It is shown that the robust stability of an interval 2‐D polynomial can ensure the stability of the quasipolynomial, and the algebraic test algorithm for 2‐D s‐z interval polynomials is provided. The stability of 2‐D s‐z vertex polynomials and 2‐D s‐z edge polynomials were tested by using a Schur Table of complex polynomials.     

Verified Computation of Fast Decreasing Polynomials

In this paper the problem of verified numerical computation of algebraic fast decreasing polynomials approximating the Dirac delta function is considered. We find the smallest degree of the polynomials and give precise estimates for this degree. It is shown that the computer algebra system Maple does not always graph such polynomials reliably because of evaluating the expressions in usual floating-point arithmetic. We propose a procedure for verified computation of the polynomials and use it to produce their correct graphic presentation in Maple.     

Root regions of bounded coefficient polynomials

For fixed degree monic polynomials with real coefficients, the direct and inverse problems of root inclusion are denned. The direct problem is to find the infimal region Γ¹ _γof the C-plane that contains all roots of the bounded coefficient norm polynomials. The inverse problem is to define the maximal subregion Γ² _γ of Γ¹ _γ such that all polynomials with their roots in Γ² _γ are bounded norm coefficient polynomials. Those two problems are solved in the case of stable and totally unstable polynomials and the boundaries of Γ¹ _γ,Γ² _γ: regions are defined as branches of an algebraic function defined by the norm bound y. For general polynomials, the inverse problem is also solved for polynomials of any degree, whereas the direct problem is discussed in the case of third-order polynomials.     

A contribution to the efficient solution of extensive symbolic computations

The solving of extensive algebraic problems with computers is difficult owing to the large amount of storage and computing time needed. This paper describes a method that obviates these difficulties in many cases by using appropriate operators instead of extensive polynomials. This calculation is prompted by the question whether a class of algebraic equilibria in plasma physics governed by magneto-hydrodynamic theory is stable. Great accuracy is called for in treating this problem.We describe the calculation of a three-dimensional integral containing Fourier, Taylor and trigonometric series.     

Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications

In this paper, we consider the problem of the simultaneous determination of time-dependent coefficients in a one-dimensional partial differential equation. The main aim is to apply the tau technique to determine unknown coefficients in a time-dependent partial differential equation. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented 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.     

Approximation of algebraic and trigonometric polynomials by feedforward neural networks

This paper presents a function approximation to a general class of polynomials by using one-hidden-layer feedforward neural networks(FNNs). Both the approximations of algebraic polynomial and trigonometric polynomial functions are discussed in details. For algebraic polynomial functions, an one-hidden-layer FNN with chosen number of hidden-layer nodes and corresponding weights is established by a constructive method to approximate the polynomials to a remarkable high degree of accuracy. For trigonometric functions, an upper bound of approximation is therefore derived by the constructive FNNs. In addition, algorithmic examples are also included to confirm the accuracy performance of the constructive FNNs method. The results show that it improves efficiently the approximations of both algebraic polynomials and trigonometric polynomials. Consequently, the work is really of both theoretical and practical significance in constructing a one-hidden-layer FNNs for approximating the class of polynomials. The work also paves potentially the way for extending the neural networks to approximate a general class of complicated functions both in theory and practice.     

A Computational Method for Determining Strong Stabilizability of n-D Systems

This paper describes an algorithm for the following problem: given two multivariate complex or real polynomials f and g , decide whether there exist complex or real polynomials h and k such that both k and fh + gk have no zero in the unit polydisc. This problem, known as strong stabilizability, is fundamental in control theory, with important applications in designing stable feedback systems with a stable compensator. Our algorithm for solving the problem is formulated based on the cylindrical algebraic decomposition(cad) of an algebraic variety. While recent applications of cad to systems and control have been focused on those problems which have a quantifier elimination formulation, our method is novel in that it explicitly computes some topological properties of an algebraic variety based on the cad to solve the problem for which a quantifier elimination formulation is not readily available.     

On the power of algebraic branching programs of width two

We show that there are families of polynomials having small depth-two arithmetic circuits that cannot be expressed by algebraic branching programs of width two. This clarifies the complexity of the problem of computing the product of a sequence of two-by-two matrices, which arises in several settings.     

The Durand-Kerner method for trigonometric and exponential polynomials

The problem of finding all roots of an exponential or trigonometric equation is reduced to the determination of zeros of algebraic polynomials where the well-known Durand-Kerner algorithm can be applied. This transformation of the problem has the additional advantage that the periodicity of the original functions is eliminated and the choice of starting values is simplified.     

Tropical algebraic geometry in Maple: A preprocessing algorithm for finding common factors for multivariate polynomials with approximate coefficients

Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view of this problem leads to efficient preprocessing techniques, applying polyhedral methods to the exact exponents and numerical techniques to the approximate coefficients. With Maple we will illustrate our use of tropical algebraic geometry.     

Trajectory sensitivity analysis using discrete Legendre orthogonal polynomials

A method of finite series expansion using discrete Legendre orthogonal polynomials (DLOPs) is applied to the problem of state trajectory sensitivity analysis. The method presented has a distinct advantage in that it reduces the problem of determining the trajectory sensitivity function to that of solving a set of algebraic equations. In addition, the method is simple and amenable to computer computation.     

Motion planning for control-affine systems satisfying low-order controllability conditions

This paper is devoted to the motion planning problem for control-affine systems by using trigonometric polynomials as control functions. The class of systems under consideration satisfies the controllability rank condition with the Lie brackets up to the second order. The approach proposed here allows to reduce a point-to-point control problem to solving a system of algebraic equations. The local solvability of that system is proved, and formulas for the parameters of control functions are presented. Our local and global control design schemes are illustrated by several examples.     

On multivariate interpolation by generalized polynomials on subsets of grids

This note may be regarded as a complement to a paper of H. Werner [17] who has carried over Newton's classical interpolation formula to Hermite interpolation by algebraic polynomials of several real variables on certain subsets of grids. Here generalized polynomials of several real or complex variables are treated. Recursive procedures are presented showing that interpolation by generalized multivariate polynomials is performed nearly as simply as interpolation by algebraic polynomials. Having in general the same approximation power, generalized polynomials may be better adapted to special situations. In particular, the results of this note can be used for constructing nonpolynomial finite elements since in that case the interpolation points usually are rather regular subsystems of grids. Though the frame is more general than in [17] some of our proofs are simpler. As an alternative method to evaluate multivariate generalized interpolation polynomials for rectangular grids a Neville-Aitken algorithm is presented.     

一类半正定多项式的平方和分解及其表达式的自动生成

建立了一个把半正定稀疏多项式表为多项式平方和的算法.这一算法依赖于Hilbert第17问题的一系列经典研究结果以及实闭域上量词消去的柱形代数剖分算法.该算法的机器实现为一类代数不等式可读性证明的自动生成提供了一种非常自然的途径.     

Functional Renewal of Behavior of Systems: Numerical Methods

The principles of functional renewal of the behavior of technical systems based on the functional capabilities in the design are stated. A finite deterministic automaton is used as the model. The functional renewal problem is solved within the framework of the theory of universal automata. A class of automata admitting modeling by a family of polynomials is described. The numerical model of automaton behavior is helpful in applying algebraic tools to solve the functional behavior renewal problem. For this automaton class, a universal enumerator is designed and analyzed, and a method of constructing a renewal sequence is described.