On Rutishauser's approach to self-similar flows

Certain variants of the Toda flow are continuous analogues of the $QR$ algorithm and other algorithms for calculating eigenvalues of matrices. This was a remarkable discovery of the early eighties. Until very recently contemporary researchers studying this circle of ideas have been unaware that continuous analogues of the quotient-difference and $LR$ algorithms were already known to Rutishauser in the fifties. Rutishauser's continuous analogue of the quotient-difference algorithm contains the finite, nonperiodic Toda flow as a special case. A nice feature of Rutishauser's approach is that it leads from the (discrete) eigenvalue algorithm to the (continuous) flow by a limiting process. Thus the connection between the algorithm and the flow does not come as a surprise. In this paper it is shown how Rutishauser's approach can be generalized to yield large families of flows in a natural manner. The flows derived include continuous analogues of the $LR$, $QR$, $SR$, and $HR$ algorithms.