Optimal rates of convergence of matrices with applications

2014/07/02 by Heinz H. Bauschke, Bauschke, Heinz H., Yunier Bello-Cruz +7 · 1 citation
Computer Science · Mathematics · #65F10 #65F15 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.1407.0671

openalex publication_date 2014/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a systematic study on the linear convergence rates of the powers of (real or complex) matrices. We derive a characterization when the optimal convergence rate is attained. This characterization is given in terms of semi-simpleness of all eigenvalues having the second-largest modulus after 1. We also provide applications of our general results to analyze the optimal convergence rates for several relaxed alternating projection methods and the generalized Douglas-Rachford splitting methods for finding the projection on the intersection of two subspaces. Numerical experiments confirm our convergence analysis.

Citations

• Splitting Algorithms for the Sum of Two Nonlinear Operators

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