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Validated numerics for algebraic path tracking

Alexandre Guillemot, Pierre Lairez

TL;DR

Using validated numerical methods, interval arithmetic and Taylor models, a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter is proposed and a Rust implementation shows tremendous improvement over existing software for certified path tracking.

Abstract

Using validated numerical methods, interval arithmetic and Taylor models, we propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. We provide a Rust implementation which shows tremendous improvement over existing software for certified path tracking.

Validated numerics for algebraic path tracking

TL;DR

Using validated numerical methods, interval arithmetic and Taylor models, a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter is proposed and a Rust implementation shows tremendous improvement over existing software for certified path tracking.

Abstract

Using validated numerical methods, interval arithmetic and Taylor models, we propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. We provide a Rust implementation which shows tremendous improvement over existing software for certified path tracking.
Paper Structure (23 sections, 10 theorems, 22 equations, 1 figure, 1 table, 4 algorithms)

This paper contains 23 sections, 10 theorems, 22 equations, 1 figure, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Moore's criterion
  3. Data structures
  4. Arithmetic circuits
  5. Intervals
  6. Interval extensions, adaptive precision
  7. Moore boxes
  8. Refinement of Moore boxes
  9. Algorithm
  10. Analysis
  11. Path tracking
  12. Setting
  13. Algorithm
  14. Analysis
  15. Predictors
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $f : V \to V$ be a continuously differentiable map and let $\rho \in (0,1)$. Let $x\in V$, $r > 0$, and let $A: V\to V$ be a linear map. Assume that for any $u, v\in rB$, Then there is a unique $\zeta \in x + rB$ such that $f(\zeta) = 0$. Moreover, and for any $y \in x + rB$,

Figures (1)

  • Figure 1: Number of iterations performed by algpath (this work) and HomotopyContinuation.jl (noncertified path tracking) in four path tracking problems. We observe that algpath performs typically no more than 5 times more iterations than a state-of-the-art noncertified numerical solver. The ratio is close to 2 on well-conditioned examples (green,0.7804;blue,0.9098] plot[mark=*] coordinates (0,0); and ) but there is much more variability on poor conditioning (, and ).

Theorems & Definitions (12)

  • Theorem 2.1
  • Proposition 3.1
  • Lemma 3.2: Correctness of Algorithm \ref{['algo:M']}
  • Remark 4.1
  • Remark 4.2
  • Theorem 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6
  • Theorem 5.1
  • ...and 2 more