A Further Extension of the Routh-Hurwitz Stability Criteria for Quintic and Sextic Equations

Abstract

Recently, Duncan ' 4 applied these test functions to the sextic and octic equations in determining their stability. I t is immediately apparent that after reducing the complexity of the stability problem, there still remains a tedious amount of necessary work in solving higher order equations. In a further effort at simplification, Higgins and Hogan employ the Hurwitz determinants in conjunction with algebraic manipulation to reduce the necessary number of parameters for a quintic equation to four. Even then, an innumerable number of graphs must be plotted in order to determine the stability of the quintic equations. Duncan derived simple relations between certain expressed polynomials and determinantal forms and tended to further minimize the number of effective parameters. Since there are no general parameters which hold for all degrees of polynomials, the stability determination becomes more difficult as the power of the polynomial increases. In Eq. (1), let