2015/01/07 by Sĩ Tiệp Đinh, Dinh, Si Tiep, Tiến-Sơn Phạm +1 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1501.01419
openalex publication_date 2015/01/07 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Let F(x) := (fij(x))i,j=1,…,p, be a real symmetric polynomial matrix of order p and let f(x) be the largest eigenvalue function of the matrix F(x). We denote by ∂^∘ f(x) the Clarke subdifferential of f at x. In this paper, we first give the following \em nonsmooth version of Łojasiewicz gradient inequality for the function f with an explicit exponent: For any x∈ \Bbb Rn there exist c > 0 and ε> 0 such that we have for all ‖x - x‖ < ε, inf \ ‖ w ‖ : w ∈ ∂^∘ f(x) \ ≥ c |f(x) - f( x)|^1 - \frac1\mathscrR(2n+p(n+1),d+3), where d:=maxi,j = 1, …, p°fi j and \mathscrR is a function introduced by D'Acunto and Kurdyka: \mathscrR(n, d) := d(3d - 3)n-1 if d ≥ 2 and \mathscrR(n, d) := 1 if d = 1. Then we establish error bounds with explicitly determined exponents, local and global, for the largest eigenvalue function f(x) of the matrix F(x).