In this paper we present a new stable algorithm for the parallel QR-decomposition of ”tall and skinny” matrices. The algorithm has been developed for the dense symmetric eigensolver ELPA, whereat the QR-decomposition of tall and skinny matrices represents an important substep. Our new approach is based on the fast but unstable CholeskyQR algorithm [1]. We show the stability of our new algorithm...

v Acknowledgments vi Chapter

Let Ω be an open, simply connected, and bounded region in R, d ≥ 2, and assume its boundary ∂Ω is smooth. Consider solving the eigenvalue problem Lu = λu for an elliptic partial differential operator L over Ω with zero values for either Dirichlet or Neumann boundary conditions. We propose, analyze, and illustrate a ‘spectral method’ for solving numerically such an eigenvalue problem. This is an...

We model a nanoMOSFET by a mesoscopic, time-dependent, coupled quantumclassical system based on a sub-band decomposition and a simple scattering operator. We first compute the sub-band decomposition and electrostatic force field described by a Schrödinger-Poisson coupled system solved by a Newton-Raphson iteration using the eigenvalue/eigenfunction decomposition. The transport in the classical ...

Abstract: In many practical systems, limit cycles can be predicted with suitable precision by frequency domain methods using describing functions. Within such an approach, limit cycles can be predicted using the “eigenvalue method” [7]. This contribution presents a novel and advantageous implementation of this method, using singular value instead of eigenvalue calculations, and enhancing comput...

In this paper, the modified linearized Phillips-Heffron model is utilized to theoretically analyze asingle-machine infinite-bus (SMIB) installed with SSSC. Then, the results of this analysis are used forassessing the potential of an SSSC supplementary controller to improve the dynamic stability of apower system. This is carried out by measuring the electromechanical controllability through sing...

In [23] we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of n-by-n matrices can be done by any algorithm in O(n) operations for any η > 0, then it can be done stably in O(n) operations for any η > 0. Here we extend this result to show that essentially all standard linear algebra operations, includin...

Hessenberg decomposition is the basic tool used in computational linear algebra to approximate the eigenvalues of a matrix. In this article, we generalize Hessenberg decomposition to continuous matrix fields over topological spaces. This works in great generality: the space is only required to be normal and to have finite covering dimension. As applications, we derive some new structure results...

Many modern high-resolution spectral estimators in signal processing and control make use of the subspace information afforded by the singular value decomposition of the data matrix, or the eigenvalue decomposition of the covariance matrix. The derivation of these estimators involves some form of matrix decomposition. In this paper, new computational techniques for obtaining eigenvalues and eig...

0. Systems theory I. Part 1: Convergence A. Two approaches to convergence B. A mathematical formulation of convergence C. Convergence is a Markov process D. Starting configuration doesn’t matter II. Part 2: Analysis A. Background and mathematical foundation 1. Stochastic matrices 2. The Eigenvalue problem 3. Spectral Decomposition 4. Spectral Decomposition of M B. The Algebraic View 1. Partitio...