Shift policies in QR-like algorithms and feedback control of self-similar flows

Abstract

FG algorithms. FG algorithms are generalizations of the well-known QR algorithm for calculating the eigenvalues of a matrix. Let F and g two closed subgroups of the general linear group GL _n (F) (F = ℝ or ℂ). Assuming F ∩ g = {I}, each matrix A ∈ GL _n (F)) has at most one factorization of the form A = FG, where F ∈ F and G ∈ g. Starting from a given matrix B ₀ ∈ GL _n (F)), the FG algorithm produces a sequence of matrices B _m , m = 1,2,…, as follows: B _i is factored into a product B _i = F _i+1 G _i+1 and this product is reversed to define B _{i +1}:= G _{i +1} F _{i +1}. Thus

$${B_i} = {F_{i + 1}}{G_{i + 1}} \Rightarrow {B_{i + 1}}: = {G_{i + 1}}{F_{i + 1}}.$$

((46.1))