A Preconditioned Squared Smith Method for Continuous-Time Lyapunov Equations

This paper proposes a preconditioned squared Smith algorithm to solve the continuous-time Lyapunov matrix equations AX+XA^T+BB^T=0 numerically.The method first uses the alternating directional implicit (ADI) method and transforms the original equations to the equivalent symmetric Stein matrix equations with some ADI parameters.Then we adopt the squared Smith algorithm to seek solutions of the Stein equations by generating the squared Smith iterations in some low-rank forms with the Krylov subspaces.And we give some numerical experiments to show the efficiency of this algorithm finally.