Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control

We consider iterative methods for finding the maximal Hermitian solution of a matrix Riccati equation arising in stochastic control. Newton’s method is very expensive when the size of the problem is large. A much less expensive iteration is introduced and shown to have several convergence properties similar to those of Newton’s method. In ordinary situations, the convergence of the new iteration is linear while the convergence of Newton’s method is quadratic. In extreme cases, the convergence of the new iteration may be sublinear while the convergence of Newton’s method may be linear. We also show how the performance of Newton’s method can be improved when its convergence is not quadratic.