On nonlinear matrix equations from the first standard form

In numerically solving nonlinear matrix equations, including algebraic Riccati equations, that are associated with the eigenspaces of certain regular matrix pencils by the doubling algorithms, the matrix pencils must first be brought into one of the two standard forms. Conversely, each standard form leads to a kind of nonlinear matrix equations, which are of interest in their own right. In this paper, we are concerned with the nonlinear matrix equations associated with the first standard form (SF1). Under the nonnegativeness assumption, we investigate solution existence and the convergence of the doubling algorithm. We obtain several results that resemble the ones for SF1 derived from an -matrix algebraic Riccati equation.