N. Del Buono

Computation of the Exponential of Large Sparse Skew-Symmetric Matrices

In this paper we consider methods for evaluating both exp(A) and

On the Low-Rank Approximation of Data on the Unit Sphere

exp(\tau A)q_1

Geometric Integration on Manifold of Square Oblique Rotation Matrices

where

Explicit methods based on a class of four stage fourth order Runge-Kutta methods for preserving quadratic laws

{\rm exp}(\cdot)

Nonnegative matrix factorizations performing object detection and localization

is the exponential function, A is a sparse skew-symmetric matrix of large dimension, q1 is a given vector, and

Computation of functions of Hamiltonian and skew-symmetric matrices

\tau

Computation of few Lyapunov exponents by geodesic based algorithms

is a scaling factor. The proposed method is based on two main steps: A is factorized into its tridiagonal form H by the well-known Lanczos iterative process, and then exp(A) is derived making use of an effective Schur decomposition of H. The procedure takes full advantage of the sparsity of A and of the decay behavior of exp(H). Several applications and numerical tests are also reported.

Numerical Integration of a Class of Ordinary Differential Equations on the General Linear Group of Matrices

In various applications, data in multidimensional space are normalized to unit length. This paper considers the problem of best fitting given points on the m-dimensional unit sphere Sm-1 by k-dimensional great circles with k much less than m. The task is cast as an algebraically constrained low-rank matrix approximation problem. Using the fidelity of the low-rank approximation to the original data as the cost function, this paper offers an analytic expression of the projected gradient which, on one hand, furnishes the first order optimality condition and, on the other hand, can be used as a numerical means for solving this problem.

A differential approach to solve the inverse eigenvalue problem derived from a neural network

In recent years there has been a growing interest in the dynamics of matrix differential systems on a smooth manifold. Research effort extends to both theory and numerical methods, particularly on the manifolds of orthogonal and symplectic matrices. This paper concerns dynamical systems on the manifold

Differential approaches for computing Euclidean diagonal norm balanced realizations in control theory

{\cal OB}(n)