Faster Numerical Univariate Polynomial Root-Finding by Means of Subdivision Iterations

Luan, Qi; Pan, Victor Y.; Kim, Wongeun; Zaderman, Vitaly

doi:10.1007/978-3-030-60026-6_25

Cited by 5 publications

(4 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The subdivision root-finders of Weyl 1924, Henrici 1974, Renegar 1987, rely on ET, RC and root radii sub-algorithms and heavily use the coefficients of p. Design and analysis of subdivision root-finders for a black box p have been continuing since 2018 in [15] (now over 150 pages), followed by [5,6,9,13,14] and this paper. This relies on the novel idea and techniques of compression of a disc and on novel ET, RC and root radii sub-algorithms.…”

Section: Related Workmentioning

confidence: 99%

“…Alternative derivation and analysis of subdivision in [15] (yielding a little stronger results but presently not included) relies on Schröder's iterations, extended from [11]. The algorithms are analyzed in [9,13,14,15], under the model for black box polynomial root-finding of [8]. [5,6] complement this study with some estimates for computational precision and Boolean complexity.…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Accelerated Subdivision for Clustering Roots of Polynomials given by Evaluation Oracles

Imbach¹,

Pan²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

In our quest for the design, the analysis and the implementation of a subdivision algorithm for finding the complex roots of univariate polynomials given by oracles for their evaluation, we present subalgorithms allowing substantial acceleration of subdivision for complex roots clustering for such polynomials. We rely on Cauchy sums which approximate power sums of the roots in a fixed complex disc and can be computed in a small number of evaluations -polylogarithmic in the degree. We describe root exclusion, root counting, root radius approximation and a procedure for contracting a disc towards the cluster of root it contains, called ε-compression. To demonstrate the efficiency of our algorithms, we combine them in a prototype root clustering algorithm. For computing clusters of roots of polynomials that can be evaluated fast, our implementation competes advantageously with user's choice for root finding, MPsolve.

show abstract

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Accelerated Subdivision for Clustering Roots of Polynomials given by Evaluation Oracles

Imbach¹,

Pan²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Design and analysis of subdivision root-finders for a black box polynomial began in 2018 in preprint [22], extended to 139 pages by August 2022 and partly covered in [20,5,21,11,6,7], and in this and upcoming papers. That study has extended the NR cost estimates of the pioneering work [12] to a much more general input class and removed its restriction on the output.…”

mentioning

confidence: 99%

Fast Approximation of Polynomial Zeros and Matrix Eigenvalues

Pan¹,

Go²,

Luan³

et al. 2023

Preprint

View full text Add to dashboard Cite

Given a black box (oracle) for the evaluation of a univariate polynomial p(x) of a degree d, we seek its zeros, that is, the roots of the equation p(x) = 0. At FOCS 2016 Louis and Vempala approximated a largest zero of such a real-rooted polynomial within 1/2 b , by performing at NR cost of the evaluation of Newton's ratio p(x) p ′ (x) at O(b log(d)) points x. They readily extended this root-finder to record fast approximation of a largest eigenvalue of a symmetric matrix under the Boolean complexity model. We apply distinct approach and techniques to obtain more general results at the same computational cost. Let a square on the complex plane be reasonably well isolated from the external roots of p(x) = 0 and let it contain at most m roots counted with multiplicities, for a fixed m ≤ d. Then we approximate all internal roots at NR cost O(m 3 b log(d)) and extend this to approximation of all eigenvalues of a matrix in such a square at record Boolean cost. For any fixed pair of a real γ ≥ 1 and an integer v ≥ 1, our alternative Las Vegas randomized root-finder runs at NR cost in O(m 2 γvb log(d)), and outputs errors with a probability at most 1/γ v . We prove that O(log(d)) bit-precision is sufficient in our root-finders provided that oracle evaluates p(x) with relative error of order log(d) at points x sufficiently isolated from the zeros of p(x). For a constant m we match the cost bounds of Louis and Vempala for approximation of roots and eigenvalues but relax the restrictions on input and output. Our use of Cauchy integrals and randomization are pioneering and nontrivial as well as our fast approximation of the root radii (that is, the distances to the roots) from any fixed complex point.

show abstract

“…Root-finding for a black box polynomial with the goal of minimizing the number of evaluations of the ratio p ′ (x) p(x) or p(x) p ′ (x) has been a well-known research subject for years but remained in stalemate since the paper[15] of 2016, which covered the history and the State of the Art of that time. The papers[30],[16], and[31] have reported new significant progress. Moreover,[31] is a comprehensive study of black box polynomial root-finders, which covers subdivision and various functional iterations: Newton's, Schröder's, Weierstrass's, aka Durand -Kerner's, and Ehrlich's, aka Aberth's.…”

mentioning

confidence: 99%

New Progress in Classic Area: Polynomial Root-squaring and Root-finding

Pan¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The DLG root-squaring iterations, due to Dandelin 1826 and rediscovered by Lobachevsky 1834 and Gräffe 1837, have been the main approach to root-finding for a univariate polynomial p(x) in the 19th century and beyond, but not so nowadays because these iterations are prone to severe numerical stability problems. Trying to avoid these problems we have found simple but novel reduction of the iterations applied for Newton's inverse ratio −p ′ (x)/p(x) to approximation of the power sums of the zeros of p(x) and its reverse polynomial. The resulting polynomial root-finders can be devised and performed independently of DLG iterations -based on Newton's identities or Cauchy integrals. In the former case the computation involve a set of leading or tailing coefficients of an input polynomial. In the latter case we must scale the variable and increase the arithmetic computational cost to ensure numerical stability. Nevertheless the cost is still manageable, at least for fast root-refinement, and the algorithms can be applied to a black box polynomial p(x) -given by a black box for the evaluation of the ratio p ′ (x) p(x) rather than by its coefficients. This enables important computational benefits, including (i) efficient recursive as well as concurrent approximation of a set of zeros of p(x) or even all of its zeros, (ii) acceleration where an input polynomial can be evaluated fast, and (iii) extension to approximation of the eigenvalues of a matrix or a polynomial matrix, being efficient if the matrix can be inverted fast, e.g., is data sparse. We also recall our recent fast algorithms for approximation of the root radii, that is, the distances to the roots from the origin or any complex value to the zeros of p(x), and propose to apply it for fast black box initialization of polynomial rootfinding by means of functional iterations such as Newton's, Ehrlich's, and Weierstrass's.

show abstract

Faster Numerical Univariate Polynomial Root-Finding by Means of Subdivision Iterations

Cited by 5 publications

References 18 publications

Accelerated Subdivision for Clustering Roots of Polynomials given by Evaluation Oracles

Accelerated Subdivision for Clustering Roots of Polynomials given by Evaluation Oracles

Fast Approximation of Polynomial Zeros and Matrix Eigenvalues

New Progress in Classic Area: Polynomial Root-squaring and Root-finding

Contact Info

Product

Resources

About