Certifying solutions to a square analytic system
Coauthors Certifying regular roots (The 44 th ISSAC) Michael Burr Anton Leykin Clemson University Georgia Tech Certifying multiple roots (arXiv:1904.07937) Nan Li Lihong Zhi Shenzhen University Chinese Academy of Sciences
Certifying (regular) Solutions
Certifying (regular) Solutions
Certifying (regular) Solutions
Certifying (regular) Solutions
Certifying (regular) Solutions
Certifying (regular) Analytic System Solutions
Certifying (regular) Analytic System Solutions
Certifying Solutions to Analytic Systems
Certifying Solutions to Analytic Systems
Previous Polynomial ingredients Implementations Hauenstein and Sottile (2012) Exponential function ingredients Hauenstein and Levandovskyy (2017) Both implemented in alphaCertified
Two Krawczyk method α -Theory Paradigms
Two Krawczyk method α -Theory Paradigms
Two Krawczyk method Paradigms
Two Krawczyk method Paradigms
Two Krawczyk method Paradigms
Two Krawczyk method Paradigms
Two Krawczyk method Paradigms
Two Krawczyk method Paradigms
Two Krawczyk method Over the Real Paradigms
Two Krawczyk method Paradigms
Two α -Theory Paradigms
Two α -Theory Paradigms
Two α -Theory Paradigms
Two α -Theory Paradigms
Two Krawczyk method α -Theory Paradigms
Two Krawczyk method α -Theory Paradigms 1) How to evaluate analytic functions at points (or over an interval)? 2) How to find the radius of convergence ?
Two D-finite functions Oracles D-finite functions
Two Analytic Continuation Majorant Series Mezzarobba and Salvy Oracles van der Hoeven (1999) (2010) present algorithm provides analytic D-finite functions to compute the majorant continuation algorithm to series of D-finite functions, approximate the value of which provides the radius a D-finite function. of convergence Implementation numGfun(Maple), ore_algebra.analytic(SageMath)
Two Analytic Continuation Majorant Series Mezzarobba and Salvy Oracles van der Hoeven (1999) (2010) present algorithm provides analytic D-finite functions to compute the majorant continuation algorithm to series of D-finite functions, approximate the value of which provides the radius a D-finite function. of convergence Implementation numGfun(Maple), ore_algebra.analytic(SageMath) We can certify a root of systems with D-finite functions
Experiments Optimization Problem
Experiments Optimization Problem
Experiments Optimization Problem
Experiments Optimization Problem
Experiments Optimization Problem
Experiments Comparison between two methods
Experiments Comparison between two methods
Numerical Multiple Roots
Numerical Multiple Roots Cluster of (two regular) Roots
Numerical Multiple Roots Cluster of (two regular) Roots
Numerical Multiple Roots Cluster of (two regular) Roots
Numerical Multiple Roots Cluster of (two regular) Roots
Numerical Multiple Roots
Numerical Multiple Roots
Numerical Multiple Roots
Numerical Multiple Roots
Numerical Multiple Roots Multiplicity 2? 3?
Numerical Multiple Roots Multiplicity 2? 3?
Separation Bound (for multiple roots)
Separation Bound (for multiple roots)
Separation Bound (for multiple roots)
Separation Bound (for multiple roots)
Previous Dedieu and Shub (2001) : multiplicity 2 Works Hao, Jiang, Li and Zhi (2019) : dim ker F′(x ∗ ) = 1
Simple Multiple Root
Simple Multiple Root
Simple Multiple Root
Simple Multiple Root
Simple Multiple Root
Simple Multiple Root
Simple Multiple Root
Simple Multiple Root
Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root (**)
Isolating Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root
Isolating Simple Multiple Root (**)
Certifying Multiple Roots
Certifying Multiple Roots
Certifying Multiple Roots
Certifying Multiple Roots
Certifying Multiple Roots
Certifying Multiple Roots
Certifying Multiple Roots
Future Oracles for other analytic functions Directions holonomic functions (i.e., multivariate setting) majorant series (van der Hoeven 2003) D- module theory Pfaffian functions
Future Newton iteration for multiple roots Directions How to define Newton iteration map NF(z) converges quadratically?
Future Newton iteration for multiple roots Directions
Thanks for your attention!
Recommend
More recommend
