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H. Górecki and S. Białas

Relations between roots and coefficients of the transcendental equations

It is proved that there exist the relations between coefficients of the transcendental equations and the infinite number of their roots, similar to Vieta's formulae.

These relations may be obtained for the entire analytic functions using theorems of residues and argument principle. In particular the meromorphic functions will be considered.

Open access

S. Białas and H. Górecki

Generalization of Vieta's formulae to the fractional polynomials, and generalizations the method of Graeffe-Lobachevsky

Two problems concerning polynomials are considered. For the first problem it is proved that the zeroes of the fractional polynomials of rational order fulfil relations similar to the Vieta's formulae for the polynomials.

In the second problem it is presented the iterative method of generalization of the Graeffe-Lobachevsky method to solution of the algebraic equations.

Open access

S. Białas and M. Góra

Abstract

In this paper, a system of Lyapunov equations

A*i P + PAi = −Qi (i = 1, . . . ,m), (A)

is considered in which Ai are given n × n complex matrices, Qi are unknown n × n Hermitian positive definite matrices and P, if any, is a common solution to the Lyapunov equations (A). Both sufficient and necessary and sufficient conditions are derived for the existence of such a matrix P. Examples are presented to illustrate the results.