Vector continued fractions using a generalized inverse

A real vector space combined with an inverse (involution) for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse permits construction of vector analogues of the Jacobi continued fraction. These vector Jacobi fractions are related to vector and scalar-valued polynomial functions of the vectors, which satisfy recurrence relations similar to those of orthogonal polynomials. The vector Jacobi fraction has strong convergence properties which are demonstrated analytically, and illustrated numerically. PACS numbers: 02.30.Mv, 02.10.Kn, 02.90.+p 1. Continued fractions An ordinary continued fraction is defined as repeated division and addition of numbers A/B + C/D + E/F + · · · (1) where the division is by everything to the right of the slash. Continued fractions have attracted interest because they can provide rapidly convergent approximations for various arithmetic quantities. For a survey of continued fractions see [1]. Inclusion of a variable in continued fractions produces functions of a complex variable, for example a Jacobi continued fraction with the complex variable z, when the {an} are all real and the {βn} are all positive in, 1/z− a0 − β1/z− a1 − β2/z− a2 − β3/z− · · · (2) This continued fraction is closely related to orthogonal polynomials [2] and Gaussian quadrature [3]. The broad motivation for this work is the approximation of distributions and functions on vector spaces, analogous to the approximation of weight distributions and smooth functions by Jacobi fractions and the polynomials associated with them. In this work, we develop a 3 Permanent address: 31 Roft Street, Oswestry SY11 2EP, UK. 0305-4470/04/010161+12$30.00 © 2004 IOP Publishing Ltd Printed in the UK 161 162 R Haydock and C M M Nex vector continued fraction which we show has convergence properties similar to those of the Jacobi fraction. While vector division might seem necessary for a vector continued fraction, we find interesting properties using only a generalized inverse of a vector. It seems that a real vector space together with an inverse for vectors is the minimum algebraic structure necessary to define a continued fraction. This paper begins with the definition of a generalized inverse for vectors and then applies this to the vector generalization of continued fractions of the Jacobi type. The next three sections develop polynomial functions of vectors, the associated Christofel–Darboux identity, and a theory of the convergence of these fractions. The penultimate section of the paper contains analytic and numerical examples of convergence and other properties of these vector continued fractions, while the final section contains remarks about quadrature and geometric algebra. 2. Definition of an inverse for a vector The inverse (involution) 1/Z of a vector Z has the property that the inverse of the inverse, 1/(1/Z) is the original vector Z. A simple multiplicative inverse of a vector Z can be defined in terms of the magnitude |Z| of Z and a self-inverse (symmetric orthogonal) transformation σ ,