We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operator A defined on a finite dimensional real Hilbert space V . In our applications, the dimension of V is large and the cost of inverting A is prohibitive. In this paper, we shall develop an effective parallelizable te...

In a recent paper Melman 12] derived upper bounds for the smallest eigen-value of a real symmetric Toeplitz matrix in terms of the smallest roots of rational and polynomial approximations of the secular equation f() = 0, the best of which being constructed by the (1; 2)-Pad e approximation of f. In this paper we prove that this bound is the smallest eigenvalue of the projection of the given eig...

We study spectra of directed networks with inhibitory and excitatory couplings. We investigate in particular eigenvector localization properties of various model networks for different values of correlation among their entries. Spectra of random networks with completely uncorrelated entries show a circular distribution with delocalized eigenvectors, whereas networks with correlated entries have...

Very often, inverse iteration is used with shifts to accelerate convergence to an eigenvector. In this paper, it is shown that, if the eigenvalue belongs to a nonlinear elementary divisor, the vector sequences may diverge even when the shift sequences converge to the eigenvalue. The local behavior is discussed through a 2 X 2 example, and a sufficient condition for the convergence of the vector...

Abstract. Consider a symmetric matrix A(v) ∈ Rn×n depending on a vector v ∈ Rn and satisfying the property A(αv) = A(v) for any α ∈ R\{0}. We will here study the problem of finding (λ, v) ∈ R × Rn\{0} such that (λ, v) is an eigenpair of the matrix A(v) and we propose a generalization of inverse iteration for eigenvalue problems with this type of eigenvector nonlinearity. The convergence of the ...

We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences to the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue prob...

We present a Chebyshev-Davidson method to compute a few smallest positive eigenvalues and corresponding eigenvectors of the linear response eigenvalue problem. The method is actually applicable to the slightly more general linear response eigenvalue problem where purely imaginary eigenvalues may occur. For the Chebyshev filter, a tight upper bound is obtained by a computable bound estimator con...

The pairwise comparisonsmethod is a convenient tool used when the relative order among different concepts (alternatives) needs to be determined. One popular implementation of the method is based on solving an eigenvalue problem for the pairwise comparisons matrix. In such cases the ranking result the principal eigenvector of the pairwise comparison matrix is adopted, whilst the eigenvalue is us...

Scales in RNA, based on geometrical considerations, can be exploited for the analysis and prediction of RNA structures. By using spectral decomposition, geometric information that relates to a given RNA fold can be reduced to a single positive scalar number, the second eigenvalue of the Laplacian matrix corresponding to the tree-graph representation of the RNA secondary structure. Along with th...

For a given subspace, the q-Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the q-Rayleigh-Ritz method defines the q-Ritz values and the q-Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the...