With The Matrix Sign Function And The Pad¨¦ Approximation To Solve The Invariant Subspace

This paper deals with the typical problem of matrix eigenvalue: compute theinvariant subspace of a given matrix associated with its preindicated eigenvalues.There are many classical algorithms for solving the eigenvalue problem, whenusing these classical algorithms to compute invariant subspace, there is so muchdifficulty, unfeasible sometimes. The matrix sign function introduced in the early1970s is widely used in control system, and can be used to compute invariantsubspace. The paper mainly studies the algorithms using matrix sign functionto compute invariant subspace.In this paper, we introduce the algebraic and geometrical definition of thematrix sign function, together with its characteristics at first. We briefly re-view the Newton iteration and the rational iteration for approximating the signfunction, including the convergence of Newton iteration and improvement of therational iteration. The main work of this paper is Pade iterative method forsolving invariant subspace corresponding to preindicated eigenvalues. At first,we discuss the Pade iteration approach for approximating the sign function. Acareful analysis is given that shows the quintic convergent rate of second-orderPade iteration. The convergence rate of the Pade iteration in higher order is alsoestimated. The computed amount increases only a little. The Krylov subspacemethod is very useful for solving the system of linear equations and eigenvalueof matrices in large scale. In view of the difficulty of computing the inverse oflarge-scale matrix, We use Krylov subspace to overcome it. together with therestarting technique and the application of Pade iterative method for the relatedsmaller matrices. Numerical experiments are reported to show the numericalefficiency of our proposed algorithms for computing invariant subspace.