Marilena Mitrouli

A matrix pencil based numerical method for the computation of the GCD of polynomials

The paper presents a new numerical method for the computation of the greatest common divisor (GCD) of an m-set of polynomials of R[s], P/sub m,d/, of maximal degree d. It is based on a previously proposed theoretical procedure (Karcanias, 1989) that characterizes the GCD of P/sub m,d/ as the output decoupling zero polynomial of a linear system S(A/spl circ/,C/spl circ/) that may be associated with P/sub m,d/. The computation of the GCD is thus reduced to finding the finite zeros of the pencil sW-AW, where W is the unobservable subspace of S(A/spl circ/,C/spl circ/). If k=dim W, the GCD is determined as any nonzero entry of the kth compound C/sub k/(sW-A/spl circ/W). The method defines the exact degree of GCD, works satisfactorily with any number of polynomials and evaluates successfully approximate solutions. >

A compound matrix algorithm for the computation of the Smith form of a polynomial matrix

In the present paper is presented a numerical method for the exact reduction of a singlevariable polynomial matrix to its Smith form without finding roots and without applying unimodular transformations. Using the notion of compound matrices, the Smith canonical form of a polynomial matrixM(s)∈ℝnxn[s] is calculated directly from its definition, requiring only the construction of all thep-compound matricesCp(M(s)) ofM(s), 1<p≤n. This technique produces a stable and accurate numerical algorithm working satisfactorily for any polynomial matrix of any degree.

Moments of a linear operator, with applications to the trace of the inverse of matrices and the solution of equations

SUMMARY Let H be a real finite dimensional Hilbert space and A an invertible linear operator on it. In this paper, we are interested in obtaining estimations of Tr(A−1) and of the norm of the error when solving the equation Ax = f ∈ H. These estimates are obtained by extrapolation of the moments of A. Numerical results are given, and applications are discussed. Copyright

The Maximal Determinant and Subdeterminants of ±1 Matrices

Abstract In this paper we study the maximal absolute values of determinants and subdeterminants of ±1 matrices, especially Hadamard matrices. It is conjectured that the determinants of ±1 matrices of order n can have only the values k·p, where p is specified from an appropriate procedure. This conjecture is verified for small values of n. The question of what principal minors can occur in a completely pivoted ±1 matrix is also studied. An algorithm to compute the (n−j)×(n−j) minors, j=1,2,…, of Hadamard matrices of order n is presented, and these minors are determined for j=1,…,4.

The growth factor of a Hadamard matrix of order 16 is 16

In 1968 Cryer conjectured that the growth factor of an n × n Hadamard matrix is n. In 1988 Day and Peterson proved this only for the Hadamard–Sylvester class. In 1995 Edelman and Mascarenhas proved that the growth factor of a Hadamard matrix of order 12 is 12. In the present paper we demonstrate the pivot structures of a Hadamard matrix of order 16 and prove for the first time that its growth factor is 16. The study is divided in two parts: we calculate pivots from the beginning and pivots from the end of the pivot pattern. For the first part we develop counting techniques based on symbolic manipulation for specifying the existence or non-existence of specific submatrices inside the first rows of a Hadamard matrix, and so we can calculate values of principal minors. For the second part we exploit sophisticated numerical techniques that facilitate the computations of all possible (n − j) × (n − j) minors of Hadamard matrices for various values of j. The pivot patterns are obtained by utilizing appropriately the fact that the pivots appearing after the application of Gaussian elimination on a completely pivoted matrix are given as quotients of principal minors of the matrix. Copyright

Numerical Computation of the Least Common Multiple of a Set of Polynomials

The paper presents a number of properties of the least common multiple (LCM) m(s) of a given set of polynomials P. These results lead to the formulation of a new procedure for computing the LCM that avoids the computation of roots. This procedure involves the computation of the greatest common divisor (GCD) z(s) of a set of polynomials T derived from P, and the factorisation of the product of the original set P, p(s) as p(s) = m(s)·z(s). The symbolic procedure leads to a numerical one, where robust methods for the computation of GCD are first used. In this numerical method the approximate factorisation of polynomials is an important part of the overall algorithm. The latter problem is handled by studying two associated problems: evaluation of order of approximation and the optimal completion problem. The new method provides a robust procedure for the computation of LCM and enables the computation of approximate values, when the original data are inaccurate.

Growth in Gaussian elimination for weighing matrices W(n,n-1)

Abstract We consider the values for large minors of a skew-Hadamard matrix or conference matrix W of order n and find that maximum n×n minor equals to (n−1) n/2 , maximum (n−1)×(n−1) minor equals to (n−1) (n/2)−1 , maximum (n−2)×(n−2) minor equals to 2(n−1) (n/2)−2 , and maximum (n−3)×(n−3) minor equals to 4(n−1) (n/2)−3 . This leads us to conjecture that the growth factor for Gaussian elimination (GE) of completely pivoted (CP) skew-Hadamard or conference matrices and indeed any CP weighing matrix of order n and weight n−1 is n−1 and that the first and last few pivots are (1,2,2,3 or 4 ,…,n−1 or (n−1)/2,(n−1)/2,n−1) for n>14. We show the unique W(6,5) has a single pivot pattern and the unique W(8,7) has at least two pivot structures. We give two pivot patterns for the unique W(10,9) .

Numerical performance of the matrix pencil algorithm computing the greatest common divisor of polynomials and comparison with other matrix-based methodologies

This paper presents a new numerical algorithm for the computation of the greatest common divisor (GCD) of several polynomials, based on system-theoretic properties. The specific algorithm, characterizes the GCD as the output decoupling zero polynomial of an appropriate linear system associated with the given polynomial set. The computation of the GCD is thus reduced to specifying a nonzero entry of a vector forming the compound matrix of a matrix pencil directly produced from the associated linear system. A detailed description of the implementation of the algorithm is presented and analytical proofs of its stability are also developed. The MATLAB code of the algorithm is also described in the appendix.

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.