On Generalizing Trace Minimization
Ky Fan’s trace minimization principle is extended along the line of the Brockett cost function tr(DXAX) in X on the Stiefel manifold, where D of an apt size is positive definite. Specifically, we…
Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint
- MathematicsJ. Glob. Optim.
A complete characterization of the solution set to the constrained minimization problem in terms of the eigenspace of the matrix pencil is provided.
Linear pencils and quadratic programming problems with a quadratic constraint
Given bounded selfadjoint operators A and B acting on a Hilbert space H, consider the linear pencil P (λ) = A+ λB, λ ∈ R. The set of parameters λ such that P (λ) is a positive (semi)definite operator…
Extensions of Wielandt’s min–max principles for positive semi-definite pencils
There are numerous min–max principles about the eigenvalues of a Hermitian matrix. The most general ones are Wielandt’s min–max principles which include the Courant–Fischer min–max principles and the…
An indefinite variant of LOBPCG for definite matrix pencils
- Mathematics, Computer ScienceNumerical Algorithms
The authors' new method can be seen as a variant of the popular LOBPCG method operating in an indefinite inner product and also turns out to be a generalization of the recently proposed LOBP4DCG method by Bai and Li for solving product eigenvalue problems.
Quadratic (Weakly) Hyperbolic Matrix Polynomials: Direct and Inverse Spectral Problems
Let L be a monic quadratic weakly hyperbolic or hyperbolic n × n matrix polynomial. We solve some direct spectral problems: We prove that the eigenvalues of a compression of L to an (n −…
Symplectic eigenvalue problem via trace minimization and Riemannian optimization
We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson’s theorem. It is…
Preconditioned gradient iterations for the eigenproblem of definite matrix pairs
- Computer ScienceETNA - Electronic Transactions on Numerical Analysis
A unifying framework of preconditioned gradient iterations for definite generalized eigenvalue problems with indefinite B, which compute a few eigenvalues closest to the definiteness interval and the corresponding eigenvectors of definite matrix pairs (A,B), that is, pairs having a positive definite linear combination.
Convergence Analysis of Extended LOBPCG for Computing Extreme Eigenvalues
- Mathematics, Computer ScienceArXiv
The algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate.
SHOWING 1-7 OF 7 REFERENCES
A Jacobi eigenreduction algorithm for definite matrix pairs
SummaryWe propose a Jacobi eigenreduction algorithm for symmetric definite matrix pairsA, J of small to medium-size real symmetric matrices withJ2=I,J diagonal (neitherJ norA itself need be…
Trace Minimization Algorithm for the Generalized Eigenvalue Problem
- Computer SciencePPSC
An inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace, and the rate of convergence of the method is established, and a technique for improving it is discussed.
The Symmetric Eigenvalue Problem
According to Parlett, 'Vibrations are everywhere, and so too are the eigenvalues associated with them. As mathematical models invade more and more disciplines, we can anticipate a demand for…