Development of matrix algorithms for solving linear and non-linear structured eigenvalue problems with high relative accuracy or with the best accuracy possible. We will particularly focus on problems which reduce to eigenvalue problems for Hamiltonian and skew-Hamiltonian problems since those appear in important applications. We will combine standrad algorithms for such problems with known high relative acuracy methods for some other matrix problems in orther to develop high relative accuracy algorithms for some of the structured problems. We will also investigate newly propsed method for clustering almost stochastic matrices which is used for the identification of meta-stable states of Markov chains. In this case high relative accuracy requirement reduces to compuation of correct signs of the singular vectors of the second largest singular value, but as fast as possible.
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