ERES Methodology and Approximate Algebraic Computations

The area of approximate algebraic computations is a fast growing area in modern computer algebra which has attracted many researchers in recent years. Amongst the various algebraic computations, the computation of the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of a set of polynomials are challenging problems that arise from several applications in applied mathematics and engineering. Several methods have been proposed for the computation of the GCD of polynomials using tools and notions either from linear algebra or linear systems theory. Amongst these, a matrix-based method which relies on the properties of the GCD as an invariant of the original set of polynomials under elementary row transformations and shifting elements in the rows of a matrix, shows interesting properties in relation to the problem of the GCD of sets of many polynomials. These transformations are referred to as Extended-Row-Equivalence and Shifting (ERES) operations and their iterative application to a basis matrix, which is formed directly from the coefficients of the given polynomials, formulates the ERES method for the computation of the GCD of polynomials and establishes the basic principles of the ERES methodology. The main objective of the present thesis concerns the improvement of the ERES methodology and its use for the efficient computation of the GCD and LCM of sets of several univariate polynomials with parameter uncertainty, as well as the extension of its application to other related algebraic problems. New theoretical and numerical properties of the ERES method are defined in this thesis by introducing the matrix representation of the Shifting operation, which is used to change the position of the elements in the rows of a matrix. This important theoretical result opens the way for a new algebraic representation of the GCD of a set polynomials, the remainder, and the quotient of Euclid's division for two polynomials based on ERES operations. The principles of the ERES methodology provide the means to develop numerical algorithms for the GCD and LCM of polynomials that inherently have the potential to efficiently work with sets of several polynomials with inexactly known coefficients. The present new implementation of the ERES method, referred to as the ``Hybrid ERES Algorithm", is based on the effective combination of symbolic-numeric arithmetic (hybrid arithmetic) and shows interesting computational properties concerning the approximate GCD and LCM problems. The evaluation of the quality, or ``strength", of an approximate GCD is equivalent to an evaluation of a distance problem in a projective space and it is thus reduced to an optimisation problem. An efficient implementation of an algorithm computing the strength bounds is introduced here by exploiting some of the special aspects of the respective distance problem. Furthermore, a new ERES-based method has been developed for the approximate LCM which involves a least-squares minimisation process, applied to a matrix which is formed from the remainders of Euclid's division by ERES operations. The residual from the least-squares process characterises the quality of the obtained approximate LCM. Finally, the developed framework of the ERES methodology is also applied to the representation of continued fractions to improve the stability criterion for linear systems based on the Routh-Hurwitz test

A Symbolic-Numeric Software Package for the Computation of the GCD of Several Polynomials

This survey is intended to present a package of algorithms for the computation of exact or approximate GCDs of sets of several polynomials and the evaluation of the quality of the produced solutions. These algorithms are designed to operate in symbolic-numeric computational environments. The key of their effectiveness is the appropriate selection of the right type of operations (symbolic or numeric) for the individual parts of the algorithms. Symbolic processing is used to improve on the conditioning of the input data and handle an ill-conditioned sub-problem and numeric tools are used in accelerating certain parts of an algorithm. A sort description of the basic algorithms of the package is presented by using the symbolic-numeric programming code of Maple

The ERES method for computing the approximate GCD of several polynomials

The computation of the greatest common divisor (GCD) of a set of polynomials has interested the mathematicians for a long time and has attracted a lot of attention in recent years. A challenging problem that arises from several applications, such as control or image and signal processing, is to develop a numerical GCD method that inherently has the potential to work efficiently with sets of several polynomials with inexactly known coefficients. The presented work focuses on: (i) the use of the basic principles of the ERES methodology for calculating the GCD of a set of several polynomials and defining approximate solutions by developing the hybrid implementation of this methodology. (ii) the use of the developed framework for defining the approximate notions for the GCD as a distance problem in a projective space to develop an optimization algorithm for evaluating the strength of different ad-hoc approximations derived from different algorithms. The presented new implementation of ERES is based on the effective combination of symbolic–numeric arithmetic (hybrid arithmetic) and shows interesting computational properties for the approximate GCD problem. Additionally, an efficient implementation of the strength of an approximate GCD is given by exploiting some of the special aspects of the respective distance problem. Finally, the overall performance of the ERES algorithm for computing approximate solutions is discussed

Approximate least common multiple of several polynomials using the ERES division algorithm

In this paper a numerical method for the computation of the approximate least common multiple (ALCM) of a set of several univariate real polynomials is presented. The most important characteristic of the proposed method is that it avoids root finding procedures and computations involving the greatest common divisor (GCD). Conversely, it is based on the algebraic construction of a special matrix which contains key data from the original set of polynomials and leads to the formulation of a linear system which provides the degree and the coefficients of the ALCM using low-rank approximation techniques and numerical optimization tools particularly in the presence of inaccurate data. The numerical stability and complexity of the method are analysed, and a comparison with other methods is provided

Numerical and Symbolical Methods for the GCD of Several Polynomials

The computation of the Greatest Common Divisor (GCD) of a set of polynomials is an important issue in computational mathematics and it is linked to Control Theory very strong. In this paper we present different matrix-based methods, which are developed for the efficient computation of the GCD of several polynomials. Some of these methods are naturally developed for dealing with numerical inaccuracies in the input data and produce meaningful approximate results. Therefore, we describe and compare numerically and symbolically methods such as the ERES, the Matrix Pencil and other resultant type methods, with respect to their complexity and effectiveness. The combination of numerical and symbolic operations suggests a new approach in software mathematical computations denoted as hybrid computations. This combination offers great advantages, especially when we are interested in finding approximate solutions. Finally the notion of approximate GCD is discussed and a useful criterion estimating the strength of a given approximate GCD is also developed

Nearest common root of a set of polynomials: A structured singular value approach

The paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients. In particular, we seek the minimum-magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the solution of a structured singular value (μ) problem arising in robust control for which numerous techniques are available. It is also shown that the method can be extended to the calculation of an “approximate GCD” of fixed degree by introducing the notion of the generalized structured singular value of a matrix. The work generalizes previous results by the authors involving the calculation of the “approximate GCD” of two polynomials, although the general case considered here is considerably harder and relies on a matrix-dilation approach and several preliminary transformations

Approximate zero polynomials of polynomial matrices and linear systems

This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials". The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of "approximate decoupling polynomials"

Flow of evaporating, gravity-driven thin liquid films over topography

The effect of topography on the free surface and solvent concentration profiles of an evaporating thin film of liquid flowing down an inclined plane is considered. The liquid is assumed to be composed of a resin dissolved in a volatile solvent with the associated solvent concentration equation derived on the basis of the well-mixed approximation. The dynamics of the film is formulated as a lubrication approximation and the effect of a composition-dependent viscosity is included in the model. The resulting time-dependent, nonlinear, coupled set of governing equations is solved using a full approximation storage multigrid method. The approach is first validated against a closed-form analytical solution for the case of a gravity-driven, evaporating thin film flowing down a flat substrate. Analysis of the results for a range of topography shapes reveal that although a full-width, spanwise topography such as a step-up or a step-down does not affect the composition of the film, the same is no longer true for the case of localized topography, such as a peak or a trough, for which clear nonuniformities of the solvent concentration profile can be observed in the wake of the topography

Student understanding of Taylor series expansions in statistical mechanics

One goal of physics instruction is to have students learn to make physical meaning of specific mathematical expressions, concepts, and procedures in different physical settings. As part of research investigating student learning in statistical physics, we are developing curriculum materials that guide students through a derivation of the Boltzmann factor using a Taylor series expansion of entropy. Using results from written surveys, classroom observations, and both individual think-aloud and teaching interviews, we present evidence that many students can recognize and interpret series expansions, but they often lack fluency in creating and using a Taylor series appropriately, despite previous exposures in both calculus and physics courses

Approximate controllability and observability measures in control systems design

The selection of systems of inputs and outputs (input and output structure) forms part of early system design, which is important since it preconditions the potential for control design. Existing methodologies for input, output structure selection rely on criteria expressing distance from uncontrollability (unobservability). The thesis introduces novel measures for evaluating and estimating the distance to uncontrollability and relatively unobservability. At first, the modal measuring approach is studied in detail, providing a framework for the ”best” structure selection. Although controllability (observability) is invariant under state feedback (output injection), the corresponding degrees expressing distance from uncontrollability (unobservability) are not. Hence, the thesis introduces new criteria for the distance problem from uncontrollability (unobservability) which is invariant under feedback transformations. The approach uses the restricted input-state (state-output) matrix pencil and then deploys exterior algebra that reduces the overall problem to the standard problem of distance of a set of polynomials from non-coprimeness. Results on the distance of the Sylvester Resultants from singularity provide the new measures. Since distance to singularity of the corresponding Sylvester matrix is the key in evaluating the distance to uncontrolability it is of the particular interest in the present work. In order to find the solution two novel methods are introduced in the thesis, namely the alternating projection algorithm and a structured singular value approach. A least-squares alternating projection algorithm, motivated by a factorisation result involving the Sylvester resultant matrix, is proposed for calculating the ”best” approximate GCD of a coprime polynomial set. The properties of the proposed algorithm are investigated and the method is compared with alternative optimisation techniques which can be employed to solve the problem. It is also shown that the problem of an approximate GCD calculation is equivalent tothe solution of a structured singular value (µ) problem arising in robust control for which numerous techniques are available. Motivated by the powerful concept of the structured singular values, the proposed method is extended to the special case of an implicit system that has a wide application in the behavioural analysis of complex systems. Moreover, µ-value approach has a potential application for the general distance problem to uncontrollability that is numerically hard to obtain. Overall, the proposed framework significantly simplifies and generalises the input-output structure selection procedure and evaluates alternative solutions for a variety of distance problems that appear in Control Theory