The GPGCD algorithm with the Bézout matrix

Abstract

With the progress of algebraic computations on polynomials and matrices, we are paying more attention to approximate algebraic algorithms. Among approximate algebraic algorithms, those for calculating approximate greatest common divisor (GCD) consider a pair of given polynomials f and g that are relatively prime in general, and find f and g which are close to f and g, respectively, in the sense of polynomial norm, and have the GCD of certain degree. The algorithms can be classified into two categories: 1) for a given tolerance (magnitude) of ||f - f|| and ||g - g||, make the degree of approximate GCD as large as possible, and 2) for a given degree d, minimize the magnitude of ||f - f|| and ||g - g||.