It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, whith m < ...

Let $h$ be a complex valued multiplicative function. For any $N\in \mathbb{N}$, we compute the value of the determinant $D_N:={det}_{i|N,j|N}\left(\frac{h\left(\left(i,j\right)\right)}{ij}\right)$ where $\left(i,j\right)$ denotes the greatest common divisor of $i$ and $j$, which appear in increasing order in rows and columns. Precisely we prove that $$D_N=\prod _{p{}^l\parallel N}{\left(\frac{1}{p^{l\left(l+1\right)}}\stackrel{l}{\prod _{i=1}}\left(h\left(p^i\right)-h\left(p^{i-1}\right)\right)\right)}^{\tau \left(N/p^l\right)}.$$ This means that $D_N^{1/\tau \left(N\right)}$ is a multiplicative function of $N$. The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions $f\left(n\right)$, with $0\le f\left(p\right)<1$, as minimal values of certain...

The coefficients of the greatest common divisor of two polynomials $f$ and $g$ (GCD$(f,g)$) can be obtained from the Sylvester subresultant matrix $S_j(f,g)$ transformed to lower triangular form, where $1\le j\le d$ and $d=$ deg(GCD$(f,g)$) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of $S_j(f,g)$ for an arbitrary allowable $j$ are in details described and an algorithm for the calculation of the GCD$(f,g)$ is formulated. If inexact polynomials are given, then an approximate greatest...

The computation of the greatest common divisor (GCD) has many applications in several disciplines including computer graphics, image deblurring problem or computing multiple roots of inexact polynomials. In this paper, Sylvester and Bézout matrices are considered for this purpose. The computation is divided into three stages. A rank revealing method is shortly mentioned in the first one and then the algorithms for calculation of an approximation of GCD are formulated. In the final stage the coefficients...