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A priori bounds for certified Krawczyk homotopy tracking

Kisun Lee

TL;DR

The paper provides the first complexity analysis for Krawczyk-based certified homotopy tracking by deriving explicit a priori stepsize bounds under affine linear parameter homotopies and relating the total iteration count to a weighted path length. It leverages Smale's alpha theory to bound certification radii and demonstrates substantial practical gains through a proof-of-concept implementation that reduces interval arithmetic overhead. The results enable more predictable, efficient certified path tracking and establish a framework for comparing path-tracking methods with subdivision-based approaches. The work also points to extensions to higher-order predictors and broader applicability in certified algebraic geometry algorithms.

Abstract

We establish the first complexity analysis for Krawczyk-based certified homotopy tracking. It consists of explicit a priori stepsize bounds ensuring the success of the Krawczyk test, and an iteration count bound proportional to the weighted length of the solution path. Our a priori bounds reduce the overhead of interval arithmetic, resulting in fewer iterations than previous methods. Experiments using a proof-of-concept implementation validate the results.

A priori bounds for certified Krawczyk homotopy tracking

TL;DR

The paper provides the first complexity analysis for Krawczyk-based certified homotopy tracking by deriving explicit a priori stepsize bounds under affine linear parameter homotopies and relating the total iteration count to a weighted path length. It leverages Smale's alpha theory to bound certification radii and demonstrates substantial practical gains through a proof-of-concept implementation that reduces interval arithmetic overhead. The results enable more predictable, efficient certified path tracking and establish a framework for comparing path-tracking methods with subdivision-based approaches. The work also points to extensions to higher-order predictors and broader applicability in certified algebraic geometry algorithms.

Abstract

We establish the first complexity analysis for Krawczyk-based certified homotopy tracking. It consists of explicit a priori stepsize bounds ensuring the success of the Krawczyk test, and an iteration count bound proportional to the weighted length of the solution path. Our a priori bounds reduce the overhead of interval arithmetic, resulting in fewer iterations than previous methods. Experiments using a proof-of-concept implementation validate the results.

Paper Structure

This paper contains 15 sections, 5 theorems, 39 equations, 1 figure, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Interval arithmetic
  4. Krawczyk method
  5. Krawczyk homotopy
  6. A priori stepsize bounds
  7. Affine linear parameter homotopies
  8. Derivation
  9. Complexity analysis
  10. Iteration count bound
  11. Certification radius bounds via Smale's alpha theory
  12. Experiments
  13. Benchmark examples
  14. Univariate examples: theory validation via exact computation
  15. Computational efficiency

Key Result

Theorem 1

If $K(F,x,r,Y) \subset r\rho B$ for some $r > 0$ and invertible matrix $Y \in \mathbb{C}^{n\times n}$, then there exists a unique solution $x^\star$ to $F=0$ in $x + rB$. Moreover, $\|x-x^\star\|\leq r\rho$ and the quasi-Newton iteration $x-YF(x)$ is $\rho$-Lipschitz continuous.

Figures (1)

  • Figure 1: Stepsize distributions (log scale) for a single solution path from each benchmark system. The boxes show interquartile ranges with the median marked by the vertical lines. Whiskers extend to the minimum and maximum values.

Theorems & Definitions (9)

  • Theorem 1: see burr2025certifiedguillemot2024validatedkrawczyk1969newton
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Proposition 5
  • proof