Matlab Code for Sorted Real Schur Forms

Abstract

In Matlab, there exists a standard command to generate a real Schur form, and another command transforms a real Schur form into a complex one. In Golub and Van Loan (1996), a Matlab-like routine is sketched that sorts a complex Schur form: given a target value ? in the complex plane, the diagonal elements of the upper triangular factor T are ordered according to their distance to ? . In this note, we implement a procedure to construct sorted real Schur forms in Matlab. This implementation is based on a block-swapping procedure by Bai and Demmel (1993). Moreover, we describe how to compute a partial Schur form (see Saad (1992)) in case the matrix A is too large to compute a complete Schur form and to order it a posteriori. The sorting of real Schur forms, both partially and completely, have important applications in the computation and tracking of invariant subspaces.