Reduction of a Regular Matrix Pair (A, B) to Block Hessenberg-Triangular Form

. An algorithm for reduction of a regular matrix pair (A; B) to block Hessenberg-triangular form is presented. This condensed form Q T (A; B)Z = (H; T ), where H and T are block upper Hessenberg and upper triangular, respectively, and Q and Z orthogonal, may serve as a first step in the solution of the generalized eigenvalue problem Ax = Bx. It is shown how an elementwise algorithm can be reorganized in terms of blocked factorizations and higher level BLAS operations. Several ways to annihilate elements are compared. Specifically, the use of Givens rotations, Householder transformations, and combinations of the two. Performance results of the different variants are presented and compared to the LAPACK implementation DGGHRD, which indeed is unblocked. 1 Introduction A reliable way to solve the generalized eigenvalue problem Ax = Bx, where A and B are square real n \Theta n matrices, is to reduce (A; B) to generalized Schur (GS) form (S; T ), where S is upper quasi-triangular and ..