An extended Hamiltonian algorithm for the general linear matrix equation

A second-order learning algorithm based on differential geometry is used to numerically solve the linear matrix equation Q = x + ∑ i = 1 m A i T x A i − ∑ i = 1 n B i T x B i . An extended Hamiltonian algorithm is proposed based on the manifold of symmetric positive definite matrices. The algorithm is compared with traditional coupled fixed-point algorithm. Numerical experiments illustrate that the convergence speed of the provided algorithm is faster than that of the coupled fixed-point algorithm.