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The Nearest Polynomial with a given zero, Revisited
, 2005
"... In his 1999 Sigsam Bulletin paper [7], H. J. Stetter gave an explicit formula for finding the nearest polynomial with a given zero. This present paper revisits the issue, correcting a minor omission from Stetter’s formula and explicitly extending the results to different polynomial bases. Experiment ..."
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Cited by 2 (2 self)
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In his 1999 Sigsam Bulletin paper [7], H. J. Stetter gave an explicit formula for finding the nearest polynomial with a given zero. This present paper revisits the issue, correcting a minor omission from Stetter’s formula and explicitly extending the results to different polynomial bases. Experiments with our implementation demonstrate that the formula may not after all, fully solve the problem, and we discuss some outstanding issues: first, that the nearest polynomial with the given zero may be identically zero (which might be surprising), and, second, that the problem of finding the nearest polynomial of the same degree with a given zero may not, in fact, have a solution. A third variant of the problem, namely to find the nearest monic polynomial (given a monic polynomial initially) with a given zero, a problem that makes sense in some polynomial bases but not others, can also be solved with Stetter’s formula, and this may be more satisfactory in some circumstances. This last can be generalized to the case where some coefficients are intrinsic and not to be changed, whereas others are empiric and may safely be changed. Of course, this minor generalization is implicit in [7]; This paper 1 simply makes it explicit.
Tropical Algebraic Geometry in Maple  a preprocessing algorithm for finding common factors to multivariate polynomials with approximate coefficients
, 2009
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Testing polynomial primality with pseudozeros
 In Proceedings of the Fifth Conference on Real Numbers and Computers
, 2003
"... 52, avenue de Villeneuve ..."
unknown title
"... quantifier elimination and also aim at practicality. This is realized by utilizing the scheme for robust control design by [1]. The organization of the rest of the paper is as follows: The idea of robust control synthesis based on SDC and special QE algorithm is explained in 52. \S 3 is devoted to o ..."
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quantifier elimination and also aim at practicality. This is realized by utilizing the scheme for robust control design by [1]. The organization of the rest of the paper is as follows: The idea of robust control synthesis based on SDC and special QE algorithm is explained in 52. \S 3 is devoted to our QE approach to robust control analysis. \S 4 provides our QE approach to various synthesis problems. We show several concrete analysis and synthesis examples demonstrating the validity of our approach in \S 5. \S 6 addresses the concluding remarks. 2Parametric approach to robust control design via QE Consider afeedback control system shown in Fig 1. $\mathrm{p}=[p_{1},p_{2}, \cdots,p_{s}] $ is the vector of uncertain real parameters in the plant G. $\mathrm{x}=[x_{1}, x_{2}, \cdots,x_{t}] $ is the vector of real parameters of controller $C $. Assume that the controller considered here is of fixed order. Figure 1: Astandard feedback system The performance of the control system can often be characterized by avector $\mathrm{a}=[a_{1}, \cdots,a\iota] $ which are functions of the plant and controller parameters $\mathrm{p}$ $H_{\infty}$norm constraints, and
General Terms: Algorithms
"... When working with empirical polynomials, it is important not to introduce unnecessary changes of basis, because that can destabilize fundamental algorithms such as evaluation and rootfinding: for more details, see e.g. [3, 4]. Moreover, in ..."
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When working with empirical polynomials, it is important not to introduce unnecessary changes of basis, because that can destabilize fundamental algorithms such as evaluation and rootfinding: for more details, see e.g. [3, 4]. Moreover, in
Formally reviewed communication A note on a nearest polynomial with a given root
"... In this paper, we consider the problem of a nearest polynomial with a given root in the complex field (the coefficients of the polynomial and the root are complex numbers). We are interested in the existence and the uniqueness of such polynomials. Then we study the problem in the real case (the coef ..."
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In this paper, we consider the problem of a nearest polynomial with a given root in the complex field (the coefficients of the polynomial and the root are complex numbers). We are interested in the existence and the uniqueness of such polynomials. Then we study the problem in the real case (the coefficients of the polynomial and the root are real numbers), and in the realcomplex case (the coefficients of the polynomial are real numbers and the root is a complex number). We derive new formulas for computing such polynomials.
Numerical Analysis: the study of algorithms
, 1998
"... for problems of continuous mathematics whilst in the introduction to [2], R. Loos defines Computer Algebra: that part of computer science which designs, analyzes, implements, and applies algebraic algorithms. Clearly these definitions are similar, and the major distinction is continuous versus algeb ..."
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for problems of continuous mathematics whilst in the introduction to [2], R. Loos defines Computer Algebra: that part of computer science which designs, analyzes, implements, and applies algebraic algorithms. Clearly these definitions are similar, and the major distinction is continuous versus algebraic. Until one introduces a notion of topology, the concepts of algebra and analysis (or continuous mathematics) are entirely independent and complementary. We can therefore expect to find much rich material in their interaction. In comparing these definitions, one really needs to appeal to examples, such as the following, to make the distinctions clearer. Computing the approximate value of a definite integral by GaussChebyshev quadrature is a numerical computation: