GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Manifolds, tensor analysis, and applications: 2nd edition
Manifolds, tensor analysis, and applications: 2nd edition
The dynamics of Rayleigh quotient iteration
SIAM Journal on Numerical Analysis
On a multivariate eigenvalue problem, part I: algebraic theory and a power method
SIAM Journal on Scientific Computing
Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems
SIAM Journal on Numerical Analysis
GMRES On (Nearly) Singular Systems
SIAM Journal on Matrix Analysis and Applications
The symmetric eigenvalue problem
The symmetric eigenvalue problem
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
Krylov Subspace Methods for Saddle Point Problems with Indefinite Preconditioning
SIAM Journal on Matrix Analysis and Applications
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Trust-Region Methods on Riemannian Manifolds
Foundations of Computational Mathematics
Optimization Algorithms on Matrix Manifolds
Optimization Algorithms on Matrix Manifolds
Towards the global solution of the maximal correlation problem
Journal of Global Optimization
Third-order methods on Riemannian manifolds under Kantorovich conditions
Journal of Computational and Applied Mathematics
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The multivariate eigenvalue problem (MEP) which originally arises from the canonical correlation analysis is an important generalization of the classical eigenvalue problem. Recently, the MEP also finds applications in many other areas and continues to receive interest. However, the existing algorithms for the MEP are the generalization of the power iteration for the classical eigenvalue problem and converge slowly. In this paper, we propose a Riemannian Newton method for the MEP, which is a generalization of the classical Rayleigh quotient iteration (RQI). Under a mild condition, the local quadratic convergence can be guaranteed. We also develop the inexact implementation by employing some Krylov subspace method and establishing the preconditioning technique to obtain an inexact Riemannian Newton step efficiently. Preliminary but promising numerical experiments are reported which show a good convergence performance in terms of the proposed method's speed and global convergence.