A matrix interpretation of the extended Euclidean algorithm (Q2784646)

The extended Euclidean algorithm for polynomials and formal power series that is used for the recursive computation of Padé approximants can be viewed in various ways as a sequence of successive matrix multiplications that are applied to a Sylvester matrix with the original data. This result is presented in a general version that includes the treatment of the Cabay-Meleshko look-ahead algorithm, which generalizes the extended Euclidean algorithm and yields a weakly stable (forward stable) method for computing Padé fractions if it is combined with an appropriate rule for choosing the look-ahead step length. Moreover a particularly appealing form is choosen for the matrix interpretation where also the product of all the matrices that are applied has a meaning: this product yields at the end four Toeplitz blocks with the coefficients of polynomials (which belong to Padé forms) that are generated by the extended Euclidean algorithm in addition to those resulting from the ordinary algorithm.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00034].