Numerical solution of a quadratic eigenvalue problem
نویسنده
چکیده
We consider the quadratic eigenvalue problem (QEP) (λ2M + λG + K)x = 0, where M = MT is positive definite, K = KT is negative definite, and G = −GT . The eigenvalues of the QEP occur in quadruplets (λ, λ,−λ,−λ) or in real or purely imaginary pairs (λ,−λ). We show that all eigenvalues of the QEP can be found efficiently and with the correct symmetry, by finding a proper solvent X of the matrix equation MX2 + GX + K = 0, as long as the QEP has no eigenvalues on the imaginary axis. This solvent approach works well also for some cases where the QEP has eigenvalues on the imaginary axis. AMS classification: 15A18; 15A24; 65F30
منابع مشابه
A Survey of the Quadratic Eigenvalue Problem
We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify the available choices of methods and catalogue a...
متن کاملA 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...
متن کاملThe Quadratic Eigenvalue Problem
We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software.
متن کاملA mathematically simple method based on denition for computing eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices
In this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to th...
متن کاملNumerical Solution of Quadratic Eigenvalue Problems with Structure-Preserving Methods
Numerical methods for the solution of large scale structured quadratic eigenvalue problems are discussed. We describe a new extraction procedure for the computation of eigenvectors and invariant subspaces of skew-Hamiltonian/Hamiltonian pencils using the recently proposed skew-Hamiltonian isotropic implicitly restarted Arnoldi method (SHIRA). As an application we discuss damped gyroscopic syste...
متن کاملQuadratic Eigenproblems Are No Problem
High-dimensional eigenproblems often arise in the solution of scientiic problems involving stability or wave modeling. In this article we present results for a quadratic eigenproblem that we encountered in solving an acoustics problem, speciically in modeling the propagation of waves in a room in which one wall was constructed of sound-absorbing material. EEcient algorithms are known for the st...
متن کامل