Open Access
The Bernstein Algorithm
Jurgen Garlo
- 01 Jan 1994
TL;DR: In this article, the problem of finding an enclosure for the range of a multivariate polynomial over a rectangular region by expanding it into Bernstein polynomials is solved.
read more
Abstract: We solve the problem of finding an enclosure for the range of a multivariate polynomial over a rectangular region by expanding the given polynomial into Bernstein polynomials. Then the coecients of the expansion provide lower and upper bounds for the range and these bounds converge monotonically if the degree of the Bernstein polynomials is elevated. To obtain a faster improvement of the bounds we use subdivision and present an economical procedure for computing the bounds on subboxes. Then we apply the results to a problem of robust control, viz. checking the (Hurwitz) stability of a polynomial with coecients depending polynomially on parameters varying inside given intervals. Numerical examples are also presented.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
Approximations for minimum and min-max vehicle routing problems
E. Arkin,Refael Hassin,Asaf Levin +2 more
- 01 Apr 2006
TL;DR: For a variety of vehicle routing problems, the goal here is to minimize λ, the maximum travel distance, and the constant ratio approximation algorithms are provided and proved their NP-hardness.
165
Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion
M. Zettler,Jürgen Garloff +1 more
TL;DR: Two algorithms are presented, both rely on the expansion of a multivariate polynomial into Bernstein polynomials and the first is an improvement of the so-called Bernstein algorithm and checks the Hurwitz determinant for positivity over the parameter set.
135
Solution of Systems of Polynomial Equations by Using Bernstein Expansion
Jiirgen Garloff,Andrew P. Smith +1 more
- 01 Jan 2001
TL;DR: Methods utilising this approach indude interval computation techniques as well as methods which apply the expansion of a multivariate polynomial into Bernstein polynomials.
•Posted Content
Linear Relaxations of Polynomial Positivity for Polynomial Lyapunov Function Synthesis
TL;DR: The LP approach is shown to be as fast as the SOS programming approach, but less prone to numerical problems, especially for finding polynomial local Lyapunov functions that prove that the system is asymptotically stable over a given bounded region containing the equilibrium.
43
An efficient algorithm for range computation of polynomials using the Bernstein form
TL;DR: A novel optimization algorithm for computing the ranges of multivariate polynomials using the Bernstein polynomial approach, which incorporates four accelerating devices, and possesses many new features, such as, the generalized matrix method for Bernstein coefficient computation, a new subdivision direction selection rule and anew subdivision point selection rule.
40
References
Global optimization using interval analysis
TL;DR: This Second Edition of Global Optimization Using Interval Analysis expands and improves various aspects of its forerunner and features significant new discussions, such as those on the use of consistency methods to enhance algorithm performance.
1.8K
•Book
On global univalence theorems
T. Parthasarathy
- 01 Jan 1983
TL;DR: In this paper, the global stability of an autonomous system on the plane has been studied in the context of univalence mapping with Leontief type Jacobians in finite dimensional spaces.
150
On the Criteria for the Stability of Small Motions
TL;DR: In this paper, the authors present a brief account of these determinants and of certain other simple forms of test function. But they do not describe the exact influence of individual factors on the stability of a system.
127