Dimitrios Triantafyllou

Two resultant based methods computing the greatest common divisor of two polynomials

In this paper we develop two resultant based methods for the computation of the Greatest Common Divisor (GCD) of two polynomials. Let S be the resultant Sylvester matrix of the two polynomials. We modified matrix S to S*, such that the rows with non-zero elements under the main diagonal, at every column, to be gathered together. We constructed modified versions of the LU and QR procedures which require only the of floating point operations than the operations performed in the general LU and QR algorithms. Finally, we give a bound for the error matrix which arises if we perform Gaussian elimination with partial pivoting to S*. Both methods are tested for several sets of polynomials and tables summarizing all the achieved results are given.

Matrix pencil methodologies for computing the greatest common divisor of polynomials: hybrid algorithms and their performance

The computation of the greatest common divisor (GCD) of several polynomials is a problem that emerges in many fields of applications. The GCD computation has a non-generic nature and thus its numerical computation is a hard problem. In this paper we examine the family of matrix pencil methods for GCD computation and investigate their performance as far as their complexity, error analysis and their effectiveness for evaluating approximate solutions. The relative merits of the various variants of such methods are examined for the different cases of sets of polynomials with varying number of elements and degree. The developed algorithms combine symbolical and numerical programming and this is what we define here as hybrid computations. The combination of numerical operations with symbolical programming can improve the nature of the methods and guarantees the stability of the algorithm. Furthermore, it emphasizes the significance of hybrid computations in complex problems such as the computation of GCD. All methods are tested thoroughly for several sets of polynomials and the results are presented in tables.

Blind image deconvolution using a banded matrix method

In this paper we study the blind image deconvolution problem in the presence of noise and measurement errors. We use a stable banded matrix based approach in order to robustly compute the greatest common divisor of two univariate polynomials and we introduce the notion of approximate greatest common divisor to encapsulate the above approach, for blind image restoration. Our method is analyzed concerning its stability and complexity resulting to useful conclusions. It is proved that our approach has better complexity than the other known greatest common divisor based blind image deconvolution techniques. Examples illustrating our procedures are given.

On rank and null space computation of the generalized Sylvester matrix

In this paper, we present new approaches computing the rank and the null space of the (mn + p)×(n + p) generalized Sylvester matrix of (m + 1) polynomials of maximal degrees n,p. We introduce an algorithm which handles directly a modification of the generalized Sylvester matrix, computing efficiently its rank and null space and replacing n by log2n in the required complexity of the classical methods. We propose also a modification of the Gauss-Jordan factorization method applied to the appropriately modified Sylvester matrix of two polynomials for computing simultaneously its rank and null space. The methods can work numerically and symbolically as well and are compared in respect of their error analysis, complexity and efficiency. Applications where the computation of the null space of the generalized Sylvester matrix is required, are also given.

The Weierstrass Canonical Form of a Regular Matrix Pencil: Numerical Issues and Computational Techniques

In the present paper, we study the derivation of the Weierstrass Canonical Form (WCF) of a regular matrix pencil. In order to compute the WCF, we use two important computational tools: a) the QZ algorithm to specify the required root range of the pencil and b) the updating technique to compute the index of annihilation. The proposed updating technique takes advantages of the already computed rank of the sequences of matrices that appears during our procedure reducing significantly the required floating-point operations. The algorithm is implemented in a numerical stable manner, giving efficient results. Error analysis and the required complexity of the algorithm are included.

A hybrid method for computing the intersection and tangency points of plane curves

Abstract In this paper we present a symbolic-numeric (hybrid) method for computing the intersection and tangency points of given plane curves. The whole procedure involves three phases: (i) implicitization, (ii) root specification, and (iii) inversion. For each one of these phases we propose an appropriate algorithm fully documented regarding its complexity and stability. A comparison with other existing methods is also provided. All the proposed methods are illustrated through examples.

Parallel QR processing of Generalized Sylvester matrices

In this paper, we develop a parallel QR factorization for the generalized Sylvester matrix. We also propose a significant faster evaluation of the QR applied to a modified version of the initial matrix. This decomposition reveals useful information such as the rank of the matrix and the greatest common divisor of the polynomials formed from its coefficients. We explicitly demonstrate the parallel implementation of the proposed methods and compare them with the serial ones. Numerical experiments are also presented showing the speed of the parallel algorithms.

Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials

Abstract This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical effciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of theGCDof sets of several polynomials with real coeffcients is provided.Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented.

Computer Graphics Techniques in Military Applications

The determination of intersection points of plane curves is a problem of Computer Graphics with many applications in Applied Mathematics, Numerical Analysis and many other scientific fields. More precisely, in military applications, the trajectories of two flying objects such as missiles, aircrafts etc, can be interpreted by two plane curves. Our scope is to find the intersection points of the given curves. The number of floating point operations (flops) of many classical methods is not satisfactory, since they demand over O(n 4) operations. Conversely, many algorithms that are fast enough, have serious problems with their numerical stability. The main objective here is to develop fast and stable algorithms computing the intersection points of plane curves. The error analysis and the computation of complexity of all the proposed methods are analysed and demonstrated through various examples.

A hybrid approach for normal factorization of polynomials

The problem considered here is an integral part of computations for algebraic control problems. The paper introduces the notion of normal factorization of polynomials and then presents a new hybrid algorithm for the computation of this factorization. The advantage of such a factorization is that it handles the determination of multiplicities and produces factors of lower degree and with distinct roots. The presented algorithm has the ability to specify the roots of the polynomials without computing them explicitly. Also it may be used for investigating the clustering of the roots of the polynomials. The developed procedure is based on the use of algorithms determining the greatest common divisor of polynomials. The algorithm can be implemented symbolically for the specification of well separated roots and numerically for the specification of roots belonging in approximate clusters.