| Computing eigenpairs of a matrix corresponding to a specific geometry in the complex plane is an important topic in real time signal processing, pattern recognition, spectral analysis, systems theory, radar, sonar, and geophysics. We have studied the matrix sign and matrix sector function iterations to extract the eigenpairs belonging to various geometries without resorting to computationally expensive eigenanalysis methods. We propose a parallelization of an existing matrix sign function algorithm, which was implemented on a Meiko CS-2 multiprocessor. We obtain a fast and stable algorithm for computing the matrix sector functions using Halley's generalized iteration formula for solving nonlinear equations. We propose a parallel iterative algorithm to compute the principal nth root of a positive definite matrix using Gauss-Legendre integration formula. Furthermore computing functions of square matrices is also an important topic in linear algebra, engineering, and applied mathematics. A parallelization of Parlett's algorithm for computing arbitrary functions of upper triangular matrices is introduced. We propose a block-recursive and a parallel algorithm for fast and efficient computation of functions of triangular matrices. The parallel complexity and cache efficiency of these algorithms for computers with two levels of memory are also analyzed. |
