Algorithm 921: alphaCertified: Certifying solutions to polynomial systems. Smale’s α-theory uses estimates related to the convergence of Newton’s method to certify that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements algorithms based on α-theory to certify solutions of polynomial systems using both exact rational arithmetic and arbitrary precision floating point arithmetic. It also implements algorithms that certify whether a given point corresponds to a real solution, and algorithms to heuristically validate solutions to overdetermined systems. Examples are presented to demonstrate the algorithms.

This software is also peer reviewed by journal TOMS.

References in zbMATH (referenced in 30 articles )

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  1. Ayyildiz Akoglu, Tulay; Hauenstein, Jonathan D.; Szanto, Agnes: Certifying solutions to overdetermined and singular polynomial systems over (\mathbbQ) (2018)
  2. Dou , Xiaojie; Cheng , Jin-San: A heuristic method for certifying isolated zeros of polynomial systems (2018)
  3. Hauenstein, Jonathan D.; Kulkarni, Avinash; Sertöz, Emre C.; Sherman, Samantha N.: Certifying reality of projections (2018)
  4. Cheng, Jin-San; Dou, Xiaojie: Certifying simple zeros of over-determined polynomial systems (2017)
  5. Chen, Tianran; Lee, Tsung-Lin; Li, Tien-Yien: Mixed cell computation in HOM4ps (2017)
  6. Hauenstein, Jonathan D. (ed.); Sommese, Andrew J. (ed.): Foreword. What is numerical algebraic geometry? (2017)
  7. Hauenstein, Jonathan D.; Levandovskyy, Viktor: Certifying solutions to square systems of polynomial-exponential equations (2017)
  8. Hein, Nickolas; Sottile, Frank: A lifted square formulation for certifiable Schubert calculus (2017)
  9. Imbach, Rémi; Moroz, Guillaume; Pouget, Marc: A certified numerical algorithm for the topology of resultant and discriminant curves (2017)
  10. Li, Zhe; Wan, Baocheng; Zhang, Shugong: The convergence conditions of interval Newton’s method based on point estimates (2017)
  11. Zhi, Lihong: Computing multiple zeros of polynomial systems: case of breadth one (invited talk) (2017)
  12. Bates, Daniel J.; Hauenstein, Jonathan D.; Niemerg, Matthew E.; Sottile, Frank: Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials (2016)
  13. Brake, Daniel A.; Hauenstein, Jonathan D.; Liddell, Alan C.: Validating the completeness of the real solution set of a system of polynomial equations (2016)
  14. Hauenstein, Jonathan D.; Hein, Nickolas; Sottile, Frank: A primal-dual formulation for certifiable computations in Schubert calculus (2016)
  15. Martín del Campo, Abraham; Sottile, Frank: Experimentation in the Schubert calculus (2016)
  16. Szanto, Agnes: Certification of approximate roots of exact polynomial systems (2016)
  17. Timothy Duff, Cvetelina Hill, Anders Jensen, Kisun Lee, Anton Leykin, Jeff Sommars: Solving polynomial systems via homotopy continuation and monodromy (2016) arXiv
  18. Griffin, Zachary A.; Hauenstein, Jonathan D.: Real solutions to systems of polynomial equations and parameter continuation (2015)
  19. Ruatta, Olivier; Sciabica, Mark; Szanto, Agnes: Overdetermined Weierstrass iteration and the nearest consistent system (2015)
  20. Bates, Daniel J.; Decker, Wolfram; Hauenstein, Jonathan D.; Peterson, Chris; Pfister, Gerhard; Schreyer, Frank-Olaf; Sommese, Andrew J.; Wampler, Charles W.: Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety (2014)

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