Advances in Computational Mathematics

, Volume 8, Issue 3, pp 193-219

First online:

On a vector q‐d algorithm

  • D.E. RobertsAffiliated withDepartment of Mathematics, Napier University

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Using the framework provided by Clifford algebras, we consider a non‐commutative quotient‐difference algorithm for obtaining the elements of a continued fraction corresponding to a given vector‐valued power series. We demonstrate that these elements are ratios of vectors, which may be calculated with the aid of a cross rule using only vector operations. For vector‐valued meromorphic functions we derive the asymptotic behaviour of these vectors, and hence of the continued fraction elements themselves. The behaviour of these elements is similar to that in the scalar case, while the vectors are linked with the residues of the given function. In the particular case of vector power series arising from matrix iteration the new algorithm amounts to a generalisation of the power method to sub‐dominant eigenvalues, and their eigenvectors.

vector continued fraction vector Padé approximant quotient‐difference algorithm Clifford algebra cross rule power method