Christou, D. (2011).
*ERES Methodology and Approximate Algebraic Computations*.
(Unpublished Doctoral thesis, City University London)

## Abstract

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.

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