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A Matrix Interpretation of the Extended Euclidean Algorithm

by M. H. Gutknecht

(Report number 2000-05)

Abstract
The extended Euclidean algorithm for polynomials and formal power series that is used for the recursive computation of \PA s can be viewed in various ways as a sequence of successive matrix multiplications that are applied to a Sylvester matrix with the original data. Here we present this result 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, we choose for the matrix interpretation a particularly appealing form 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 the polynomials (which belong to \PF s) that are generated by the extended Euclidean algorithm in addition to those resulting from the ordinary Euclidean algorithm.

Keywords:

@Techreport{G00_264,
  author = {M. H. Gutknecht},
  title = {A Matrix Interpretation of the Extended Euclidean Algorithm},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2000-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2000/2000-05.pdf },
  year = {2000}
}

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