Fast and Accurate Algorithms for Structured Matrix Computations: Applications and Software

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Proposal #0098222Georgia State University Olshevsky, VadimNumerous applications in sciences and engineering, as well as in computational, applied and pure mathematics give rise to problems involving matrices with Toeplitz, Hankel, Vandermonde, Cauchystructures, along with many other patterns of structure. Standard mathematical software tools [e.g., MATLAB, Mathematica, MathCad, Maple, LAPACK], are based on standard [structure-ignoring]methods, and therefore their use is often not appropriate for obtaining a satisfactory solution. Here are two major reasons. First, ignoring the structure artificially squares the size of the data, which requires unnecessary storage and an extremely large amount of CPU time. Secondly, many structured matrices areextremely ill-conditioned, which means that all available standard methods may fail to produce even one correct digit in the computed solution. The only way to overcome these difficulties is to exploit the special structure of such matrices, and to design more efficient algorithms.The objective of this project is the study of several known and several new theoretical and computational problems related to structured matrices which arise in several applied areas, including signal and image processing, system theory, and control theory. This research involves the development of new accuratefast and superfast algorithms for several new classes of structured matrices arising in rational matrix interpolation and approximation problems with norm constraints (passive interpolation). This study will also focus on various stability problems for matrix polynomials, and on various problems in thetheory of error-correcting codes.