On the optimal stability of the Bernstein basis

Authors:
R. T. Farouki and T. N. T. Goodman

Journal:
Math. Comp. **65** (1996), 1553-1566

MSC (1991):
Primary 65G99; Secondary 65D17

DOI:
https://doi.org/10.1090/S0025-5718-96-00759-4

MathSciNet review:
1351201

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the Bernstein polynomial basis on a given interval is ``optimally stable,'' in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval. This result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations. We further show, by means of some low--degree examples, that the Bernstein form is not uniquely optimal in this respect. However, it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval. These ideas are illustrated by comparing the stability properties of the power, Bernstein, and generalized Ball bases.

**1.**A. A. Ball,*CONSURF part one: Introduction to conic lofting tile*, Comput. Aided Design**6**(1974), 243--249.**2.**J. M. Carnicer and J. M. Peña,*Shape preserving representations and optimality of the Bernstein basis*, Adv. Comp. Math.**1**(1993), 173--196. MR**94i:65138****3.**J. M. Carnicer and J. M. Peña,*Total positivity and optimal bases*, in Total Positivity and its Applications (M. Gasca and C. A. Micchelli, eds.), Kluwer Academic Publishers, Dordrecht, 1996, pp. 133--155.**4.**G. Farin,*Curves and Surfaces for Computer Aided Geometric Design*, Academic Press, Boston, 1993. MR**93k:65016****5.**R. T. Farouki and V. T. Rajan,*On the numerical condition of polynomials in Bernstein form*, Comput. Aided Geom. Design**4**(1987), 191--216. MR**89a:65028****6.**W. Gautschi,*Questions of numerical condition related to polynomials*, in Studies in Numerical Analysis, MAA Studies in Mathematics**24**, (G. H. Golub, ed.) 1984, 140--177. CMP**20:07****7.**T. N. T. Goodman and H. B. Said,*Properties of generalized Ball curves and surfaces*, Comput. Aided Design**23**(1991), 554--560.**8.**T. N. T. Goodman and H. B. Said,*Shape preserving properties of the generalized Ball basis*, Comput. Aided Geom. Design**8**(1991), 115--121. MR**92c:65181****9.**H. B. Said,*A generalized Ball curve and its recursive algorithm*, ACM Trans. Graphics**8**(1989), 360--371.**10.**T. V. To,*Polar Form Approach to Geometric Modeling*, Dissertation No. 92, Asian Institute of Technology, Bangkok, Thailand, 1992.**11.**J. H. Wilkinson,*The evaluation of the zeros of ill--conditioned polynomials. Parts I & II*, Numer. Math.**1**(1959), 150--166. MR**22:321****12.**J. H. Wilkinson,*Rounding Errors in Algebraic Processes*, Dover (reprint), New York, 1963. CMP**94:14**

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65G99,
65D17

Retrieve articles in all journals with MSC (1991): 65G99, 65D17

Additional Information

**R. T. Farouki**

Affiliation:
Department of Mechanical Engineering & Applied Mechanics, University of Michigan, Ann Arbor, Michigan 48109

Email:
farouki@engin.umich.edu

**T. N. T. Goodman**

Affiliation:
Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, Scotland

Email:
tgoodman@mcs.dundee.ac.uk

DOI:
https://doi.org/10.1090/S0025-5718-96-00759-4

Received by editor(s):
March 2, 1995

Received by editor(s) in revised form:
August 28, 1995

Article copyright:
© Copyright 1996
American Mathematical Society