Estimation of polynomial roots by continued fractions

One of the ways to describe dynamical characteristics of the system is the s-transfer function, which belongs to the class of the so-called external descriptions and from the mathematical point of view it represents a rational function. Decomposition into the partial fractions (sum of exponential functions in the time domain) is based on the knowledge of the roots of the polynomial defined by the denominator of the transfer function; from their location it is possible to deduce dynamical characteristics, system stability, etc. The numerator roots position indicates the phase characteristic of the system. There exists a large number of successful methods for calculating the polynomial roots (for instance Lehmer-Shur, Bairstow, Graef, Bernoullia, Newton-Raphson and many others). The method described here is based on Viskovatoff's decomposition of a rational function, and makes use of the continued fraction theory and the characteristic root loci construction proposed by Evans. In the second part of this paper two existing methods are discussed for approximate finding of the polynomial roots through the construction of characteristic root loci. Each of them is illustrated by a simple solved example. In the third part a short introducion to the analytic theory of continued fractions is given. It contains necessary conclusions for the proposed method and is also completed by a solved example. In the fourth part the general algorithm is given and a sample solution is explored. The final part provides a comparison with other methods, the possibility of algorithmization, characteristic root loci construction on microcomputers, etc.