An explicit polynomial to globalize algorithms for solving matrix polynomial equations

Abstract

Generally, the solution of matrix polynomial equations by means of a global Newton-type algorithm uses an exact linear search that leads to the minimization of a merit function in terms of the Frobenius norm, whose explicit form is known only in the quadratic case. Because of the importance of knowing explicitly this function in the minimization process, in this paper, we obtain the explicit form of the polynomial in the general case, as well as the explicit form of its derivative, and we obtain a sufficient condition to minimize on the interval 0,2. In addition, we present some numerical tests that show the advantage of having the explicit polynomial and the interval to minimize it.