Summary:
The Greatest Common Divisor (GCD) of a polynomial set is proven to be very important to many applications in applied mathematics and engineering.
Several methods have been proposed for the computation of the GCD of sets of polynomials.
Most of them are based on the Euclidean algorithm. They are designed to process two polynomials at a time and can be applied iteratively when a set of more than two polynomials is considered . Conversely, there exist efficient matrix-based methods which can compute the degree and the coefficients of the GCD by applying specific transformations to a matrix formed directly from the coefficients of the polynomials of the entire given set.
Barnett's theorems about (GCD) through Bezoutians involve Bezout-like matrices and suggest a very compact way of parametrising and representing the GCD of several univariate polynomials.
The present work introduces the application of the QR decomposition with column pivoting (QRCP) to a {Bezout} matrix, achieving the computation of the degree and the coefficients of the GCD through the range of the {Bezout} matrix ,especially when the rank deficiency of the {Bezout} matrix is high.In the beginning we construct the {Bezout}matrix of a set of polynomials ,we apply Barnett's theorems and in the end we apply the QRCP method to find the coeffiecients of the GCD.
This method provides the means for a more efficient implementation of the classical {Bezout}-QR method with less computational complexity and without compromising accuracy, and it enriches the existing framework for the computation of the GCD of several polynomials using structured matrices. The classical GCD representations through structured matrices are revisited and their computational complexity is theoretically analyzed and compared. Demonstrative examples explaining the application of each method are given.We compare the methods and their complexity.We propose the use of the {rank revealing QR with column pivoting} for the computation of the GCD of polynomials through {Bezout}-like matrices which improves the numerical behavior of the existing {Bezout}-QR algorithms.
