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Bernstein polynomial
About: Bernstein polynomial is a research topic. Over the lifetime, 2242 publications have been published within this topic receiving 32218 citations.
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12 Sep 2002
TL;DR: Polynomials in one variable Grobner bases of zero-dimensional ideals Bernstein's theorem and fewnomials as mentioned in this paper are the primary decomposition of polynomial systems in economics and statistics.
Abstract: Polynomials in one variable Grobner bases of zero-dimensional ideals Bernstein's theorem and fewnomials Resultants Primary decomposition Polynomial systems in economics Sums of squares Polynomial systems in statistics Tropical algebraic geometry Linear partial differential equations with constant coefficients Bibliography Index.
860 citations
TL;DR: One hundred years after the introduction of the Bernstein polynomial basis, this survey surveys the historical development and current state of theory, algorithms, and applications associated with this remarkable method of representing polynomials over finite domains.
451 citations
TL;DR: In this article, the convergence of P(u, x) to f(x) as u −> °o was studied and generalized analogs of known properties of S. Bernstein's approximation polynomials in a finite interval.
Abstract: The paper studies the convergence of P(u, x) to f(x) as u —> °o . The results obtained are generalized analogs, for the interval 0< x < °°, of known properties of S. Bernstein's approximation polynomials in a finite interval. 1. With a function j(t) in the closed interval [0,1], S. Bernstein in 1912 associated the polynomials
438 citations
IBM1
TL;DR: Bernstein forms for various basic polynomial procedures are developed, and are found to be of similar complexity to their customary power forms, establishing the viability of systematic computation with the Bernstein form, avoiding the need for (numerically unstable) basis conversions.
365 citations
IBM1
TL;DR: It is shown that the condition numbers of the simple real roots of a polynomial on the unit interval are always smaller in the Bernstein basis than in the power basis, and it is proved that among a large family ofPolynomial bases characterized by certain simple properties, the Bernstein base exhibits optimal root conditioning.
335 citations