An Efficient Method for Evaluating Complex Polynomials

Article

Abstract

We propose an efficient hardware-oriented method for evaluating complex polynomials. The method is based on solving iteratively a system of linear equations. The solutions are obtained digit-by-digit on simple and highly regular hardware. The operations performed are defined over the reals. We describe a complex-to-real transform, a complex polynomial evaluation algorithm, the convergence conditions, and a corresponding design and implementation. The latency and the area are estimated for the radix-2 case. The main features of the method are: the latency of about m cycles for an m-bit precision; the cycle time independent of the precision; a design consisting of identical modules; and digit-serial connections between the modules. The number of modules, each roughly corresponding to serial-parallel multiplier without a carry-propagate adder, is 2(n + 1) for evaluating an n-th degree complex polynomial. The method can also be used to compute all successive integer powers of the complex argument with the same latency and a similar implementation cost. The design allows straightforward tradeoffs between latency and cost: a factor k decrease in cost leads to a factor k increase in latency. A similar tradeoff between precision, latency and cost exists. The proposed method is attractive for programmable platforms because of its regular and repetitive structure of simple hardware operators.

Keywords

Complex polynomials Complex powers Complex-to-real transform Digit-by-digit algorithms Left-to-right evaluation 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Computer Science DepartmentUniversity of California at Los AngelesLos AngelesUSA
  2. 2.CNRS-Laboratoire LIP, projet Arénaire, InriaUniversité de Lyon, Ecole Normale Supérieure de LyonLyon Cedex 07France

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