ALGORITHM OF THE REDUCED POLYNOMIAL EQUATIONS SOLUTION USING CONTINUED FRACTIONS

Abstract

The article presents an algorithm where continued fractions are used to find the zeros of nth
degree polynomial. Our day there is a wide variety of methods and algorithms for solving such
type problems. The proposed algorithm feature is its effective possibility using for large values of
n. Besides this algorithm can be applied for complex roots. Any real number can be represented as
a continued fraction: finite or infinite. The main purpose of continuous fractions is that they allow
us to find good approximations of real numbers in the ordinary fractions form in algebraic equations
and systems solution. The purpose of this work is algorithm with continued fractions for solving
reduced polynomial equation that contains both real and complex roots, estimation of the
number of arithmetic steps in its numerical solution. The analytical expressions for polynomial
equation solutions are given in this paper. The obtained analytical expressions represent the ratio
of the Toeplitz determinants. A distinctive feature of these determinants is the presence of coefficients
of the solved algebraic equation as diagonal elements. A modified Rutishauser algorithm
was used to obtain a numerical solution. Complex roots can be found using the summation algorithm
of divergent continued fractions. As an illustration of the proposed algorithm the results of
the numerical solution for fifth degree polynomial equation are given. The advantage of the algorithm
is the small number of arithmetic operations required, the possibility of considering highdegree
polynomials, and the small error of calculations.