alphaCertified

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 20 articles )

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  1. Chen, Tianran; Lee, Tsung-Lin; Li, Tien-Yien: Mixed cell computation in HOM4ps (2017)
  2. Hauenstein, Jonathan D. (ed.); Sommese, Andrew J. (ed.): Foreword. What is numerical algebraic geometry? (2017)
  3. Hauenstein, Jonathan D.; Levandovskyy, Viktor: Certifying solutions to square systems of polynomial-exponential equations (2017)
  4. Hein, Nickolas; Sottile, Frank: A lifted square formulation for certifiable Schubert calculus (2017)
  5. Imbach, Rémi; Moroz, Guillaume; Pouget, Marc: A certified numerical algorithm for the topology of resultant and discriminant curves (2017)
  6. 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)
  7. Brake, Daniel A.; Hauenstein, Jonathan D.; Liddell, Alan C.: Validating the completeness of the real solution set of a system of polynomial equations (2016)
  8. Hauenstein, Jonathan D.; Hein, Nickolas; Sottile, Frank: A primal-dual formulation for certifiable computations in Schubert calculus (2016)
  9. Szanto, Agnes: Certification of approximate roots of exact polynomial systems (2016)
  10. Griffin, Zachary A.; Hauenstein, Jonathan D.: Real solutions to systems of polynomial equations and parameter continuation (2015)
  11. Ruatta, Olivier; Sciabica, Mark; Szanto, Agnes: Overdetermined Weierstrass iteration and the nearest consistent system (2015)
  12. Bates, Daniel J.; Niemerg, Matthew: Using monodromy to avoid high precision in homotopy continuation (2014)
  13. Hauenstein, Jonathan D.; Pan, Victor Y.; Szanto, Agnes: A note on global Newton iteration over archimedean and non-Archimedean fields (2014)
  14. Shen, Fei; Wu, Wenyuan; Xia, Bican: Real root isolation of polynomial equations based on hybrid computation (2014)
  15. Beltrán, Carlos; Leykin, Anton: Robust certified numerical homotopy tracking (2013)
  16. Hauenstein, Jonathan D.: Numerically computing real points on algebraic sets (2013)
  17. Hauenstein, Jonathan; He, Yang-Hui; Mehta, Dhagash: Numerical elimination and moduli space of vacua (2013)
  18. García-Puente, Luis D.; Hein, Nickolas; Hillar, Christopher; Martín del Campo, Abraham; Ruffo, James; Sottile, Frank; Teitler, Zach: The secant conjecture in the real Schubert calculus (2012)
  19. Hauenstein, Jonathan D.; Sottile, Frank: Algorithm 921: alphaCertified: certifying solutions to polynomial systems (2012)
  20. Mehta, Dhagash: Numerical polynomial homotopy continuation method and string vacua (2011)