The shifted QR algorithm for Hermitian matrices

An analysis of the shiftedLR algorithm

“…an argument entirely analogous to that used in Dekker and Traub[2] we can proveTheorem 5.5. Class Two shifts are zeros o/ Lanczos Polynomials.…”

“…(5.9) and (5.10) establish the transformation rule for ~s (2). Let E k =E denote the last h columns of the identity matrix.…”

“…In [2], we introduced Class Two shifts and showed that for any Hermitian matrix, the QR algorithm is convergent if second order Class Two shifts are used. Class Two shifts are also members of the family of shift strategies defined by (1.3).…”

Convergence of the shifted $QR$ algorithm for unitary Hessenberg matrices

“…of A (k) in the iterating process converges to zero; in other words, the last row of A (k) tends to a limit form λ [3,13]. As we shall see, convergence of β (k) n−1 depends on how we choose the shift sequence λ (k) , and the selection of an efficient shift strategy is of crucial importance to the implementation of the algorithm.…”

“…In the Hermitian tridiagonal case a constructive proof for the global convergence of the shifted QR algorithm was obtained by exploiting the connection between QR and inverse iteration [3,8,13]. We generalize this approach and derive a residual bound for normal Hessenberg matrices.…”