Advances in Computational Mathematics
, Volume 8, Issue 3, pp 193219
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.
 Title
 On a vector q‐d algorithm
 Journal

Advances in Computational Mathematics
Volume 8, Issue 3 , pp 193219
 Cover Date
 199804
 DOI
 10.1023/A:1018944213562
 Print ISSN
 10197168
 Online ISSN
 15729044
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

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

 D.E. Roberts ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Napier University, 219 Colinton Road, Edinburgh, EH14 1DJ, UK