Preconditioned Iteration Methods For The Matrix Equations

Matrix computation is widely used in many fields of scientific computing and engineering applications,and the numerical solution problem of Lyapunov and Sylvester equations that have played a very important role in many areas,such as design and analysis of control system,boundary value problem of partial differential equation,restoration of images and model reduction of large-scale linear dynamical system.Therefore,it is of great practical value to study the numerical solution algorithms of matrix equations.As we all know,the most common kind of method to solve the matrix equation problem with the large and sparse coefficient matrice is the iterative algorithm.However,with the increase of coefficient matrix scale,the convergence of classical iterative algorithms is relatively slow and the computation is very huge that we will apply the preconditioned technique to deal with these problems.Moreover,in the present paper,we propose a preconditioned Squared Smith iterative method to solve the large-scale matrix equations based on the preconditioned technique.We firstly construct a preconditioner by using the alternating directional implicit(ADI)method,and transform the original equation to an equivalent matrix equation whose spectral properties are better.Then we apply the Squared Smith method to generate the low-rank approximation form with a Krylov subspace,and give the estimation of errors and residuals of the preconditioned Squared Smith algorithm.Finally,we give some numerical experiments of Lyapunov and Sylvester matrix equations to show the efficiency and accuracy of this algorithm by comparing the numerical results of Matlab experiments between it with other traditional iteration methods such as Jacobi,global-Krylov subspace and block-Krylov subspace iterative algorithms.