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Tangent Graeffe Iteration

Gregorio Malajovich, Jorge P. Zubelli

TL;DR

This work advances Graeffe's univariate polynomial root solver by introducing Renormalized Tangent Graeffe Iteration, a method well-suited to floating-point computing. By renormalizing coefficients, employing a Renormalized Newton Diagram, and augmenting with 1-jet perturbation techniques, the approach achieves convergence for circle-free polynomials and enables recovery of both moduli and arguments of all roots with controlled error. The authors provide convergence proofs, error bounds decaying like $2^{-2^{N-C}}$ (for relative coordinates) and $\rho^{-2^N}$ (for tangent coordinates), along with an $O(d^2)$ per-iteration cost and $O(d)$ memory footprint. Numerical results across real/complex and ill-conditioned polynomials demonstrate stability, practical performance, and parallelizability, suggesting Renormalized Tangent Graeffe as a competitive, globally convergent root-finding framework that complements local refinement methods.

Abstract

Graeffe iteration was the choice algorithm for solving univariate polynomials in the XIX-th and early XX-th century. In this paper, a new variation of Graeffe iteration is given, suitable to IEEE floating-point arithmetics of modern digital computers. We prove that under a certain generic assumption the proposed algorithm converges. We also estimate the error after N iterations and the running cost. The main ideas from which this algorithm is built are: classical Graeffe iteration and Newton Diagrams, changes of scale (renormalization), and replacement of a difference technique by a differentiation one. The algorithm was implemented successfully and a number of numerical experiments are displayed.

Tangent Graeffe Iteration

TL;DR

This work advances Graeffe's univariate polynomial root solver by introducing Renormalized Tangent Graeffe Iteration, a method well-suited to floating-point computing. By renormalizing coefficients, employing a Renormalized Newton Diagram, and augmenting with 1-jet perturbation techniques, the approach achieves convergence for circle-free polynomials and enables recovery of both moduli and arguments of all roots with controlled error. The authors provide convergence proofs, error bounds decaying like (for relative coordinates) and (for tangent coordinates), along with an per-iteration cost and memory footprint. Numerical results across real/complex and ill-conditioned polynomials demonstrate stability, practical performance, and parallelizability, suggesting Renormalized Tangent Graeffe as a competitive, globally convergent root-finding framework that complements local refinement methods.

Abstract

Graeffe iteration was the choice algorithm for solving univariate polynomials in the XIX-th and early XX-th century. In this paper, a new variation of Graeffe iteration is given, suitable to IEEE floating-point arithmetics of modern digital computers. We prove that under a certain generic assumption the proposed algorithm converges. We also estimate the error after N iterations and the running cost. The main ideas from which this algorithm is built are: classical Graeffe iteration and Newton Diagrams, changes of scale (renormalization), and replacement of a difference technique by a differentiation one. The algorithm was implemented successfully and a number of numerical experiments are displayed.

Paper Structure

This paper contains 21 sections, 11 theorems, 113 equations, 4 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Main Result
  3. What the Graeffe Iteration is; Its historical weaknesses
  4. Renormalizing Graeffe
  5. The Renormalized Graeffe Iteration
  6. The Renormalized Newton Diagram
  7. Computation of the Newton Diagram
  8. Algorithm and Main Statements
  9. Some Estimates about Symmetric Functions
  10. A Decision Criterion
  11. Proof of Proposition \ref{['prop:cvxhull']}
  12. Tangent Graeffe Iteration
  13. Perturbation Methods, Infinitesimals, 1-Jets of Polynomials
  14. The Iteration
  15. Convergence Results
  16. ...and 6 more sections

Key Result

Theorem 1

Let $f$ be a real (resp. complex) circle free degree $d$ polynomial, not vanishing at 0. Denote by $\zeta$ the vector of all the roots of $f$ with multiplicity canonically ordered as above. Then, a total of $N$ iterations of Renormalized Tangent Graeffe (Algorithm alg:solve) produces $\zeta^{(N)} The running time for each iteration is $O(d^2)$ exact arithmetic operations (including transcendent

Figures (4)

  • Figure 1: The function $r^{(N)}(i)$, for $N = 0, 1, 2$
  • Figure 2: The function $r^{(N)}(i)$ and its Convex Hull
  • Figure 3: Real pseudo-random polynomials
  • Figure 4: Complex pseudo-random polynomials

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Definition 3
  • Definition 4
  • Proposition 1: OstrowskiOSTROWSKI
  • ...and 16 more