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The Numerical Factorization of Polynomials

Abstract

Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper formulates the notion of numerical factorization based on the geometry of polynomial spaces and the stratification of factorization manifolds. Furthermore, this paper establishes the existence, uniqueness, Lipschitz continuity, condition number, and convergence of the numerical factorization to the underlying exact factorization, leading to a robust and efficient algorithm with a MATLAB implementation capable of accurate polynomial factorizations using floating point arithmetic even if the coefficients are perturbed.

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Notes

  1. http://homepages.neiu.edu/~naclab.html.

  2. http://homepages.neiu.edu/~zzeng/NumFactorTests.zip.

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Correspondence to Zhonggang Zeng.

Additional information

Communicated by Felipe Cucker.

Wenyuan Wu: This work is partially supported by grant NSFC 11471307, 11171053.

Zhonggang Zeng: Research supported in part by NSF under Grant DMS-0715127.

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Wu, W., Zeng, Z. The Numerical Factorization of Polynomials. Found Comput Math 17, 259–286 (2017). https://doi.org/10.1007/s10208-015-9289-1

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  • DOI: https://doi.org/10.1007/s10208-015-9289-1

Keywords

  • Numerical polynomial factorization
  • Ill-posed problem
  • Factorization manifold
  • Sensitivity

Mathematics Subject Classification

  • 12Y05
  • 13P05
  • 65J20
  • 65F22
  • 65H04