We consider several nonlinear matrix equations, for example the quadraticmatrix equation
where X is an n×n unknown complex matrix and A, B and C are given matrices with complex elements, the matrix polynomial
Am, Am-1, ... , A0 and X are real n×n matrices, etc. The convergence of Newton’s method, and incorporating exact line searches and relaxation method for solving nonlinear matrix equations are also considered. We show that an elementwise minimal nonnegative solvent can be found by these methods with the zero starting matrix. Finally, functional iterations and conjugate gradient methods for computing solutions to equations Q(X) and p(X) are introduced.