Self-equivalent flows associated with the generalized eigenvalue problem

Abstract

We present a family of flows which includes continuous analogues of the unshifted and shifted LZ and QZ algorithms for the generalized eigenvalue problem. In order to do this we use elementary Lie theory to create a general family of algorithms, of which the LZ and QZ algorithms are special cases. For each such algorithm we construct a family of associated flows, some of which are interpolants of the algorithm. We do not restrict our attention to Hessenberg-triangular forms; we consider arbitrary pairs of nonsingular matrices.