Trust Region Problems and Nonsymmetric Eigenvalue Perturbations

Indefinite Trust Region Subproblems and Nonsymmetric Eigenvalue Perturbations

1 This paper extends the theory of trust region subproblems in two ways: (i) it allows indeenite inner products in the quadratic constraint and (ii) it uses a two sided (upper and lower bound) quadratic constraint. Characterizations of optimality are presented, which have no gap between necessity and suuciency. Conditions for the existence of solutions are given in terms of the deeniteness of a...

Numerical Solution of Large Nonsymmetric Eigenvalue Problems

Adaptive Model Trust Region Methods for Generalized Eigenvalue Problems

Computing a few eigenpairs of large-scale matrices is a significant problem in science and engineering applications and a very active area of research. In this paper, two methods that compute extreme eigenpairs of positive-definite matrix pencils are combined into a hybrid scheme that inherits the advantages of both constituents. The hybrid algorithm is developed and analyzed in the framework o...

A Model-Trust-Region Framework for Symmetric Generalized Eigenvalue Problems

A general inner-outer iteration for computing extreme eigenpairs of symmetric/positive-definite matrix pencils is proposed. The principle of the method is to produce a sequence of p-dimensional bases {Xk} that converge to a minimizer of a generalized Rayleigh quotient. The role of the inner iteration is to produce an “update” vector by (approximately) minimizing a quadratic model of the Rayleig...

Homotopy-determinant Algorithm for Solving Nonsymmetric Eigenvalue Problems

The eigenvalues of a matrix A are the zeros of its characteristic polynomial fiX) = dtt[A XI]. With Hyman's method of determinant evaluation, a new homotopy continuation method, homotopy-determinant method, is developed in this paper for finding all eigenvalues of a real upper Hessenberg matrix. In contrast to other homotopy continuation methods, the homotopy-determinant method calculates eigen...