Scaled Toda-like flows (Q1347227)

This very interesting paper discusses the solution of various scaled Toda-like flows of the form \(x' = [x, A\circ x]\), \(x(0) = x_ 0\), where \([ , ]\) is a Lie bracket and \(\circ\) a Hadamard product. The presence of \(A\) allows arbitrary and independent scaling for each element of the matrix \(x\). Various choices of \(A\) are considered which lead, for example, to the continuous power method and the QR algorithm as applied to \(x_ 0\). A convergence theorem is given which allows \(A\) to be chosen in such a way that the convergence of the classical Toda flow is accelerated. This paper concludes with various choices of \(A\) aimed at aggregating the eigenvalues of \(x_ 0\) into blocks.