Efficient computation of extreme eigenvalues of large-scale linear Hermitian eigenproblems can be achieved by preconditioned conjugate gradient (PCG) methods. In this paper, we study PCG methods for computing extreme eigenvalues of nonlinear Hermitian eigenproblems of the form T (λ)v = 0 that admit a nonlinear variational principle. We investigate some theoretical properties of a basic CG metho...

By means of the two-scale convergence method, we investigate the asymptotic behavior of eigenvalues and eigenfunctions of Stekloff eigenvalue problems in perforated domains. We prove a concise and precise homogenization result including convergence of gradients of eigenfunctions which improves the understanding of the asymptotic behavior of eigenfunctions. It is also justified that the natural ...

We propose an eecient implementation of the Chebyshev Galerkin spectral method for the biharmonic operator. This discretization leads to banded matrices which, compared with other methods of the same type, are also better conditioned. The eeciency of the method is illustrated on the Orr-Sommerfeld eigenvalue problem, where an improved convergence can be observed and the spurious eigenvalues are...

This paper gives max characterizations for the sum of the largest eigen-values of a symmetric matrix. The elements which achieve the maximum provide a concise characterization of the generalized gradient of the eigenvalue sum in terms of a dual matrix. The dual matrix provides the information required to either verify rst-order optimality conditions at a point or to generate a descent direction...

In this article, by extending the method of [AC] we prove a sharp estimate on the expansion modulus of the gradient of the logarithm of the parabolic kernel to the Schördinger operator with convex potential on a bounded convex domain. The result improves an earlier work of Brascamp-Lieb which asserts the log-concavity of the parabolic kernel. We also give an alternate proof to a corresponding e...

Nuisance attribute projection (NAP) was an effective method to reduce session variability in SVM-based speaker verification systems. As the expanded feature space of nonlinear kernels is usually high or infinite dimensional, it is difficult to find nuisance directions via conventional eigenvalue analysis and to do projection directly in the feature space. In this paper, two different approaches...

given four complex matrices a, b, c and d where a 2 cnn and d 2 cmm andlet the matrix(a bc d)be a normal matrix and assume that  is a given complex number that is not eigenvalue of matrix a. we present a method to calculate the distance norm (with respect to 2-norm) from d to the set of matrices x 2 cmm such that,  be a multiple eigenvalue of matrix(a bc x). we also nd the nearest matrix ...

We investigate the eecient computation of a few of the lowest eigenvalues of a symmetric eigenvalue problem occurring in quantum dynamical molecular simulations. The systems of equations that arise when applying subspace iteration or the Jacobi-Davidson algorithm with the shift-and-invert spectral transformation are solved by the preconditioned conjugate gradient method. We propose and analyze ...

To compute the smallest eigenvalues and associated eigenvectors of a real symmetric matrix, we consider the Jacobi–Davidson method with inner preconditioned conjugate gradient iterations for the arising linear systems. We show that the coe9cient matrix of these systems is indeed positive de:nite with the smallest eigenvalue bounded away from zero. We also establish a relation between the residu...