2019
DOI: 10.1007/s10208-019-09414-2
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The Real Polynomial Eigenvalue Problem is Well Conditioned on the Average

Abstract: We study the average condition number for polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with Gaussian entries are very well-conditioned on the average. 1. Polynomial Root Finding: I is the set of univariate real polynomials of degree d, O = R and V = {(f, ζ) : f (ζ) = 0}. 2. Polynomial System Solving, which we can see as the homogeneous multivariate version of Polynomial Root Finding:… Show more

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Cited by 13 publications
(17 citation statements)
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“…Smale started studying the probability that a polynomial is ill-conditioned [66]. This strategy was extended to linear algebra condition numbers [26,31,41], to systems of polynomial equations in diverse settings [42,62], to linear systems of inequalities [49], to linear and convex programming [2,68], eigenvalue and eigenvectors in the classic and other settings [7], to polynomial eigenvalue problems [6,8], and to other computational models [30], among others. As there is a substantive bibliography on this setting, we refer the reader to [28] for further details.…”
Section: The Condition Number Of Tensor Rank Decompositionmentioning
confidence: 99%
“…Smale started studying the probability that a polynomial is ill-conditioned [66]. This strategy was extended to linear algebra condition numbers [26,31,41], to systems of polynomial equations in diverse settings [42,62], to linear systems of inequalities [49], to linear and convex programming [2,68], eigenvalue and eigenvectors in the classic and other settings [7], to polynomial eigenvalue problems [6,8], and to other computational models [30], among others. As there is a substantive bibliography on this setting, we refer the reader to [28] for further details.…”
Section: The Condition Number Of Tensor Rank Decompositionmentioning
confidence: 99%
“…Since, for k = 1, we recover the well-known limit distribution of the eigenvalues of a Gaussian random matrix, within the proof we tacitly assume that k ≥ 2. Theorem 3.1 Let P n (x) be a monic n ×n complex random matrix polynomial of degree k as in (4), where the entries of each coefficient C j are i.i.d. complex random variables normally distributed with mean 0 and variance 1.…”
Section: Empirical Spectral Distribution For N × N Monic Complex Gaus...mentioning
confidence: 99%
“…can potentially be valuable to numerical analysts in the context of testing numerical methods for the solution of the PEP. Indeed, although randomly generated problems are expected not to be very challenging from the numerical point of view (by the results in [2,4]), it is common practice to use them as benchmark for minimal performance requirements; in published research papers on this subject, tests on random input are in fact often included among the numerical experiments. The analytic knowledge of the limit eigenvalue distributions that we obtain in this article can help to predict the behavior of randomly generated problems: when scrutinizing a novel algorithm, if the numerically computed eigenvalues should significantly deviate from the expectations, then this fact can raise legitimate suspicions on the accuracy of the computations.…”
Section: Introductionmentioning
confidence: 99%
“…In addition, the expression for the stochastic condition number involves a ratio of gamma functions (see Corollary 5.2 or the case r = n in Theorem 1.1 and Theorem 1.2). From (5) we get the approximation Γ (k + 1/2)/Γ (k) ≈ √ k, so that the stochastic condition number for regular polynomial eigenvalue problems satisfies…”
Section: Measure Concentration For the Directional Sensitivity Of Regmentioning
confidence: 99%