[Paper Review] Locating the eigenvalues of matrix polynomials

Dario A. Bini, Vanni Noferini+1 more|arXiv (Cornell University)|Jun 16, 2012

Matrix Theory and Algorithms15 references4 citations

TL;DR

This paper extends classical root localization techniques—Pellet's theorem and Newton polygon methods—for scalar polynomials to matrix polynomials, enabling effective initial approximation selection for the Ehrlich-Aberth iteration. By leveraging tropical algebra and eigenvalue modulus bounds, the method significantly reduces both average and maximum iterations needed for convergence in polynomial eigenvalue problems.

ABSTRACT

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by A.E. Pellet [Bulletin des Sciences Mathématiques, (2), vol 5 (1881), pp.393-395], some results of D.A. Bini [Numer. Algorithms 13:179-200, 1996] based on the Newton polygon technique, and recent results of M. Akian, S. Gaubert and M. Sharify (see in particular [LNCIS, 389, Springer p.p.291-303] and [M. Sharify, Ph.D. thesis, École Polytechnique, ParisTech, 2011]). These extensions are applied for determining effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich-Aberth method. Numerical experiments that show the computational advantage of these results are presented.

Motivation & Objective

• Address the challenge of poor convergence in polynomial eigenvalue solvers when roots have widely varying moduli.
• Improve initial approximation selection for iterative eigensolvers like Ehrlich-Aberth by leveraging root localization techniques.
• Extend scalar polynomial root bounds—Pellet’s theorem and Newton polygon analysis—to matrix polynomials.
• Develop a practical criterion for placing initial approximations that reduces computational cost in eigenvalue computation.
• Enable efficient, high-precision computation of matrix polynomial eigenvalues, analogous to the MPSolve software for scalar polynomials.

Proposed method

• Adapt Pellet’s theorem to matrix polynomials by analyzing the moduli of eigenvalues using coefficient norms and Rouché’s theorem.
• Construct the Newton polygon for matrix polynomial coefficients by plotting points (i, log||A_i||) and computing the upper convex hull.
• Use the slopes of Newton polygon segments to determine radii r_i = ||A_{k_{i-1}}|| / ||A_{k_i}||^{1/(k_i - k_{i-1})} for initial approximation placement.
• Place initial approximations uniformly on circles of radius r_i, with multiplicity m_i = k_i - k_{i-1} per radius, to match expected eigenvalue clusters.
• Apply the Ehrlich-Aberth iteration with these structured initial guesses, tracking both maximum (simul_it) and average (aver_it) iteration counts.
• Use the tropical algebra framework to interpret Newton polygon results and justify the effectiveness of radius-based clustering.

Experimental results

Research questions

• RQ1 Can Pellet’s theorem and Newton polygon techniques be generalized to matrix polynomials to bound eigenvalue moduli?
• RQ2 How can eigenvalue modulus localization improve the convergence of iterative eigensolvers like Ehrlich-Aberth?
• RQ3 What is the computational advantage of using Newton polygon-based initial approximations over uniform circular placement?
• RQ4 To what extent do these bounds remain effective for matrix polynomials with unbalanced eigenvalue moduli?
• RQ5 Can the Newton polygon method be used to design a multiprecision matrix polynomial root-finder similar to MPSolve?

Key findings

• For matrix polynomials with unbalanced eigenvalue moduli, Newton polygon-based initial approximations reduce average iterations by up to 75% compared to uniform unit circle placement.
• With m=40, the average iterations (aver_it) drop from 1466 (unit circle) to 6.1 (Newton polygon), a reduction factor of nearly 240x.
• The maximum iterations (simul_it) also decrease significantly, from 1594 to 13 for m=40, indicating faster convergence across all components.
• The method remains effective even for non-orthogonal coefficient matrices, with similar iteration reductions observed in random coefficient tests.
• The reduction in iteration count grows approximately linearly with matrix size m, suggesting scalability for large-scale problems.
• The Newton polygon method avoids placing initial approximations in regions with no eigenvalues, as confirmed by annulus-based bounds and the absence of roots in ]s_{h_j}, t_{h_j}[ intervals.