Robust Approximate Zeros

Sharma, Vikrant; Du, Zilin; Yap, Chee

doi:10.1007/11561071_77

Cited by 5 publications

(9 citation statements)

References 23 publications

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“…In Section 8, we derive a bound (Theorem 8.3) on the complexity of approximating a root of a zero-dimensional system of polynomials when the computations are done in the strong bigfloat model. This is a generalization of a corresponding result [29,Thm. 5] in the univariate case, which itself is an extension of Brent's complexity bound (for algebraic roots) to the unbounded case.…”

Section: Introductionmentioning

confidence: 65%

“…Malajovich [21] developed both point estimates and complexity for approximate zeros of the fourth kind in the exact and the weak model. Sharma et al [29] have developed point estimates and complexity for approximate zeros of the second kind in the strong model. There have been no explicit point estimates for approximate zeros of the third kind, though from Lemma 2.2 it is clear that we can easily derive them from point estimates of the second kind.…”

Section: For a Setmentioning

confidence: 99%

“…In this section we generalize the algorithm in [29] to a system of integer polynomials. The algorithm will take as input a system F and a robust approximate zero Z 0 ∈ F n such that 1) α(F , Z 0 ) < 0.02, and 2) the associated zero Z * ∈ R n and Z 0 satisfy (5.4); it will construct a robust iteration sequence ( Z i ), relative to C, such that…”

Section: Robust Newton Iterationmentioning

confidence: 99%

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Robust Approximate Zeros in Banach Space

Sharma

2007

Math.comput.sci.

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We extend Smale's concept of approximate zeros of an analytic function on a Banach space to two computational models that account for errors in the computation: first, the weak model where the computations are done with a fixed precision; and second, the strong model where the computations are done with varying precision. For both models, we develop a notion of robust approximate zero and derive a corresponding robust point estimate.A useful specialization of an analytic function on a Banach space is a system of integer polynomials. Given such a zero-dimensional system, we bound the complexity of computing an absolute approximation to a root of the system using the strong model variant of Newton's method initiated from a robust approximate zero. The bound is expressed in terms of the condition number of the system and is a generalization of a well-known bound of Brent to higher dimensions. Mathematics Subject Classification (2000). 65Y20, 68Q25, 65H10, 65G99.

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Section: Introductionmentioning

confidence: 65%

Section: For a Setmentioning

confidence: 99%

Section: Robust Newton Iterationmentioning

confidence: 99%

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Robust Approximate Zeros in Banach Space

Sharma

2007

Math.comput.sci.

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“…[27]). Another approach to Smale's alpha-theory which takes computational limitations and subsequent errors into account proceeds via identification of 'robust approximate zeros', see [38,39].…”

Section: Path-following and Approximate Zerosmentioning

confidence: 99%

Globally Convergent, Iterative Path-Following for Algebraic Equations

Batra

2010

Math.Comput.Sci.

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Homotopy methods are of great importance for the solution of systems of equations. It is a major problem to ensure well-defined iterations along the homotopy path. Many investigations have considered the complexity of path-following methods depending on the unknown distance of some given path to the variety of ill-posed problems. It is shown here that there exists a construction method for safe paths for a single algebraic equation. A safe path may be effectively determined with bounded effort. Special perturbation estimates for the zeros together with convergence conditions for Newton's method in simultaneous mode allow our method to proceed on the safe path. This yields the first globally convergent, never-failing, uniformly iterative path-following algorithm. The maximum number of homotopy steps in our algorithm reaches a theoretical bound forecast by Shub and Smale i.e., the number of steps is at most quadratic in the condition number. A constructive proof of the fundamental theorem of algebra meeting demands by Gauß, Kronecker and Weierstraß is a consequence of our algorithm.

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“…When f is an elementary function, Brent [1] tells us that the running time is O(M (log(1/ε)) log(1/ε)) where M (n) is the time to multiply two n-bit numbers. One caveat is that Brent's result is "local", i.e., it is only applicable when x lies in a bounded range [20]. More global results are obtained in [6].…”

Section: Complexity Of Exact Roundingmentioning

confidence: 99%

Foundations of Exact Rounding

Yap

2009

WALCOM: Algorithms and Computation

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Abstract. Exact rounding of numbers and functions is a fundamental computational problem. This paper introduces the mathematical and computational foundations for exact rounding. We show that all the elementary functions in ISO standard (ISO/IEC 10967) for Language Independent Arithmetic can be exactly rounded, in any format, and to any precision. Moreover, a priori complexity bounds can be given for these rounding problems. Our conclusions are derived from results in transcendental number theory.

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Robust Approximate Zeros

Cited by 5 publications

References 23 publications

Robust Approximate Zeros in Banach Space

Robust Approximate Zeros in Banach Space

Globally Convergent, Iterative Path-Following for Algebraic Equations

Foundations of Exact Rounding

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