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**On the optimal stability of the Bernstein basis.**
*(English)*
Zbl 0853.65051

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

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### References:

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