The Solution of Algebraic Equations of Continuous Fractions of Nikiports

Abstract

Analytical expressions representing all the roots of a random algebraic equation of n-th degree in terms of the equation coefficients are presented in the paper. These formulas consist of two ratios of infinite Toeplitz determinants. The diagonal elements of the determinants are the coefficients of algebraic equations. To find complex roots the method of summation of divergent continued fractions is used.Keywords: algebraic equation, infinite Toeplitz determinant, r/ϕ-algorithm, diverging continuous fractions. IntroductionTo design modern complex objects it is necessary to analyze their supposed characteristics even in the early stages of their development. Differential and integral equations as well as the classical algebraic equations are used for modeling. Algebraic equation is one of the oldest objects of research in mathematics.There are various applications of algebraic equations to scientific and technical problems. For example, algebraic equations arise is of interest in studies of equilibrium states of complex thermodynamic and mechanical systems. Algebraic equations are often used in aerodynamics. For example, the rate of climb of a plane is determined by the algebraic equation of eighth degree. Algebraic equations are used in the calculation of the flow over the wing in the Prandtl theory. The degree of the equations depends on the law of variation of the lift coefficient from the angle of attack. The problem of structural stability involves the calculation of eigenvalues of matrices. The eigenvalues are determined from the solution of algebraic equation. The equation degree is equal to the number of harmonics. Algebraic equations most often arise in various geometrical calculations, for example, in determination of the intersection points of two curves, in the design of wings, fuselages, etc. The recently published monograph [1,2] dedicated to the various aspects of the theory and practice of algebraic equations. However, internationally known expert R. Hamming gave, in the book published half a century ago [3], the following remark: "the problem of finding the roots of polynomials occurs frequently enough to warrant a thorough examination and development of special methods for its solution. A whole book can be dedicated to the different known methods of finding a valid linear and quadratic multiplier. The large verity of methods shows that no one of them is completely satisfactory". In fact, there * shmoylov40@atinfotectt.ru † vtgak@mail.ru c Siberian Federal University. All rights reserved -533 - The paper presents analytical expressions that represent all the roots of a random algebraic equation of n-th degree in terms of the coefficients of the equation. These formulas consist of two ratios of infinite Toeplitz determinants. The diagonal elements of the determinants are the coefficients of algebraic equations. To find complex roots an advanced method of summation of divergent continued fractions, called r/ϕ-algorithm, is used [7]. The algorithm is finding application in various areas of computati...