Bauer's Spectral Factorization Method for Low Order Multiwavelet Filter Design

Para-Hermitian polynomial matrices obtained by matrix spectral factorization lead to functions useful in control theory systems, basis functions in numerical methods or multiscaling functions used in signal processing. We introduce a fast algorithm for matrix spectral factorization based on Bauer$'$s method. We convert Bauer$'$ method into a nonlinear matrix equation (NME). The NME is solved by two different numerical algorithms (Fixed Point Iteration and Newton$'$s Method) which produce approximate scalar or matrix factors, as well as a symbolic algorithm which produces exact factors in closed form for some low-order scalar or matrix polynomial matrices, respectively. Convergence rates of the two numerical algorithms are investigated for a number of singular and nonsingular scalar and matrix polynomials taken from different areas. In particular, one of the singular examples leads to new orthogonal multiscaling and multiwavelet filters. Since the NME can also be solved as a Generalized Discrete Time Algebraic Riccati Equation (GDARE), numerical results using built-in routines in Maple 17.0 and 6 Matlab versions are presented.

Design of a Simple Orthogonal Multiwavelet Filter by Matrix Spectral Factorization

We consider the design of an orthogonal symmetric/antisymmetric multiwavelet from its matrix product filter by matrix spectral factorization (MSF). As a test problem, we construct a simple matrix product filter with desirable properties, and factor it using Bauer's method, which in this case can be done in closed form. The corresponding orthogonal multiwavelet function is derived using algebraic techniques which allow symmetry to be considered. This leads to the known orthogonal multiwavelet SA1, which can also be derived directly. We also give a lifting scheme for SA1, investigate the influence of the number of significant digits in the calculations, and show some numerical experiments.

A Kaczmarz Algorithm for Solving Tree Based Distributed Systems of Equations

The Kaczmarz algorithm is an iterative method for solving systems of linear equations. We introduce a modified Kaczmarz algorithm for solving systems of linear equations in a distributed environment, i.e. the equations within the system are distributed over multiple nodes within a network. The modification we introduce is designed for a network with a tree structure that allows for passage of solution estimates between the nodes in the network. We prove that the modified algorithm converges under no additional assumptions on the equations. We demonstrate that the algorithm converges to the solution, or the solution of minimal norm, when the system is consistent. We also demonstrate that in the case of an inconsistent system of equations, the modified relaxed Kaczmarz algorithm converges to a weighted least squares solution as the relaxation parameter approaches $0$.