# Computation of Minkowski Values of Polynomials over Complex Sets

## 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 value**P*_{m}(*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 value**P*_{f}(*X*), where *P* is represented as the product (**z**−**r**_{1})⋅⋅⋅(**z**−**r**_{n}) of *n* linear factors – each incurring an independent choice **z**∈*X* – and **r**_{1},...,**r**_{n} are the roots of *P*(**z**); and the *Minkowski Horner value**P*_{h}(*X*), where the evaluation is performed by “nested multiplication” and incurs *n* independent values **z**∈*X*. A new algorithm for the evaluation of *P*_{h}(*X*), when 0∉*X*, is presented.

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]R.T. Farouki and J.-C.A. Chastang, Curves and surfaces in geometrical optics, in:
*Mathematical Methods in Computer Aided Geometric Design*, Vol. II, eds. T. Lyche and L.L. Schumaker (Academic Press, New York, 1992) pp. 239–260.Google Scholar - [2]R.T. Farouki and J.-C.A. Chastang, Exact equations of “simple” wavefronts, Optik 91 (1992) 109–121.Google Scholar
- [3]R.T. Farouki, W. Gu and H.P. Moon, Minkowski roots of complex sets, in:
*Geometric Modeling and Processing 2000*(IEEE Computer Soc. Press, Los Alamitos, CA, 2000) pp. 287–300.Google Scholar - [4]R.T. Farouki and C.Y. Han, Solution of elementary equations in the Minkowski geometric algebra of complex sets, Adv. Comput. Math. (2004) to appear.Google Scholar
- [5]R.T. Farouki and H.P. Moon, Minkowski geometric algebra and the stability of characteristic polynomials, in:
*Visualization and Mathematics*, Vol. 3, eds. H.-C. Hege and K. Polthier (Springer, Berlin, 2003) pp. 163–188.Google Scholar - [6]R.T. Farouki, H.P. Moon and B. Ravani, Algorithms for Minkowski products and implicitly-defined complex sets, Adv. Comput. Math. 13 (2000) 199–229.Google Scholar
- [7]R.T. Farouki, H.P. Moon and B. Ravani, Minkowski geometric algebra of complex sets, Geometriae Dedicata 85 (2001) 283–315.Google Scholar
- [8]R.T. Farouki and H. Pottmann, Exact Minkowski products of
*N*complex disks, Reliable Computing 8 (2002) 43–66.Google Scholar - [9]I. Gargantini and P. Henrici, Circular arithmetic and the determination of polynomial zeros, Numer. Math. 18 (1972) 305–320.Google Scholar
- [10]C.G. Gibson,
*Elementary Geometry of Differentiable Curves*(Cambridge Univ. Press, Cambridge, 2001).Google Scholar - [11]M. Hauenschild, Arithmetiken für komplexe Kreise, Computing 13 (1974) 299–312.Google Scholar
- [12]M. Hauenschild, Extended circular arithmetic, problems and results, in:
*Interval Mathematics 1980*, ed. K.L.E. Nickel (Academic Press, New York, 1980) pp. 367–376.Google Scholar - [13]R.E. Moore,
*Interval Analysis*(Prentice-Hall, Englewood Cliffs, NJ, 1966).Google Scholar - [14]R.E. Moore,
*Methods and Applications of Interval Analysis*(SIAM, Philadelphia, PA, 1979).Google Scholar - [15]M.S. Petković and L.D. Petković,
*Complex Interval Arithmetic and Its Applications*(Wiley-VCH, Berlin, 1998).Google Scholar - [16]J.W. Rutter,
*Geometry of Curves*(Chapman & Hall/CRC, Boca Raton, FL, 2000).Google Scholar