Newton's method for symmetric and bisymmetric solvents of the nonlinear matrix equations (Q2870417)

The authors consider the matrix polynomial equation \(P(X)=A_0X^m+A_1X^{m-1}+ \cdots +A_m=0\), \(A_i,X\in \mathbb{R}^{n\times n}\), and its special case, the quadratic matrix equation of the form \(Q(X)=AX^2+BX+C=0\), \(A,B,C\in \mathbb{R}^{n\times n}\). Such equations can be solved using Newton's method (see [\textit{W. Kratz} and \textit{E. Stickel}, IMA J. Numer. Anal. 7, 355--369 (1987; Zbl 0631.65040)] and [\textit{G. J. Davis}, SIAM J. Sci. Stat. Comput. 2, 164--175 (1981; Zbl 0467.65021)], respectively) but under the assumption that the Fréchet derivative is not singular. The authors introduce Newton's methods for solving \(P(X)\) and \(Q(X)\) which does not rely on the singularity of the Fréchet derivative. They prove that for the symmetric or bisymmetric starting matrix \(X_0\), the methods converge to a solvent with the same properties as \(X_0\).