This paper takes another look at the convergence of the Arnoldi procedure forsolving nonsymmetric eigenvalue problems. Three different viewpoints are considered. The first usesa bound on the distance from the eigenvector to the Krylov subspace from the smallest singularvalue of matrix consisting of the Krylov basis. A second approach relies on the Schur factorization.Finally, a ...

Ulam’s method is a rigorous numerical scheme for approximating invariant densities of dynamical systems. The phase space is partitioned into connected sets and an inter-set transition matrix is computed from the dynamics; an approximate invariant density is read off as the leading left eigenvector of this matrix. When a hole in phase space is introduced, one instead searches for conditional inv...

Background Diffusion tensor imaging enables the study of cardiac microstructure in vivo, including its changes through the cardiac cycle. The left ventricle consists of myocytes with opposing helical arrangements in the epiand endocardium. Histology shows that these myocytes are arranged into small laminar “sheetlet” structures, separated by shear layers which allow the sheetlets to move and re...

We continue with F being an arbitrary field and V a finite dimensional vector space over F ; dimV = n. Conditions on F may be added later on. We assume T ∈ L(V ). Definition 1 We say that an element λ ∈ F is an eigenvalue of T iff there exists x ∈ V , x ̸= 0 such that Tx = λx. We call x ∈ V , x ̸= 0 such that Tx = λx an eigenvector of T , corresponding to the eigenvector λ. If λ is an eigenvector...

A theoretical method is developed for improving certain calculations for eigenvector derivatives in linear systems. The subject algorithms are all forward analysis, i.e., the eigenvector changes are being driven by design parameter perturbations. The new method emphasizes proper mass normalization and is most needed when iteratively computing eigenvector perturbations within a reduced-dimension...

The Fukui matrix is introduced as the derivative of the one-electron reduced density matrix with respect to a change in the number of electrons under constant external potential. The Fukui matrix extends the Fukui function concept: the diagonal of the Fukui matrix is the Fukui function. Diagonalizing the Fukui matrix gives a set of eigenvectors, the Fukui orbitals, and accompanying eigenvalues....