Certifying solutions to a square analytic system Coauthors - - PowerPoint PPT Presentation

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Certifying solutions to a square analytic system Coauthors - - PowerPoint PPT Presentation

Jul 25, 2023 436 likes •1.33k views

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

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Certifying solutions to a square analytic system

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Coauthors

Certifying regular roots (The 44th ISSAC)

Lihong Zhi Chinese Academy of Sciences Michael Burr Clemson University Anton Leykin Georgia Tech

Certifying multiple roots (arXiv:1904.07937)

Nan Li Shenzhen University

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Certifying (regular) Solutions

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Certifying (regular) Solutions

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Certifying (regular) Solutions

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Certifying (regular) Solutions

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Certifying (regular) Solutions

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Certifying (regular) Solutions

Analytic System

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Certifying (regular) Solutions

Analytic System

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Certifying Solutions to Analytic Systems

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Certifying Solutions to Analytic Systems

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Previous Implementations

Polynomial ingredients Hauenstein and Sottile (2012) Exponential function ingredients Hauenstein and Levandovskyy (2017) Both implemented in alphaCertified

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Two Paradigms

Krawczyk method α-Theory

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Two Paradigms

Krawczyk method α-Theory

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Two Paradigms

Krawczyk method

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Two Paradigms

Krawczyk method

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Two Paradigms

Krawczyk method

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Two Paradigms

Krawczyk method

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Two Paradigms

Krawczyk method

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Two Paradigms

Krawczyk method

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Two Paradigms

Krawczyk method

Over the Real

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Two Paradigms

Krawczyk method

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Two Paradigms

α-Theory

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Two Paradigms

α-Theory

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Two Paradigms

α-Theory

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Two Paradigms

α-Theory

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Two Paradigms

Krawczyk method α-Theory

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Two Paradigms

Krawczyk method α-Theory 1) How to evaluate analytic functions at points (or over an interval)? 2) How to find the radius of convergence?

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Two Oracles

D-finite functions

D-finite functions

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Two Oracles

D-finite functions

Analytic Continuation van der Hoeven (1999) provides analytic continuation algorithm to approximate the value of a D-finite function. Majorant Series Mezzarobba and Salvy (2010) present algorithm to compute the majorant series of D-finite functions, which provides the radius

  • f convergence

Implementation numGfun(Maple), ore_algebra.analytic(SageMath)

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Two Oracles

D-finite functions

Analytic Continuation van der Hoeven (1999) provides analytic continuation algorithm to approximate the value of a D-finite function. Majorant Series Mezzarobba and Salvy (2010) present algorithm to compute the majorant series of D-finite functions, which provides the radius

  • f convergence

Implementation numGfun(Maple), ore_algebra.analytic(SageMath) We can certify a root of systems with D-finite functions

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Experiments

Optimization Problem

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Experiments

Optimization Problem

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Experiments

Optimization Problem

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Experiments

Optimization Problem

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Experiments

Optimization Problem

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Experiments

Comparison between two methods

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Experiments

Comparison between two methods

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Numerical Multiple Roots

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Numerical Multiple Roots

Cluster of (two regular) Roots

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Numerical Multiple Roots

Cluster of (two regular) Roots

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Numerical Multiple Roots

Cluster of (two regular) Roots

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Numerical Multiple Roots

Cluster of (two regular) Roots

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Numerical Multiple Roots

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Numerical Multiple Roots

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Numerical Multiple Roots

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Numerical Multiple Roots

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Numerical Multiple Roots

Multiplicity 2? 3?

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Numerical Multiple Roots

Multiplicity 2? 3?

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Separation Bound

(for multiple roots)

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Separation Bound

(for multiple roots)

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Separation Bound

(for multiple roots)

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Separation Bound

(for multiple roots)

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Previous Works

Dedieu and Shub (2001) : multiplicity 2 Hao, Jiang, Li and Zhi (2019) : dim ker F′(x∗) = 1

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Simple Multiple Root

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Simple Multiple Root

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Simple Multiple Root

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Simple Multiple Root

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Simple Multiple Root

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Simple Multiple Root

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Simple Multiple Root

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Simple Multiple Root

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Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

(**)

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

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Isolating Simple Multiple Root

(**)

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Certifying Multiple Roots

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Certifying Multiple Roots

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Certifying Multiple Roots

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Certifying Multiple Roots

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Certifying Multiple Roots

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Certifying Multiple Roots

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Certifying Multiple Roots

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Future Directions

Oracles for other analytic functions

holonomic functions (i.e., multivariate setting) majorant series (van der Hoeven 2003) D-module theory Pfaffian functions

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Future Directions

Newton iteration for multiple roots

How to define Newton iteration map NF(z) converges quadratically?

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Future Directions

Newton iteration for multiple roots

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Thanks for your attention!