Geometric Means of HPD GLT Matrix-Sequences: Beyond Invertibility Assumptions and Convergence Properties

Author : Muhammad Faisal Khan, Asim Ilyas, Stefano Serra-Capizzano

Abstract :In this work, we extend our study on the spectral distribution of the geometric mean of matrix sequences formed by Hermitian Positive Definite (HPD) matrices, assuming that all input matrix sequences belong to the same Generalized Locally Toeplitz (GLT) *-algebra. Building upon our previous results [1], we further examine whether the assumption that at least one of the input GLT symbols is invertible almost everywhere is necessary. Since inversion is mainly required due to non commutativity matrices, we aim to investigate whether this assumption can be removed when the GLT symbols commute for every d,r≥1. Numerical experiments are conducted to support this idea. Additionally, we investigate the Karcher mean of more than two HPD GLT matrix-sequences and study its GLT nature. In particular, we analyze how starting with an initial guess that is already a GLT matrix-sequence affects the convergence of the iterative process. Our goal is not just to compute the geometric mean, but to observe how quickly the iterations reach the final result when a suitable initial guess is used. The study also extends the proof for the block multilevel case (r=1 and d≥1), while refining numerical validation in broader settings

Keywords :Geometric Mean, Matrix-Sequences, Karcher Mean, Generalized Locally Toeplitz (GLT) Algebra, Spectral Distribution

Conference Name :International Conference on Numerical analysis (ICNA-25)

Conference Place London, UK

Conference Date 29th Apr 2025