Tropical Roots as Approximations to Eigenvalues of Matrix Polynomials

Abstract

Abstract. The tropical roots of t × p(x) = max 0≤i≤ A i x i are points at which the maximum is attained for at least two values of i for some x. These roots, which can be computed in only O( ) operations, can be good approximations to the moduli of the eigenvalues of the matrix polynomial P (λ) = i=0 λ i A i , in particular when the norms of the matrices A i vary widely. Our aim is to investigate this observation and its applications. We start by providing annuli defined in terms of the tropical roots of t × p(x) that contain the eigenvalues of P (λ). Our localization results yield conditions under which tropical roots offer order of magnitude approximations to the moduli of the eigenvalues of P (λ). Our tropical localization of eigenvalues is less tight than eigenvalue localization results derived from a generalized matrix version of Pellet's theorem but they are easier to interpret. Tropical roots are already used to determine the starting points for matrix polynomial eigensolvers based on scalar polynomial root solvers such as the Ehrlich-Aberth method and our results further justify this choice. Our results provide the basis for analyzing the effect of Gaubert and Sharify's tropical scalings for P (λ) on (a) the conditioning of linearizations of tropically scaled P (λ) and (b) the backward stability of eigensolvers based on linearizations of tropically scaled P (λ). We anticipate that the tropical roots of t × p(x), on which the tropical scalings are based will help designing polynomial eigensolvers with better numerical properties than standard algorithms for polynomial eigenvalue problems such as that implemented in the MATLAB function polyeig.

Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

“…Given the norms P j of the matrix coefficients P j of the matrix polynomial P (z) = d j=0 P j z j , we define the tropical (or max-times) scalar polynomial t × p(x) = max 0≤j≤d P j x j , where x takes nonnegative real values. As mentioned in the introduction, the tropical roots of t × p are points at which the maximum is attained for at least two values of j for some x (see for example [18,19] for a precise definition of tropical roots together with their multiplicities).…”

“…The tropical roots of t × p are the points at which the maximum is attained for at least two values of i for some x. They can be computed in only O(d) operations and they are known to offer order of magnitude approximations to the moduli of the eigenvalues of P , in particular when the norms of the matrices P i vary widely [6,18,19]. Matrix polynomials with such property occur frequently in applications-see the NLEVP collection of test problems [5].…”

Incomplete LU Preconditioner Based on Max-Plus Approximation of LU Factorization

“…Max-plus algebra is the analogue of linear algebra developed for the binary operations max and plus over the real numbers together with −∞, the latter playing the role of additive identity. Max-plus algebraic techniques have already been used in numerical linear algebra to, for example, approximate the orders of magnitude of the roots of scalar polynomials [19], approximate the moduli of the eigenvalues of matrix polynomials [1,10,14], and approximate singular values [9]. These approximations have been used as starting points for iterative schemes and in the design of preprocessing steps to improve the numerical stability of standard algorithms [3,6,7,14,20].…”

“…Max-plus algebraic techniques have already been used in numerical linear algebra to, for example, approximate the orders of magnitude of the roots of scalar polynomials [19], approximate the moduli of the eigenvalues of matrix polynomials [1,10,14], and approximate singular values [9]. These approximations have been used as starting points for iterative schemes and in the design of preprocessing steps to improve the numerical stability of standard algorithms [3,6,7,14,20]. Our aim is to show how max-plus algebra can be used to approximate the sizes of the entries in the LU factors of a complex or real matrix A and how these approximations can subsequently be used in the construction of an incomplete LU (ILU) factorization preconditioner for A.…”

“…where u is the unit roundoff and ρ n (A) is the growth factor for A defined by (14) ρ n (A) = max 0≤k≤n−1 max k≤i,j≤n |u…”