On the optimal stability of the Bernstein basis.

*(English)*Zbl 0853.65051Summary: 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.

##### MSC:

65G99 | Error analysis and interval analysis |

65D17 | Computer-aided design (modeling of curves and surfaces) |

##### Keywords:

optimal stability; Bernstein polynomial basis; nonnegative basis transformations; generalized Ball bases; power bases; comparisons; interval arithmetic##### Software:

CONSURF
PDF
BibTeX
XML
Cite

\textit{R. T. Farouki} and \textit{T. N. T. Goodman}, Math. Comput. 65, No. 216, 1553--1566 (1996; Zbl 0853.65051)

Full Text:
DOI

##### References:

[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. Comput. Math. 1 (1993), no. 2, 173 – 196. · Zbl 0832.41013 |

[3] | J. M. Carnicer and J. M. Peña, Total positivity and optimal bases, in Total Positivity and its Applications , Kluwer Academic Publishers, Dordrecht, 1996, pp. 133–155. · Zbl 0892.15002 |

[4] | Gerald Farin, Curves and surfaces for computer aided geometric design, 3rd ed., Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1993. A practical guide; With 1 IBM-PC floppy disk (5.25 inch; DD). · Zbl 0850.68323 |

[5] | R. T. Farouki and V. T. Rajan, On the numerical condition of polynomials in Bernstein form, Comput. Aided Geom. Design 4 (1987), no. 3, 191 – 216. · Zbl 0636.65012 |

[6] | W. Gautschi, Questions of numerical condition related to polynomials, in Studies in Numerical Analysis, MAA Studies in Mathematics 24, 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. · Zbl 0749.65009 |

[8] | T. N. T. Goodman and H. B. Said, Shape preserving properties of the generalised Ball basis, Comput. Aided Geom. Design 8 (1991), no. 2, 115 – 121. · Zbl 0729.65006 |

[9] | H. B. Said, A generalized Ball curve and its recursive algorithm, ACM Trans. Graphics 8 (1989), 360–371. · Zbl 0746.68101 |

[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. I, II, Numer. Math. 1 (1959), 150 – 180. · Zbl 0202.43701 |

[12] | J. H. Wilkinson, Rounding Errors in Algebraic Processes, Dover (reprint), New York, 1963. CMP 94:14 · Zbl 1041.65502 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.