The Real Polynomial Eigenvalue Problem is Well Conditioned on the Average

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-019-09414-2 The Real Polynomial Eigenvalue Problem is Well Conditioned on the Average Carlos Beltrán1 · Khazhgali Kozhasov2,3,4 Received: 20 February 2018 / Revised: 10 January 2019 / Accepted: 29 January 2019 © The Author(s) 2019 Abstract We study the average condition number for polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average. Keywords Condition number · Polynomial eigenvalue problem · Random matrices Mathematics Subject Classification 14Q20 · 15A18 · 15A22 · 15B52 · 65F15 Introduction Following the ideas in [3,7,19], we note that many different numerical problems can be described within the following simple general framework. We consider a space of inputs and a space of outputs denoted by I and O, respectively, and some equation of Communicated by Alan Edelman. Carlos Beltrán was supported by the Spanish “Ministerio de Economía y Competitividad” under Projects MTM2017-83816-P and MTM2017-90682-REDT (Red ALAMA), as well as by the Banco Santander and Universidad de Cantabria under Project 21.SI01.64658. B Khazhgali Kozhasov Carlos Beltrán 1 Universidad de Cantabria, Av. de los Castros, s/n 39005, Santander, Spain 2 SISSA, Via Bonomea 265, 34136 Trieste, Italy 3 Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany 4 Present Address: TU Braunschweig, Universitätsplatz 2, 38106 Braunschweig, Germany 123 Foundations of Computational Mathematics the form ev(i, o) = 0 stating when an output is a solution for a given input. Both I and O, and the solution variety V = {(i, o) ∈ I × O : o is an output to i} = {(i, o) ∈ I × O : ev(i, o) = 0} are frequently real algebraic or just semialgebraic sets. The numerical problem to be solved can then be written as “given i ∈ I, find o ∈ O such that (i, o) ∈ V,” or “find all o ∈ O such that (i, o) ∈ V.” One can have in mind the following examples: 1. Polynomial Root Finding: I is the set of univariate real polynomials of degree d, O = R and V = {( f , ζ ) : f (ζ ) = 0}. If we look for, say, the largest real zero, then the solution variety V = {( f , ζ ) : f (ζ ) = 0, η > ζ : f (η) = 0} is semialgebraic. 2. Polynomial System Solving, which we can see as the homogeneous multivariate version of Polynomial Root Finding: I is the projective space of (dense or structured) systems of n real homogeneous polynomials of degrees d1 , . . . , dn in variables x0 , . . . , xn , O = RPn is the real projective space of dimension n and V = {( f , ζ ) : f (ζ ) = 0}. 3. EigenValue Problem: I = Rn×n , O = R and V = {(A, λ) : det(A − λ Id) = 0}. 4. (Homogeneous) Polynomial EigenValue Problem (in the sequel called PEVP): I is the set of tuples of d + 1 real n × n matrices A = (A0 , . . . , Ad ), O = RP1 and V = {(A, [α : β]) : P(A, α, β) = det(α 0 β d A0 + α 1 β d−1 A1 + · · · + α d β 0 Ad ) = 0}. One can force some of the matrices to be symmetric and/or positive definite (which leads to a semialgebraic solution variety), a particularly important case in applications, or consider other structured problems, see [10,13,18,21]. In the cases d = 1 and d = 2 polynomial eigenvalues are often referred to as generalized eigenvalues and quadratic eigenvalues, respectively. If n = 1, we recover the homogeneous version of 1. In this paper, we prove a general theorem computing exactly the expected value of the condition number in a wide collection of problems, including problem 4 above. We start by recalling the general geometric definition of the condition number, which is usually thought of as “a measure of the sensitivity of the solution o under an infinitesimal perturbation of the input i.” A Finsler structure on a differentiable manifold M is a smooth field of norms  ·  p : T p M → R, p ∈ M on M (see [3, p. 223] for more details). In particular, a Riemannian structure ·, · on M defines  a Finsler structure on it by  ṗ p =  ṗ, ṗ p , p ∈ M, ṗ ∈ T p M. Definition 1 (Condition number in the algebraic setting) Let I, O and V ⊆ I × O be real algebraic varieties such that the smooth loci of I, O are endowed with Finsler structures and let (i, o) ∈ V be a smooth point of V such that i ∈ I, o ∈ O are smooth points of I and O, respectively. Moreover, assume that D(i,o) p1 : T(i,o) V → Ti I is invertible. Then, the condition number μ(i, o) of (i, o) ∈ V is defined as     μ(i, o) =  D(i,o) p2 ◦ D(i,o) p1−1  , op 123 Foundations of Computational Mathematics where p1 : V → I, p2 : V → O are the projections and  · op is the operator norm. For points (i, o) ∈ V not satisfying the above assumptions, the condition number is set to ∞. See [7, Sec. 14.1] for more on this geometric approach to the condition number. Remark 1 Definition 1 is intrinsic in I, i.e., changing I to some subvariety I ⊂ I leads (in general) to different, smaller, value of the condition number, since perturbations of the input are only allowed in the direction of the tangent space to the input set. Note also that the condition number depends on choices of Finsler structures on I and O. Example 1 The classical Turing’s condition number μ(A) = Aop A−1 op for matrix inversion corresponds to the following setting: – O = I = M(n, R) is the set of n × n real matrices endowed with the Finsler structure associated to relative errors in operator norm:  Ȧ A =  Ȧop /Aop . – V = {(A, B) : AB = Id} = {(A, B) : B = A−1 }. In the PEVP, the input space I is endowed with the following Riemannian structure:  Ȧ, Ḃ A = (( Ȧ0 , Ḃ0 ) + · · · + ( Ȧd , Ḃd ))/((A0 , A0 ) + · · · + (Ad , Ad )), where (·, ·) is the Frobenius inner product (A, B) = trace(B t A), A = (A0 , . . . , Ad ) and Ȧ = ( Ȧ0 , . . . , Ȧd ), Ḃ = ( Ḃ0 , . . . , Ḃd ) ∈ T A I. The output space O = RP1 possesses the standard metric, and the solution variety V = {(A, [α : β]) : P(A, α, β) = 0} is endowed with the induced product Riemannian structure. An explicit formula for the condition number for the Homogeneous PEVP was derived in [10, Th. 4.2] (we write here the relative condition number version): 1/2  d  r  2k 2d−2k A, (1) α β μ(A, (α, β)) = |t v| k=0 where A = (A0 , . . . , Ad ), (α, β) ∈ R2 is a (representative of a) polynomial eigenvalue of A, r and  are the corresponding right and left eigenvectors and v=β ∂ ∂ P(A, α, β)r − α P(A, α, β)r . ∂α ∂β A given tuple A can have up to nd real isolated polynomial eigenvalues. We define the condition number of A simply as the sum of the condition numbers over all these PEVs:  μ(A) = μ(A, (α, β)). [α:β]∈RP1 is a PEV of A (If A = (A0 , . . . , Ad ) has infinitely many polynomial eigenvalues, we have μ(A) = ∞). The most important result in this paper is a very general theorem which is designed to provide exact formulas for the expected value (...truncated)