  The Toda equation is one of the most famous integrable systems, and its
time-discretization is simply the recursion formula of the quotient-difference
(qd) algorithm for computing eigenvalues of tridiagonal matrices. An extension
of the Toda equation is the q-Toda equation, which is derived by replacing
standard derivatives with the so-called q-derivatives involving a parameter q
such that 0 < q < 1. In our previous paper, we showed that a discretization of
the q-Toda equation is shown to be also applicable to computing tridiagonal
eigenvalues. In this paper, we consider another extension of the q-discrete
Toda equation and find an application to computing eigenvalues of Hessenberg
totally nonnegative (TN) matrices, which are matrices where all minors are
nonnegative. There are two key components to our approach. First, we consider
the extended q-discrete equation from the perspective of shifted LR
transformations, similarly to the discrete Toda and its q-analogue cases.
Second, we clarify asymptotic convergence as discrete-time goes to infinity in
the qdiscrete Toda equation by focusing on TN properties. We also present two
examples to numerically verify convergence to Hessenberg TN eigenvalues
numerically.
