We consider several nonlinear matrix equations, for example the quadraticmatrix equation
Q(X)=AX2+BX=C=0,
where  X   is an n×n  unknown complex matrix and  A, B  and  are  given matrices with complex elements, the matrix polynomial

P(X)=A0Xm+A1Xm-1+...+Am=0,
Am, Am-1, ... , A0  and  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.

kim_hyunmin.pdf