mNEPv - Monotone Nonlinear Eigenvector Problems

Dated 04-20-2023

This folder contains the MATLAB implementation of the algorithms used in the research paper

Variational Characterization of Monotone Nonlinear Eigenvector Problems and Geometry of Self-consistent Field Iteration by Zhaojun Bai and Ding Lu, 2022. (Manuscript available at: https://doi.org/10.48550/arXiv.2211.05179 )

Description

A Monotone Nonlinear Eigenvector problems (mNEPv) is defined as $$H(x) x = \lambda x,$$ where $H(x)$ is a Hermitian matrix-valued function of the form $$H(x):= \sum_{i=1}^m h_i(x^HA_ix) A_i,$$ and $A_1,\dots,A_m$ are $n$ by $n$ Hermitian matrices, $h_1,\dots,h_m$ are differentiable and non-decreasing functions over $\mathbb R$. The goal is to find a unit-length vector $x\in\mathbb C^n$ and a scalar $\lambda\in\mathbb R$ satisfying $H(x)x=\lambda x$ and, furthermore, $\lambda (= x^H H(x) x)$ is the largest eigenvalue of $H(x)$.

The algorithm implemented is the self-consistent-field iteration (SCF) with local acceleration, which ensures global convergence to a solution, as described in the paper. This package contains all the example data and routines used in the study to facilitate its reproducibility.

Contents

• mNEPv solver

  • scf.m: SCF iteration with and without local acceleration

• Examples

  • numrd2d.m: mNEPv from numerical radius computation
  • dhdae1.m: mNEPv from distance problems of linear dHDAE systems
  • dhdae2.m: mNEPv from distance problems of quadratic dHDAE systems
  • tensorappr.m: mNEPv from Tensor rank-1 approximation

• Other files: numrange.m (plot numerical range), goescf.m (geometry of SCF), and data files.

External

The MATLAB toolbox Manopt is used in dhdae2 for comparison. The comparison is deactivated by default. To use it please first download and install the Manopt.

Contact

For questions, please contact Ding.Lu@uky.edu