# Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property

@article{Betcke2017RestartingIP, title={Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property}, author={Marta M. Betcke and Heinrich Voss}, journal={Numerische Mathematik}, year={2017}, volume={135}, pages={397 - 430} }

In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration. In contrast to global numbering which requires including all the previously computed eigenvectors in the search subspace, the proposed local numbering only requires a presence of one eigenvector in the search subspace…

## 8 Citations

### Preconditioned Eigensolvers for Large-Scale Nonlinear Hermitian Eigenproblems with Variational Characterizations. II. Interior Eigenvalues

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2015

Numerical experiments demonstrate that PLMR methods provide a rapid and robust convergence toward interior eigen Values, and the approach is also shown to be efficient and reliable for computing a large number of extreme eigenvalues, dramatically outperforming standard preconditioned conjugate gradient methods.

### On the perturbation of an $L^2$-orthogonal projection

- Mathematics
- 2018

New perturbation bounds for the L^2-orthogonal projection onto the column space of a matrix, which involve upper (lower) bounds and combined upper ( lower) bounds are established.

### A survey on variational characterizations for nonlinear eigenvalue problems

- MathematicsETNA - Electronic Transactions on Numerical Analysis
- 2021

Variational principles are very powerful tools when studying self-adjoint linear operators on a Hilbert space H. Bounds for eigenvalues, comparison theorems, interlacing results, and monotonicity of…

### NEP-PACK: A Julia package for nonlinear eigenproblems - v0.2

- Computer ScienceArXiv
- 2018

The package provides a framework to represent NEPs, as well as efficient implementations of many state-of-the-art algorithms, and makes full use of the efficiency of Julia, yet maintains usability, and integrates well with other software packages.

### The infinite Lanczos method for symmetric nonlinear eigenvalue problems

- Mathematics
- 2019

A new iterative method for solving large scale symmetric nonlineareigenvalue problems is presented. We firstly derive an infinite dimensional symmetric linearization of the nonlinear eigenvalue pro…

### On restarting the tensor infinite Arnoldi method

- Computer Science
- 2016

An extension of TIAR which corresponds to generating the Krylov space using not only polynomials, but also structured functions, which are sums of exponentials and polynmials, while maintaining a memory efficient tensor representation is considered.

### Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues

- Computer ScienceMath. Comput.
- 2016

Numerical experiments demonstrate that PLMR provide a rapid and robust convergence towards interior eigenvalues, and is shown to be ecient and reliable for computing a large number of extreme eigen values, dramatically outperforming standard preconditioned conjugate gradient methods.

## References

SHOWING 1-10 OF 44 REFERENCES

### Locking and Restarting Quadratic Eigenvalue Solvers

- Computer ScienceSIAM J. Sci. Comput.
- 2001

A link between methods for solving quadratic eigenvalue problems and the linearized problem is shown by employing a locking and restarting scheme based on the Schur form of thelinearized problem inquadratic residual iteration and Jacobi--Davidson.

### A LOCAL RESTART PROCEDURE FOR ITERATIVE PROJECTION METHODS FOR NONLINEAR SYMMETRIC EIGENPROBLEMS

- Mathematics, Computer Science
- 2004

This paper proposes a localized version of safeguarded iteration which is able to cope with nonlinear eigenvalue problems and hits their limitation if a large number of eigenvalues are required.

### Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems

- MathematicsNumerische Mathematik
- 2013

It is shown that the inexact algorithms can achieve the same order of convergence as the exact methods if appropriate sequences of tolerances are applied to the inner solves, and the use of a nonlinear Rayleigh functional is shown to be fundamental in achieving higher order of converge rates.

### Nonlinear Eigenvalue Problems: Newton-type Methods and Nonlinear Rayleigh Functionals

- Mathematics
- 2008

Nonlinear eigenvalue problems arise in many fields of natural and engineering sciences. Theoretical and practical results are scattered in the literature and in most cases they have been developed…

### ITERATIVE PROJECTION METHODS FOR LARGE-SCALE NONLINEAR EIGENVALUE PROBLEMS ∗

- Computer Science
- 2010

This presentation reviews iterative projection methods for sparse nonlinear eigenvalue problems which have proven to be very efficient and nonlinear Arnoldi method, the Jacobi–Davidson method, and the rational Krylov method.

### An Arnoldi Method for Nonlinear Symmetric Eigenvalue Problems

- Computer Science
- 2003

This approach to sparse nonlinear eigenvalue problems is generalized by nesting the linearization of problem (1) by Regula falsi and the solution of the resulting linear eigenproblem by Arnoldi’s method, where the Regulas falsi iteration and the Arnoldi recursion are knit together.

### Jacobi-Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils

- Computer ScienceSIAM J. Sci. Comput.
- 1998

Two algorithms, JDQZ for the generalized eigen problem and JDQR for the standard eigenproblem, that are based on the iterative construction of a (generalized) partial Schur form are presented, suitable for the efficient computation of several eigenvalues and the corresponding eigenvectors near a user-specified target value in the complex plane.

### Jacobi-Davidson style QR and QZ algorithms for the partial reduction of matrix pencils

- Computer Science
- 1996

Two algorithms are presented, JDQR for the standard eigenproblem, and JDQZ for the generalized eigen problem, that are based on the iterative construction of the (generalized) partial Schur form with the Jacobi-Davidson approach, suitable for the computation of several eigenvalues and the corresponding eigenvectors near a user-speci ed target value in the complex plane.

### A linear eigenvalue algorithm for the nonlinear eigenvalue problem

- Mathematics, Computer ScienceNumerische Mathematik
- 2012

This paper characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted B and the resulting algorithm is completely equivalent to the standard Arnoldi method and inherits many of its attractive properties.

### Computing a Partial Schur Factorization of Nonlinear Eigenvalue Problems Using the Infinite Arnoldi Method

- Computer Science, MathematicsSIAM J. Matrix Anal. Appl.
- 2014

A technique to compute a partial Schur factorization of a nonlinear eigenvalue problem (NEP) using the infinite Arnoldi method and shows that the invariant pairs of the operator are equivalent to invariant pair of the NEP.