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 37 articles )

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  1. Breiding, Paul; Sturmfels, Bernd; Timme, Sascha: 3264 conics in a second (2020)
  2. Cheng, Jin-San; Dou, Xiaojie; Wen, Junyi: A new deflation method for verifying the isolated singular zeros of polynomial systems (2020)
  3. Lairez, Pierre: Rigid continuation paths. I: Quasilinear average complexity for solving polynomial systems (2020)
  4. Brake, Danielle A.; Hauenstein, Jonathan D.; Vinzant, Cynthia: Computing complex and real tropical curves using monodromy (2019)
  5. Hauenstein, Jonathan D.; Oeding, Luke; Ottaviani, Giorgio; Sommese, Andrew J.: Homotopy techniques for tensor decomposition and perfect identifiability (2019)
  6. Ayyildiz Akoglu, Tulay; Hauenstein, Jonathan D.; Szanto, Agnes: Certifying solutions to overdetermined and singular polynomial systems over (\mathbbQ) (2018)
  7. Dou , Xiaojie; Cheng , Jin-San: A heuristic method for certifying isolated zeros of polynomial systems (2018)
  8. Hauenstein, Jonathan D.; Kulkarni, Avinash; Sertöz, Emre C.; Sherman, Samantha N.: Certifying reality of projections (2018)
  9. Hauenstein, Jonathan D.; Regan, Margaret H.: Adaptive strategies for solving parameterized systems using homotopy continuation (2018)
  10. Cheng, Jin-San; Dou, Xiaojie: Certifying simple zeros of over-determined polynomial systems (2017)
  11. Chen, Tianran; Lee, Tsung-Lin; Li, Tien-Yien: Mixed cell computation in HOM4ps (2017)
  12. Hauenstein, Jonathan D. (ed.); Sommese, Andrew J. (ed.): Foreword. What is numerical algebraic geometry? (2017)
  13. Hauenstein, Jonathan D.; Levandovskyy, Viktor: Certifying solutions to square systems of polynomial-exponential equations (2017)
  14. Hein, Nickolas; Sottile, Frank: A lifted square formulation for certifiable Schubert calculus (2017)
  15. Imbach, Rémi; Moroz, Guillaume; Pouget, Marc: A certified numerical algorithm for the topology of resultant and discriminant curves (2017)
  16. Li, Zhe; Wan, Baocheng; Zhang, Shugong: The convergence conditions of interval Newton’s method based on point estimates (2017)
  17. Zhi, Lihong: Computing multiple zeros of polynomial systems: case of breadth one (invited talk) (2017)
  18. 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)
  19. Brake, Daniel A.; Hauenstein, Jonathan D.; Liddell, Alan C.: Validating the completeness of the real solution set of a system of polynomial equations (2016)
  20. Hauenstein, Jonathan D.; Hein, Nickolas; Sottile, Frank: A primal-dual formulation for certifiable computations in Schubert calculus (2016)

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