Approximate polynomial greatest common divisors and nearest singular polynomials

Karmarkar, Narendra; Lakshman, Y. N.

doi:10.1145/236869.236892

Cited by 69 publications

(61 citation statements)

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“…We found that the naive algorithm fails completely when the degree e of the input polynomials and the degree r of the GCD satisfy the following relation: 17) Other relations between the degrees work fine. While the advanced algorithm remains numerically stable no matter what the relations between the degrees are.…”

Section: Numerical Experimentsmentioning

confidence: 97%

“…That many algorithms to solve problems with exact data turn out to be numerically unstable is a growing concern in computer algebra, which has led to hybrid symbolicnumeric computation [6]. In particular, the approximate GCD problem has received a lot of research attention in recent years, see for instance [2], [8], [13], [17], [24], [37] and [38].…”

Section: Introductionmentioning

confidence: 99%

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Computing Dynamic Output Feedback Laws

Verschelde

Wang

2004

IEEE Trans. Automat. Contr.

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The pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed-loop system has a desired set of eigenvalues. Converting this problem into a question of enumerative geometry, efficient numerical homotopy algorithms to solve this problem for general Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While dynamic feedback laws offer a wider range of use, the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before. In this paper we present symbolic-numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine. We report on numerical experiments with our publicly available software and illustrate its application on various control problems from the literature. 2000 Mathematics Subject Classification. Primary 93B55. Secondary 14Q99, 65H10, 68W30, 93B27.

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Section: Numerical Experimentsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Computing Dynamic Output Feedback Laws

Verschelde

Wang

2004

IEEE Trans. Automat. Contr.

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“…In the same paper [11], Corless et al also suggest seeking the minimum distance. In 1996/1998 Karmarkar and Lakshman [52,53] formulated two numerical GCD problems: "The nearest GCD problem" and the "highest degree approximate common divisor problem" in detail, with the latter specifying requirements of backward nearness, highest degree, and minimum distance. The formulation in Definition 1.2.5 differs somewhat with Karmarkar-Lakshman's.…”

Section: The Three-strikes Principle For Removing Ill-posednessmentioning

confidence: 99%

Regularization and Matrix Computation in Numerical Polynomial Algebra

Zeng

2009

Texts and Monographs in Symbolic Computation

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“…The nearest singular polynomial [16,17,18] of f HxL is the nearest polynomial f êê ê HxL that has a double root, minimizes °f HxL -f êê ê HxL¥, and has the same degree as f HxL. A similar problem that finds the nearest polynomial with constrained roots has been studied in [19,20].…”

Section: · Nearest Singular Polynomial (Univariate Case)mentioning

confidence: 99%

Symbolic-Numeric Algebra for Polynomials

Nagasaka¹

2007

TMJ

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Symbolic-Numeric Algebra for Polynomials (SNAP) is a prototype package that includes basic functions for computing approximate algebraic properties, such as the approximate GCD of polynomials. Our aim is to show how the unified tolerance mechanism we introduce in the package works. Using this mechanism, we can carry out approximate calculations under certified tolerances. In this article, we demonstrate the functionality of the currently released package (Version 0.2), which is downloadable from wwwmain.h.kobe-u.ac.jp/~nagasaka/research/snap/index.phtml.en. ‡ Introduction Recently, there have been many results in the area of symbolic-numeric algorithms, especially for polynomials (for example, approximate GCD for univariate polynomials [1,2,3,4] and approximate factorization for bivariate polynomials [5,6,7,8]). We think those results have practical value, which should be combined and implemented into one integrated computer algebra system such as Mathematica. In fact, Maple has such a special package called SNAP (SymbolicNumeric Algorithms for Polynomials). Currently, ours is the only such package available for Mathematica.We have been developing our SNAP package for Mathematica, which is an abbreviation for Symbolic-Numeric Algebra for Polynomials. We use algebra to mean continuous capabilities of approximate operations; for example, computing an approximate GCD between an empirical polynomial and the nearest singular polynomial computed by SNAP functions of another empirical polynomial. This continuous applicability is more important for practical computations than the number of algorithms that are already implemented, especially for average users.Our aim is to provide practical implementations of SNAP functions with a unified tolerance mechanism for polynomials like Mathematica's floating-point numbers or Kako and Sasaki's effective floating-point numbers [9]. Our idea is very simple. We only have to add new data structures representing such polynomials with tolerances and basic calculation routines and SNAP functions for the structures. At this time, only simple tolerance representations (l 2 -norm, l 1 -norm, and l ¶ -norm) for coefficient vectors of polynomials and a few SNAP functions are implemented, but the package is an ongoing project (Version 1.0 should be released in March 2007).

show abstract

Approximate polynomial greatest common divisors and nearest singular polynomials

Cited by 69 publications

References 10 publications

Computing Dynamic Output Feedback Laws

Computing Dynamic Output Feedback Laws

Regularization and Matrix Computation in Numerical Polynomial Algebra

Symbolic-Numeric Algebra for Polynomials

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