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Polynomial Root-Finding and Algebraic Eigenvalue Problem

Victor Y. Pan

TL;DR

The paper develops near-optimal root-finders for univariate polynomials, including black-box polynomials, and extends these techniques to approximate matrix eigenvalues. It introduces fast subdivision methods built on soft exclusion/inclusion tests, notably rooted in Newton’s inverse ratio and root-lifting, with robust error-detection, randomized guarantees, and compression-based acceleration. Key contributions include Main Theorems 1–4 establishing Las Vegas bit-complexity bounds for Problems 0/0*, 1/1*, and 2/2*; a framework to extend these methods to eigenvalues and to handle black-box inputs efficiently, plus detailed guidance on precision, stability, and multi-point polynomial evaluation. The work significantly improves practical root-finding performance (especially for high-degree polynomials) and provides foundational techniques applicable to polynomial factorization, root isolation, and the initialization of functional iterations, with broad implications for numerical algebra and computational algebra systems.

Abstract

Univariate polynomial root-finding has been studied for four millennia and very intensively in the last decades. Our new near-optimal root-finders approximate all zeros of a polynomial p almost as fast as one accesses its coefficients with the precision required for the solution within a prescribed error bound. Furthermore, our root-finders can be applied to a black box polynomial, defined by an oracle (black box subroutine) for its evaluation rather than by its coefficients. Due to this feature our root-finders support approximation of the eigenvalues of a matrix in a record Las Vegas expected bit operation time and are particularly fast for a polynomial that can be evaluated fast such as the sum of a few shifted monomials or a Mandelbrot-like polynomial defined by a recurrence. Our divide and conquer algorithm of ACM STOC 1995 is the only other known near-optimal polynomial root-finder, but it extensively uses the coefficients, is quite involved, and has never been implemented, while according to extensive numerical experiments with standard test polynomials, already a slower initial implementation of our new root-finders competes with user's choice package of root-finding subroutines MPSolve and supersedes it more and more significantly as the degree of a polynomial grows large. We elaborate upon the design and analysis of our algorithms, comment on their potential heuristic acceleration, and briefly cover polynomial root-finding by means of functional iterations. Our techniques can be of independent interest.

Polynomial Root-Finding and Algebraic Eigenvalue Problem

TL;DR

The paper develops near-optimal root-finders for univariate polynomials, including black-box polynomials, and extends these techniques to approximate matrix eigenvalues. It introduces fast subdivision methods built on soft exclusion/inclusion tests, notably rooted in Newton’s inverse ratio and root-lifting, with robust error-detection, randomized guarantees, and compression-based acceleration. Key contributions include Main Theorems 1–4 establishing Las Vegas bit-complexity bounds for Problems 0/0*, 1/1*, and 2/2*; a framework to extend these methods to eigenvalues and to handle black-box inputs efficiently, plus detailed guidance on precision, stability, and multi-point polynomial evaluation. The work significantly improves practical root-finding performance (especially for high-degree polynomials) and provides foundational techniques applicable to polynomial factorization, root isolation, and the initialization of functional iterations, with broad implications for numerical algebra and computational algebra systems.

Abstract

Univariate polynomial root-finding has been studied for four millennia and very intensively in the last decades. Our new near-optimal root-finders approximate all zeros of a polynomial p almost as fast as one accesses its coefficients with the precision required for the solution within a prescribed error bound. Furthermore, our root-finders can be applied to a black box polynomial, defined by an oracle (black box subroutine) for its evaluation rather than by its coefficients. Due to this feature our root-finders support approximation of the eigenvalues of a matrix in a record Las Vegas expected bit operation time and are particularly fast for a polynomial that can be evaluated fast such as the sum of a few shifted monomials or a Mandelbrot-like polynomial defined by a recurrence. Our divide and conquer algorithm of ACM STOC 1995 is the only other known near-optimal polynomial root-finder, but it extensively uses the coefficients, is quite involved, and has never been implemented, while according to extensive numerical experiments with standard test polynomials, already a slower initial implementation of our new root-finders competes with user's choice package of root-finding subroutines MPSolve and supersedes it more and more significantly as the degree of a polynomial grows large. We elaborate upon the design and analysis of our algorithms, comment on their potential heuristic acceleration, and briefly cover polynomial root-finding by means of functional iterations. Our techniques can be of independent interest.

Paper Structure

This paper contains 108 sections, 85 theorems, 139 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Keywords:
  2. 2020 Math. Subject Classification:
  3. State of the Art and our Main Results
  4. Polynomial Root-finding: Problems, Impact, and Earlier Studies
  5. Reduction to the case of the unit disc
  6. Black box root-finders and approximation of matrix eigenvalues
  7. More on Black Box Polynomial Root-finders
  8. Why we should use them
  9. Progress at FOCS 2016 and its limitations
  10. Our complexity estimates
  11. Fast subdivision root-finding: overview and soft e/i tests
  12. Subdivision root-finders and reduction to e/i tests
  13. Subdivision iterations with soft e/i tests
  14. Fast soft e/i tests
  15. Background: Newton's Inverse Ratio
  16. ...and 93 more sections

Key Result

Proposition 1.1

The reciprocal of an approximation within $\epsilon$ to a zero of $p_{\rm rev}(x)$ in the disc $D(0,1)$ approximates the associated zero of $p(x)$ lying outside that disc within a relative error at most $\epsilon$.

Figures (8)

  • Figure 1:
  • Figure 2: Roots (asterisks) define compact components. A subdivision step halves their diameters and accordingly strengthens isolation of their minimal covering squares and discs.
  • Figure 3: Five blue marked centers of suspect squares converge to a green-marked root cluster with linear rate; two red marked Schröder's iterates of (\ref{['eqSCHR']}) converge with quadratic rate.
  • Figure 5: Roots (asterisks) define compact components. A subdivision step halves their diameters and at least doubles isolation of their minimal covering squares and discs.
  • Figure 6: Two discs superscribing two non-adjacent suspect squares are not minimal but do not overlap.
  • ...and 3 more figures

Theorems & Definitions (154)

  • Example 1.1
  • Example 1.2
  • Proposition 1.1
  • Remark 1.1
  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof
  • ...and 144 more