Abstract: In this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its block is not zero and has perturbations. Computable formula of backward errors are derived. The expressions are useful for testing the stability of practical algorithms. [Copyright &y& Elsevier]
Abstract: In this paper, algorithms are developed for the problems of spectral factorization and sum of squares of polynomial matrices with n indeterminates, and a natural interpretation of the tools employed in the algorithms is given using ideas from the theory of lossless and dissipative systems. These algorithms are based on the calculus of 2n-variable polynomial matrices and their associated quadratic differential forms, and share the common feature that the problems are lifted from the original n-variable polynomial context to a 2n-variable polynomial context. This allows to reduce the spectral factorization problem and the sum of squares problem to linear matrix inequalities (LMI’s), to the feasibility of a semialgebraic set or to a linear eigenvalue problem. [Copyright &y& Elsevier]