The authors consider a method for the solution of the algebraic Riccati equation \[ -XNX+XA+A^ TX+K=0, \] where A, K, N, X are real \(n\times n\) matrices, \(N=N^ T\) and \(K=K^ T\). A solution X is obtained as \(X=- ZY^{-1}\), where the matrices Z, Y are also \(n\times n\) and the columns of \(\left[ \begin{matrix} Z \\ Y \end{matrix} \right]\) form an invariant subspace of the Hamiltonian matrix \[ M=\left[ \begin{matrix} A&N \\ K&-A^ T \end{matrix} \right]. \] It is proposed a variant of the SR-Algorithm of the first and the second author [IEEE Trans. Autom. Control, AC-31, 1104-1113 (1986; Zbl 0616.65048)] combined with the square reduced algorithm of C. Van Loan [Linear Algebra Appl. 61, 233-251 (1984; Zbl 0565.65018)] which is used to compute good approximations to the eigenvalues of M. The numerical results and comparison with other algorithms are also presented.