Mathematical Sciences: Matrix Inequalities and Matrix Perturbation Theory

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The investigator continues research in matrix inequalities and their applications. The objectives of this project are to (a) extend the perturbation theory for functions of matrices, and develop efficient algorithms to compute the condition number for these functions, (b) derive good perturbation bounds for matrix factorizations, (c) extend the theory of matrix norms, in particular unitary similarity invariant norms, and the induced norms of Hadamard multipliers. Projects (a) and (b) have applications in numerical linear algebra. The problems of (c) are of interest in their own right and are relevant to obtaining the results in (a) and (b). There are some unavoidable errors in numerical calculations. For example, whenever calculations are done by computer there is the possibility that small roundoff errors will be incurred -- for example, 1/3 = 0.333333... may approximated by 0.3333. Another source of errors is that the input data may be the result of measurements that are not exact. It is possible that these small errors can cause a large error in the final answer -- so large that it is useless. The investigator studies the effects of small errors in numerical linear algebraic calculations. This will help one to decide whether a particular method of calculation produces reliable results, and to know in advance what accuracy is needed in measurements to ensure a given accuracy in the final answer.