Asymptotics of Divide-And-Conquer Recurrences Via Iterated Function Systems - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2012

## Asymptotics of Divide-And-Conquer Recurrences Via Iterated Function Systems

#### Abstract

Let $k≥2$ be a fixed integer. Given a bounded sequence of real numbers $(a_n:n≥k)$, then for any sequence $(f_n:n≥1)$ of real numbers satisfying the divide-and-conquer recurrence $f_n = (k-mod(n,k))f_⌊n/k⌋+mod(n,k)f_⌈n/k⌉ + a_n, n ≥k$, there is a unique continuous periodic function $f^*:\mathbb{R}→\mathbb{R}$ with period 1 such that $f_n = nf^*(\log _kn)+o(n)$. If $(a_n)$ is periodic with period $k, a_k=0$, and the initial conditions $(f_i:1 ≤i ≤k-1)$ are all zero, we obtain a specific iterated function system $S$, consisting of $k$ continuous functions from $[0,1]×\mathbb{R}$ into itself, such that the attractor of $S$ is $\{(x,f^*(x)): 0 ≤x ≤1\}$. Using the system $S$, an accurate plot of $f^*$ can be rapidly obtained.

### Dates and versions

hal-01197237 , version 1 (11-09-2015)

### Identifiers

• HAL Id : hal-01197237 , version 1
• DOI :

### Cite

John C. Kieffer. Asymptotics of Divide-And-Conquer Recurrences Via Iterated Function Systems. 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'12), 2012, Montreal, Canada. pp.55-66, ⟨10.46298/dmtcs.2983⟩. ⟨hal-01197237⟩

### Export

BibTeX XML-TEI Dublin Core DC Terms EndNote DataCite

90 View