Article citationsMore>>

W. G. Horner, “A New Method of Solving Numerical Equations of All Orders, by Continuous Approximation,” Philosophical Transactions of Royal Society of London, Vol. 109, 1819, pp. 308-335. doi:10.1098/rstl.1819.0023

has been cited by the following article:

• TITLE: Large-Integer Multiplication Based on Homogeneous Polynomials

  AUTHORS: Boris S. Verkhovsky

  KEYWORDS: Homogeneous Polynomials; Toom-Cook Algorithm; Multidigit Integers; Multi-Stage Multiplication; Generalized Horner Rule; Large-Integer Multiplication

  JOURNAL NAME: International Journal of Communications, Network and System Sciences, Vol.5 No.8, August 28, 2012

  ABSTRACT: Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations and elimination of necessity to evaluate polynomials in points with larger coordinates. It is demonstrated that a two-stage implementation of the proposed and Toom-Cook algorithms asymptotically require twice as many standard multiplications than their direct implementation. A multistage implementation of these algorithms is also less efficient than their direct implementation. Although the proposed algorithms as well as the corresponding Toom-Cook algorithms require numerous algebraic additions, the Generalized Horner rule for evaluation of homogeneous polynomials, provided in the paper, decrease this number twice.