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 50 articles )

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  1. Duff, Timothy; Hein, Nickolas; Sottile, Frank: Certification for polynomial systems via square subsystems (2022)
  2. Frohmader, Andrew; Heaton, Alexander: Epsilon local rigidity and numerical algebraic geometry (2022)
  3. Hauenstein, Jonathan D.; Manivel, Laurent; Szendrői, Balázs: On the equations defining some Hilbert schemes (2022)
  4. Brysiewicz, Taylor: Necklaces count polynomial parametric osculants (2021)
  5. Brysiewicz, Taylor; Rodriguez, Jose Israel; Sottile, Frank; Yahl, Thomas: Solving decomposable sparse systems (2021)
  6. Timme, Sascha: Mixed precision path tracking for polynomial homotopy continuation (2021)
  7. Breiding, Paul; Sturmfels, Bernd; Timme, Sascha: 3264 conics in a second (2020)
  8. Brustenga I Moncusì, Laura; Timme, Sascha; Weinstein, Madeleine: 96120: the degree of the linear orbit of a cubic surface (2020)
  9. Cheng, Jin-San; Dou, Xiaojie; Wen, Junyi: A new deflation method for verifying the isolated singular zeros of polynomial systems (2020)
  10. Do, Ngoc; Kuchment, Peter; Sottile, Frank: Generic properties of dispersion relations for discrete periodic operators (2020)
  11. Lairez, Pierre: Rigid continuation paths. I: Quasilinear average complexity for solving polynomial systems (2020)
  12. Shirinkam, Sara; Alaeddini, Adel; Gross, Elizabeth: Identifying the number of components in Gaussian mixture models using numerical algebraic geometry (2020)
  13. Sottile, Frank: General witness sets for numerical algebraic geometry (2020)
  14. Brake, Danielle A.; Hauenstein, Jonathan D.; Vinzant, Cynthia: Computing complex and real tropical curves using monodromy (2019)
  15. Burr, Michael; Lee, Kisun; Leykin, Anton: Effective certification of approximate solutions to systems of equations involving analytic functions (2019)
  16. Duff, Timothy; Hill, Cvetelina; Jensen, Anders; Lee, Kisun; Leykin, Anton; Sommars, Jeff: Solving polynomial systems via homotopy continuation and monodromy (2019)
  17. Hauenstein, Jonathan D.; Oeding, Luke; Ottaviani, Giorgio; Sommese, Andrew J.: Homotopy techniques for tensor decomposition and perfect identifiability (2019)
  18. Ayyildiz Akoglu, Tulay; Hauenstein, Jonathan D.; Szanto, Agnes: Certifying solutions to overdetermined and singular polynomial systems over (\mathbbQ) (2018)
  19. Dou , Xiaojie; Cheng , Jin-San: A heuristic method for certifying isolated zeros of polynomial systems (2018)
  20. Hauenstein, Jonathan D.; Kulkarni, Avinash; Sertöz, Emre C.; Sherman, Samantha N.: Certifying reality of projections (2018)

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