Using the matrix sign function to compute invariant subspaces. (English) Zbl 0914.65035

Bai, Zhaojun; Demmel, James

SIAM J. Matrix Anal. Appl. 19, No. 1, 205-225 (1998).

Authors’ abstract: The matrix sign function has several applications in system theory and matrix computations. However, the numerical behavior of the matrix sign function, and its associated divide-and-conquer algorithm for computing invariant subspaces, are still not completely understood. In this paper, we present a new perturbation theory for the matrix sign function, the conditioning of its computation, the numerical stability of the divide-and-conquer algorithm, and iterative refinement schemes. Numerical examples are also presented. An extension of the matrix-sign-function-based algorithm to compute left and right deflating subspaces for a regular pair of matrices is also described.

Reviewer: A.Meister (Hamburg)

MSC:

65F15	Numerical computation of eigenvalues and eigenvectors of matrices
65F35	Numerical computation of matrix norms, conditioning, scaling

Keywords:

numerical examples; matrix sign function; Newton’s method; eigenvalue problem; invariant subspace; defacting subspaces; divide-and-conquer algorithm; conditioning; numerical stability; iterative refinement

Software:

EISPACK; LAPACK