On a nonlinear matrix equation arising in nano research (Q2903121)

Solution methods for the nonlinear matrix equations \({X + A^TX^{-1}A = Q}\) and \(X + A^TX^{-1}A = Q + i \eta I\) are studied, where \(A\) is a real square matrix and \(Q\) is a real symmetric matrix. At first, the convergence of some modified fixed-point iterations is proved. Furthermore, it is shown that the imaginary part \(X_I\) of the matrix \(X_\ast\) is positive semidefinite, where \(X_\ast\) is the limit of the unique stabilizing solution \(X_\eta\) of the equation \(X + A^TX^{-1}A = Q + i \eta I\) as \(\eta \rightarrow 0^+\). The rank of \(X_I\) in terms of the number of unimodular eigenvalues of the quadratic pencil \(\lambda^2 A^T -\lambda Q +A\) is determined. A structure-preserving algorithm which is directly applied to the equation \({X + A^TX^{-1}A = Q}\) is proposed and analysed. Finally, some numerical experiments are presented, e.g.~the structure-preserving algorithm and the QZ algorithm are compared.