Resumen de Truncation error bounds for the composition of limit-periodic linear fractional transformations

C. Baltus

• Baltus, C., Truncation error bounds for the composition of limit-periodic linear fractional transformations, Journal of Computational and Applied Mathematics 46 (1993) 395–404.

  Suppose (i) {tn(w} is a sequence of linear fractional transformations, tn(w) → t(w) = (aw + b)/(w + d), ad ≠ b or a = b = 0 ≠ d, where fixed points un → u, vn → v, with |(d + v)/(d + u)| < 1, and (ii) Tn(w) ≔ t1 ∘ t2 ∘ t3 ∘ ⋯ ∘ tn(w). In the first case, lim Tn(w) is known to exist and be the same for all w ≠ v; proof is given for the second case. The value region concept, familiar in the study of continued fractions, is employed to show that {Tn(u)| converges faster than any other {Tn(w)} and to develop bounds on the truncation error |lim Tn(u) − Tn(u|. In an example (the even and odd parts of a Gaussian continued fraction) the bounds are sharp and hold for small n.