The Generalized Toda Flow, the QR Algorithm and the Center Manifold Theory

Abstract

A continuous version of the classical QR algorithm, known as the Toda flow, is generalized to complex-valued, full and nonsymmetric matrices. It is shown that this generalized Toda flow, when sampled at integer times, gives the same sequence of matrices as the OR algorithm applied to the matrix exp $(G(X_0 ))$. When $G(X) = X$, global convergence is deduced for the case of distinct real eigenvalues. This convergence property can also be understood locally by the center manifold theory. It is shown that the manifold of upper triangular matrices with decreasing main diagonal entries is the stable center manifold for the Toda flow. One interesting example is given to demonstrate geometrically the dynamical behavior of this flow.