A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalues and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding near-breakdown and discuss maintaining biorthogonality. A system of linear equations can be solved ...

A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. Some reorthogonalization is ne...

The almost Mathieu operator is the discrete Schrodinger Hα,β,θ on \(\ell ^2(\mathbb {Z})\) defined via \((H_{\alpha ,\beta ,\theta }f)(k) = f(k + 1) - \beta \cos {}(2\pi \alpha k \theta ) f(k)\). We derive explicit estimates for eigenvalues at edge of spectrum finite-dimensional \({H^{(n)}_{\alpha }}\). furthermore show that (properly rescaled) m-th Hermite function ϕm an approximate eigenvecto...

This part describes the results obtained in the simulations and experiments reported in the previous sessions of this report. A) Statistical comparison of the principal component analysis in various spaces The cumulative contributions of the eigenvectors in each space are summarized for multiple-of-three numbers of eigenvectors in Tables I to III. Table I. Cumulative contribution of the eigenve...

The ability of humans to perceive the spatial orientation of an occluded arm was investigated. It was hypothesized that this ability is tied to the arm's inertial eigenvectors, invariant mechanical parameters corresponding to a limb's axes of rotational symmetry. By breaking the coincidence between the eigenvectors of the arm and its longitudinal axis, 3 experiments were directed at the possibi...

The complete knowledge of Laplacian eigenvalues and eigenvectors of complex networks plays an outstanding role in understanding various dynamical processes running on them; however, determining analytically Laplacian eigenvalues and eigenvectors is a theoretical challenge. In this paper, we study the Laplacian spectra and their corresponding eigenvectors of a class of deterministically growing ...

Spatial-correlation properties of the wave functions (eigenvectors) of a spin-one eigenproblem for dipole interaction is studied for random geometries of the underlying system. This problem describes, in particular, polar excitations (“plasmons”) of large clusters. In contrast to Berry’s conjecture of quantum chaos for massive particles, we have found long-range spatial correlations for wave fu...

For the one-dimensional XXX model under the periodic boundary conditions, we discuss two types of eigenvectors, regular eigenvectors which have finite-valued rapidities satisfying the Bethe ansatz equations, and non-regular eigenvectors which are descendants of some regular eigenvectors under the action of the SU(2) spin-lowering operator. It was pointed out by many authors that the non-regular...