Numerical Algorithms

, Volume 36, Issue 1, pp 13–29

# Computation of Minkowski Values of Polynomials over Complex Sets

• Rida T. Farouki
• Chang Yong Han

## Abstract

As a generalization of Minkowski sums, products, powers, and roots of complex sets, we consider the Minkowski value of a given polynomial P over a complex set X. Given any polynomial P(z) with prescribed coefficients in the complex variable z, the Minkowski value P(X) is defined to be the set of all complex values generated by evaluating P, through a specific algorithm, in such a manner that each instance of z in this algorithm varies independently over X. The specification of a particular algorithm is necessary, since Minkowski sums and products do not obey the distributive law, and hence different algorithms yield different Minkowski value sets P(X). When P is of degree n and X is a circular disk in the complex plane we study, as canonical cases, the Minkowski monomial valuePm(X), for which the monomial terms are evaluated separately (incurring $$\frac{1}{2}$$n(n+1) independent values of z) and summed; the Minkowski factor valuePf(X), where P is represented as the product (zr1)⋅⋅⋅(zrn) of n linear factors – each incurring an independent choice zX – and r1,...,rn are the roots of P(z); and the Minkowski Horner valuePh(X), where the evaluation is performed by “nested multiplication” and incurs n independent values zX. A new algorithm for the evaluation of Ph(X), when 0∉X, is presented.

Minkowski geometric algebra complex variables polynomial value sets Horner's method families of curves envelopes

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## Copyright information

© Kluwer Academic Publishers 2004

## Authors and Affiliations

• Rida T. Farouki
• 1
• Chang Yong Han
• 1
1. 1.Department of Mechanical and Aeronautical EngineeringUniversity of CaliforniaDavisUSA