Journal of Mathematical Sciences, Vol. 120, No. 1, 2004 AN APPLICATION OF CONTROL-THEORETIC METHODS TO DIGITAL ARITHMETIC ALGORITHMS E. Rocha, A. Sarychev, A. Pereira, and R. Rodrigues UDC 517.977 Digital arithmetic has been an active field of research over the past two decades. A large number of algorithms have been published for the hardware implementation of the basic arithmetic operations and also the elementary transcendental functions. However, a solid mathematical framework seems to be still missing. This paper concerns normalization (multiplicative or additive) methods over generalized redundant number systems and the search for a global proof of convergence. In our presentation, we model redundant normalization algorithms as discrete-time, time-variant dynamical control systems with the strings of digits being treated as integer-valued controls. We use results on time-variant feedback stabilization for control systems in order to prove the convergence of algorithms for the elementary functions. 1. Introduction We consider redundant signed-digit number systems [3], which are positional radix-r number repre- sentation systems with a consecutive integer digit set A = {−α,...,β} and r ≥ 2. The redundancy factor of such a system is ρ = α + β r − 1 . We assume that systems are redundant, i.e., ρ> 1, and that positive and negative numbers are representable, i.e., α ≥ 1 and β ≥ 1. The redundancy plays a crucial role in the design of hardware algorithms for the basic operations of addition and subtraction, allowing for the removal of carry/borrow propagation chains typical of pure binary adders. Also, it proved to be a valuable factor in algorithms for division, providing the design of simple quotient-digit-selection schemes. In fact, it is well known that whenever long sequences of simple basic operations or operands are involved, the initial and final conversion and reconversion time from redundant to nonredundant forms is greater than compensated by the gain in computation speed achieved by operations performed with redundant number systems. We consider the class of online digit recurrence algorithms based on the evaluation of sequences of products (multiplicative normalization) or sums (additive normalization) in order to approximate the value of some function. Whenever one of the sequences converges to some fixed point, the other approaches the desired function value. These algorithms are designed in such a way that each step is reduced to computing a few sums, shifts, and small table lookup. The drawback is the linear rate of convergence, typically each iteration computes one radix-r digit of the result. Many algorithms have been described in the literature, generally focusing on particular digit sets, usually of small radix and symmetric, such as radix-2 {−1, 0, 1}, radix-4 {−2, −1, 0, 1, 2}, or radix-4 {−3, −2, −1, 0, 1, 2, 3}. Recently, some proposals for higher radix (16 and 32) implementations appeared. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 107, Aveiro Seminar on Control, Optimization, and Graph Theory, 2002. 1072–3374/04/1201–0995 c  2004 Plenum Publishing Corporation 995