### Construction and features of the optimum rational function used in the ADI-method

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We show how the strict spectral approximation can be used to obtain characterizations and properties of solutions of some problems in the linear space of matrices. Namely, we deal with (i) approximation problems with singular values preserving functions, (ii) the Moore-Penrose generalized inverse. Some properties of approximation by positive semi-definite matrices are commented.

We give formulas for the coefficients of a double Chebyshev series for a hypergeometric function of two variables x and y. We express these coefficients in terms of other hypergeometric functions of two variables. In particular, for hypergeometric functions expressed in terms of corresponding hypergeometric functions of one variable with an argument of the form x+y, the Chebyshev coefficients are values of another hypergeometric function of one variable. In Section 1 we give basic information on...

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In the paper we review the numerical methods for computing the polar decomposition of a matrix. Numerical tests comparing these methods are included. Moreover, the applications of the polar decomposition and the most important its properties are mentioned.

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