Variational Characterization of Monotone Nonlinear Eigenvector Problems and Geometry of Self-Consistent-Field Iteration

2022/11/09 by Zhaojun Bai, Ding Lu, Bai, Zhaojun +1 · 1 citation
Computer Science · Engineering · Mathematics · #65F15 #65H17 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2211.05179

openalex publication_date 2022/11/09 · openalex created_date 2022/11/16 · openalex updated_date 2026/07/28

Abstract

This paper concerns a class of monotone eigenvalue problems with eigenvector nonlinearities (mNEPv). The mNEPv is encountered in applications such as the computation of joint numerical radius of matrices, best rank-one approximation of third-order partial symmetric tensors, and distance to singularity for dissipative Hamiltonian differential-algebraic equations. We first present a variational characterization of the mNEPv. Based on the variational characterization, we provide a geometric interpretation of the self-consistent-field (SCF) iterations for solving the mNEPv, prove the global convergence of the SCF, and devise an accelerated SCF. Numerical examples from a variety of applications demonstrate the theoretical properties and computational efficiency of the SCF and its acceleration.

Cited by

• From eigenvector nonlinearities with quadratic structure to eigenvalue nonlinearities with algebraic structure

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