Numerical solutions for large sparse quadratic eigenvalue problems

Numerical methods for large eigenvalue problems

Over the past decade considerable progress has been made towards the numerical solution of large-scale eigenvalue problems, particularly for nonsymmetric matrices. Krylov methods and variants of subspace iteration have been improved to the point that problems of the order of several million variables can be solved. The methods and software that have led to these advances are surveyed.

A numerical method for quadratic eigenvalue problems of gyroscopic systems

We consider the quadratic eigenvalues problem (QEP) of gyroscopic systems ðlMþ lGþ KÞx 1⁄4 0, where M 1⁄4 M>;G 1⁄4 G> and K 1⁄4 K> 2 R n with M being positive definite. Guo [Numerical solution of a quadratic eigenvalue problem, Linear Algebra and its Applications 385 (2004) 391–406] showed that all eigenvalues of the QEP can be found by solving the maximal solution of a nonlinear matrix equatio...

General Solutions for a Class of Inverse Quadratic Eigenvalue Problems

Based on various matrix decompositions, we compare different techniques for solving the inverse quadratic eigenvalue problem, where n× n real symmetric matrices M , C and K are constructed so that the quadratic pencil Q(λ) = λM + λC + K yields good approximations for the given k eigenpairs. We discuss the case where M is positive definite for 1≤ k ≤ n, and a general solution to this problem for...

The Numerical Treatment of Large Eigenvalue Problems

This paper surveys techniques for calculating eigenvalues and eigenvectors of very large matrices.

Numerical Solution of Large Nonsymmetric Eigenvalue Problems