Solving composite sum of powers via Padé approximation and orthogonal polynomials with application to optimal PWM problem

摘要

This paper presents methods for solving the polynomial system∑j=1kxji-∑j=k+1nxji=pi,i=1,2,…,n,which is called the composite sum of powers. It is shown that these polynomial equation can be reduced to a single-variable polynomial equations by exploiting the modified Newton’s identities. In this paper we generalize this identity and solve it via Padé approximation theory and the related theory of formal orthogonal polynomials (FOPs). Because the solution forms the roots of FOPs we present several interesting computational procedures, such as the use of three-term reccurence formulas, determinantal formulations and the computation of the eigenvalues of tridiagonal matrices. The computation of this special polynomial system arise in practical engineering task of solving optimal odd symmetry single-phase pulse-width modulated (PWM) problem.