Integrable Systems Related to Matrix LR Transformations

Abstract

The discrete Toda (dToda) equation, which is a representative integrable system, is the recursion formula of the well-known quotient-difference algorithm for computing the eigenvalues of tridiagonal matrices. In other words, the dToda equation is related to the LR transformations of tridiagonal matrices. In this chapter, by extending the application of LR transformations from tridiagonal to Hessenberg matrices, we capture the discrete hungry Toda (dhToda) and discrete relativistic Toda (drToda) equations, which are extensions of the dToda equation from the perspective of LR transformations. From the LR perspective, we identify further extensions of the dhToda equations, and clarify the relationship between the drToda equation and the discrete hungry Lotka–Volterra system. We also demonstrate that ultradiscrete versions of discrete integrable systems related to the LR transformations can be used to compute the eigenvalues of matrices over min-plus algebra, as can discrete integrable systems over linear algebra. These ultradiscrete integrable systems are expected to be equations of motion for box-and-ball systems (BBSs), which are cellular automata that describe mobility phenomena. Thus, we present an example of utilizing the LR perspective to design a new BBS.

1 Introduction

The definition of integrable systems has been the subject of various discussions. However, in applied mathematics, integrable systems can be regarded as a general term for systems that precisely grasp some concept, as well as for systems whose solutions can be written explicitly. One important development in the study of such integrable systems is soliton theory [4, 5, 12], which made great strides in the latter half of the twentieth century. During an interaction, soliton waves change their shape and velocity when the fast wave overtakes the slow wave. However, after an interaction, both waves return to their pre-interaction states. Soliton equations that represent these interactions can be discretized with respect to the time variable by skillful manners specific to integrable systems. The motions of soliton waves are typically drawn on a computer. The discrete soliton equations can be further simplified by quantization on the velocity of the soliton. This procedure, called the ultradiscretization, has resulted in the so-called box-and-ball systems (BBSs) [31], a type of cellular automaton. Cellular automata summarize various discrete phenomena by defining state-change rules for cells packed in a lattice. For example, they are useful for analyzing traffic flows to capture the motion of multiple cars. Therefore, we expect that BBSs will enrich the mathematical representation of mobility phenomena.

Fig. 3.1

An example of discrete-time evolution from $n=0$ to $n=6$ of the BBS

In the first proposed BBS [31], each ball, in order from left to right, moves to the nearest empty box to its right along an infinite number of boxes, arranged in a straight line without gaps. An example of the discrete-time evolutions of this BBS is shown in Fig. 3.1. Interestingly, the velocity of each soliton translates into the number of successive balls in each ball group, which can be determined by observation. We can describe the motion of m ball groups under a discrete-time evolution from n to $n+1$ using the ultradiscrete Toda (udToda) equation:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle Q_k^{(n+1)}=\min \left( \sum _{i=1}^kQ_i^{(n)}-\sum _{i=1}^{k-1}Q_i^{(n+1)},E_k^{(n)}\right) , \quad k=1,2,\dots ,m,\\[12pt] Q_k^{(n+1)}+E_k^{(n+1)}=Q_{k+1}^{(n)}+E_k^{(n)},\quad k=1,2,\dots ,m-1,\\ E_0^{(n)}:=+\infty ,\quad E_m^{(n)}:=+\infty , \end{array}\right. \end{aligned}$$

(3.1)

where $Q_k^{(n)}$ and $E_k^{(n)}$ are the number of successive balls in the kth ball group from the left and the number of successive boxes in the box array between the kth and $(k+1)$th ball groups at discrete-time n, respectively. The udToda equation (3.1) is derived from the ultradiscretization of the discrete Toda (dToda) equation [13]:

$$\begin{aligned} \left\{ \begin{array}{l} q_k^{(n+1)}+e_{k-1}^{(n+1)}=q_k^{(n)}+e_k^{(n)},\quad k=1,2,\dots ,m,\\ q_k^{(n+1)}e_k^{(n+1)}=q_{k+1}^{(n)}e_k^{(n)},\quad k=1,2,\dots ,m-1,\\ e_0^{(n)}:=0,\quad e_m^{(0)}:=0, \end{array}\right. \end{aligned}$$

(3.2)

which is a famous discrete integrable system and has branches in fields other than mathematical physics. The soliton behavior in the BBS ascribes conserved quantities that remain unchanged under the discrete-time evolution in the udToda equation (3.1). Of course, these conserved quantities are derived from the dToda equation (3.2). We can capture conserved quantities of the dToda equation (3.2) in the following matrix representation that the dToda equation (3.2) satisfies, which is called the Lax representation:

$$\begin{aligned} & L^{(n+1)}R^{(n+1)}=R^{(n)}L^{(n)},\end{aligned}$$

(3.3)

$$\begin{aligned} & L^{(n)}:=\left[ \begin{array}{cccc} 1 &{} &{} &{} \\ e_1^{(n)} &{} 1 &{} &{} \\ &{} \ddots &{} \ddots &{} \\ {} &{} &{} e_{m-1}^{(n)} &{} 1 \end{array}\right] ,\quad R^{(n)}:=\left[ \begin{array}{cccc} q_1^{(n)} &{} 1 &{} &{} \\ {} &{} q_2^{(n)} &{} \ddots &{} \\ &{} &{} \ddots &{} 1 \\ {} &{} &{} &{} q_m^{(n)} \end{array}\right] . \end{aligned}$$

(3.4)

Using the equality of each entry in the Lax representation (3.3), we can easily obtain the dToda equation (3.2). Note that $R^{(n)}$ is nonsingular. Consider $A^{(n)}:=L^{(n)}R^{(n)}$ as a tridiagonal matrix. Then we obtain $A^{(n+1)}=R^{(n)}A^{(n)}(R^{(n)})^{-1}$, which implies that the dToda equation yields the matrix transformations from $A^{(n)}$ to $A^{(n+1)}$ without changing the eigenvalues. Thus, Lax representations are a good starting point for understanding other soliton BBSs.

Recapturing the Lax representation (3.3) in terms of a numerical technique in linear algebra can be regarded as an LR transformation [24], which first decomposes $A^{(n)}$ into a product of lower and upper bidiagonal matrices and then generates $A^{(n+1)}$ by inverting the product. The dToda equation can thus generate LR transformations of tridiagonal matrices. In fact, the dToda equation (3.2) is the recursion formula of the well-known quotient-difference (qd) algorithm [24], which computes symmetric tridiagonal eigenvalues. The qd algorithm was originally designed based on a sequence of tridiagonal LR transformations.

The discrete Lotka–Volterra (dLV) system [33] also has a close relationship with tridiagonal LR transformations. This is consistent with the dToda equation and dLV system being linked by a variable transformation called the Bäcklund transformation (for the dToda-dLV case, this is specifically the Miura transformation). The dLV system is a time-discretization of the predator-prey Lotka–Volterra (LV) system, and is expressed using the time-discretization parameter $\delta ^{(n)}$ as follows:

$$\begin{aligned} \left\{ \begin{array}{l} u_k^{(n+1)}(1+\delta ^{(n+1)}u_{k-1}^{(n+1)})=u_k^{(n)}(1+\delta ^{(n)}u_{k+1}^{(n)}),\quad k=1,2,\dots ,2m-1,\\ u_0^{(n)}:=0,\quad u_{2m}^{(n)}:=0, \end{array}\right. \end{aligned}$$

(3.5)

where $u_k^{(n)}$ corresponds to the number of members of the kth species at discrete-time n. For the kth species, the dLV system (3.5) assumes that the $(k-1)$th species is the predator and the $(k+1)$th species is the prey. A time-delay version of the dLV system (3.5) has been formulated by enforcing a time delay before preying and being preyed on; this is related to multifold LR transformations of tridiagonal matrices [25]. Similar to dToda, the dLV and delayed dLV systems can be applied to compute the singular values of bidiagonal matrices, which is the mathematical equivalent of computing the eigenvalues of symmetric tridiagonal matrices. Here, we emphasize that the LR transformation perspective transforms nonlinear phenomena into linear algebra.

We now return to the topic of the BBSs. An extension of a simple BBS is a BBS with numbered balls, where each ball performs a more complicated motion than in the simple BBS [32]. This is called a numbered BBS (nBBS) and differs from the simple BBS because it is not designed based on the time-discretization of an integrable system and its ultradiscretization. The discrete integrable system associated with the nBBS was actually derived from the inverse ultradiscretization of its equation of motion. Considering the hungry extensions of the LV system [2, 14, 30] and its discretization, the resulting equation was named the discrete hungry Toda (dhToda) equation. Hungry LV (hLV) systems, sometimes referred to as Bogoyavlenskii lattices, describe multiple predator-prey interactions, which describes one species preying on multiple species because it is hungrier.

The LR transformation perspective plays a key role in determining the Bäcklund transformation between the dhToda equation and one discrete hLV (dhLV) system, as does formulating another type of dhToda equation that is different from the nBBS-origin equation and is linked to the other dhLV system. In addition, the LR transformation perspective is useful to clarify continuous-time analogues of dhToda equations [27]. Note that the Lax representation is equivalent to the related LR transformation for dToda, but not for dhTodas. With either approach, the dToda equation is translated into a single-matrix equation. In contrast, Lax representations of the dhToda equations are a single-matrix equation but LR transformations related to the dhToda equations are expressed using a multiple-matrix equation. Moreover, by combining the LR transformations with another matrix transformation, we can naturally derive nonautonomous versions of the dhToda equations, which are the dhToda equations with an arbitrary parameter.

The remainder of this chapter is organized as follows. In Sect. 3.2, we describe discrete hungry integrable systems, which are dhToda equations, their nonautonomous versions, and dhLV systems, using the framework of LR transformation. We also describe the relationship between discrete hungry integrable systems and computing matrix eigenvalues. In Sect. 3.3, we relate the discrete relativistic Toda (drToda) equation to LR transformations, and clarify the link between the drToda equation and dhLV system by taking advantage of the flexibility of the LR transformation perspective. In Sect. 3.4, by considering ultradiscrete analogues of the L and R factors in LR transformations, we demonstrate that the udToda and udhToda equations generate matrix transformations that preserve an eigenvalue over min-plus algebra and can be used to compute the eigenvalues of the min-plus matrices. These properties correspond to the simple BBS and nBBS having conserved quantities under discrete-time evolution. In Sect. 3.5, we present an nBBS based on a dhToda equation derived from the LR perspective. Finally, in Sect. 3.6, we give concluding remarks.

2 Discrete Hungry Integrable Systems

In this section, we first relate two types of dhToda equations to the LR transformations of Hessenberg matrices. We then derive nonautonomous extensions of the dhToda equations by introducing shifts in the LR transformations and show their relationship to dhLV systems. We also describe the asymptotic convergence of discrete hungry integrable systems as discrete-time tends to infinity.

2.1 Discrete Hungry Toda Equations

An extension of the dToda equation, the dhToda equation [32] is written as

$$\begin{aligned} \left\{ \begin{array}{ll} q_k^{(n+M)}+e_{k-1}^{(n+1)}=q_k^{(n)}+e_k^{(n)},\quad k=1,2,\dots ,m,\\ q_k^{(n+M)}e_{k}^{(n+1)}=q_{k+1}^{(n)}e_k^{(n)},\quad k=1,2,\dots ,m-1,\\ e_0^{(n)}:=0,\quad e_m^{(n)}:=0,\quad n=0,1,\dots , \end{array}\right. \end{aligned}$$

(3.6)

where M is a positive integer. We refer to (3.6) as the dhToda-I equation to distinguish it from other dhToda equations that will appear later. Note that the dhToda-I equation with $M=1$ coincides with the dToda equation (3.2). In general, the values of q and e are uniquely determined at every discrete-time n if $\{q_k^{(0)}\}_{k=1}^m,\{q_k^{(1)}\}_{k=1}^m,\dots ,\{q_k^{(M-1)}\}_{k=1}^m$ and $\{e_k^{(0)}\}_{k=1}^{m-1}$ are given. Using (3.4), we obtain the Lax representation of the dhToda-I equation:

$$\begin{aligned} L^{(n+1)}R^{(n+M)}=R^{(n)}L^{(n)},\quad n=0,1,\dots . \end{aligned}$$

(3.7)

By applying the discrete-time evolutions generated by (3.7) to the upper Hessenberg matrix $A^{(n)}:=L^{(n)}R^{(n+M-1)}R^{(n+M-2)}\cdots R^{(n)}$, we derive

$$\begin{aligned} A^{(n+1)}=R^{(n)}A^{(n)}(R^{(n)})^{-1},\quad n=0,1,\dots . \end{aligned}$$

(3.8)

This implies that the dhToda-I equation generates $A^{(1)},A^{(2)},\dots$ with the same eigenvalues as those of $A^{(0)}$. We can rewrite the similarity transformation (3.8) as

$$\begin{aligned} \left\{ \begin{array}{l} A^{(n)}=L^{(n)}(R^{(n+M-1)}R^{(n+M-2)}\cdots R^{(n)}),\\ (R^{(n+M-1)}R^{(n+M-2)}\cdots R^{(n)})L^{(n)}=A^{(n+M)}. \end{array}\right. \end{aligned}$$

(3.9)

Thus, the dhToda-I equation first decomposes $A^{(n)}$ into the product of a lower bidiagonal L and multiple upper bidiagonal R factors, and then constructs $A^{(n+M)}$ by swapping the L and R factors M times. Moreover, by observing the entries of $A^{(n)}$, the asymptotic convergence as $n\rightarrow \infty$ of this sequence of LR transformations can be used to compute the eigenvalues of $A^{(0)}$.

Theorem 3.1

(Fukuda et al. [7]) Suppose that $A^{(0)}$ is totally nonnegative (TN), that is, $q_k^{(j)}>0$ for $j=0,1,\dots ,M-1$ and $e_k^{(0)}>0$. Then, $A^{(n)}$ converges to an upper triangular matrix as $n\rightarrow \infty$ and

$$\begin{aligned} & \lim _{n\rightarrow \infty }q_k^{(n)}q_k^{(n-1)}\cdots q_k^{(n-M+1)}=\lambda _k(A^{(0)}),\quad k=1,2,\dots ,m,\\ & \lim _{n\rightarrow \infty }e_k^{(n)}=0,\quad k=1,2,\dots ,m-1, \end{aligned}$$

where $\lambda _k(A^{(0)})$ denotes the kth largest eigenvalue of $A^{(0)}$.

Another hungry extension of the dToda equation is the dhToda-II equation [21] and is expressed as

$$\begin{aligned} \left\{ \begin{array}{l} q_k^{(n+1)}+e_{k-1}^{(n+M)}=q_k^{(n)}+e_k^{(n)},\quad k=1,2,\dots ,m,\\ q_k^{(n+1)}e_k^{(n+M)}=q_{k+1}^{(n)}e_k^{(n)},\quad k=1,2,\dots ,m-1,\\ e_0^{(n)}:=0,\quad e_m^{(n)}:=0,\quad n=0,1,\dots . \end{array}\right. \end{aligned}$$

(3.10)

As for the dhToda-I equation, the dhToda-II equation with $M=1$ is the dToda equation (3.2). However, the dhToda-II equation differs from the dhToda-I equation in that the initial values required to uniquely determine all q and e are $\{q_k^{(0)}\}_{k=1}^m$ and $\{e_k^{(0)}\}_{k=1}^{m-1},\{e_k^{(1)}\}_{k=1}^{m-1},\dots ,\{e_k^{(M-1)}\}_{k=1}^{m-1}$ rather than $\{q_k^{(0)}\}_{k=1}^m,\{q_k^{(1)}\}_{k=1}^m,\dots ,\{q_k^{(M-1)}\}_{k=1}^m$ and $\{e_k^{(0)}\}_{k=1}^{m-1}$. Additionally, in the Lax representation of the dhToda-II equation, an arbitrary M appears in the superscript of the L factor instead of the R factor, as follows:

$$\begin{aligned} L^{(n+M)}R^{(n+1)}=R^{(n)}L^{(n)},\quad n=0,1,\dots . \end{aligned}$$

(3.11)

Preparing the lower Hessenberg matrix $A^{(n)}:=L^{(n)}L^{(n+1)}\cdots L^{(n+M-1)}R^{(n)}$ shows that the dhToda-II equation yields a sequence of matrices with the same eigenvalues as those of $A^{(0)}$. This is because (3.11) leads to $A^{(n+1)}={(L^{(n)})}^{-1} A^{(n)}L^{(n)}$ for $n=0,1,\dots$. Moreover, (3.11) leads to

$$\begin{aligned} \left\{ \begin{array}{l} A^{(n)}=(L^{(n)}L^{(n+1)}\cdots L^{(n+M-1)})R^{(n)},\\ R^{(n)}(L^{(n)}L^{(n+1)}\cdots L^{(n+M-1)})=A^{(n+M)}, \end{array}\right. \end{aligned}$$

where the number of L and R factors is reversed compared with in (3.9). Here, we emphasize that (3.10) was originally formulated from an LR transformation rather than realistic dynamics. As $n\rightarrow \infty$, the dhToda-II equation exhibits asymptotic convergence similar to that described in Theorem 3.1.

Theorem 3.2

(Nishiyama et al. [21]) Suppose that $A^{(0)}$ is TN, that is, $q_k^{(0)}>0$ and $e_k^{(j)}>0$ for $j=0,1,\dots ,M-1$. Then, $A^{(n)}$ converges to a lower triangular matrix as $n\rightarrow \infty$ and

$$\begin{aligned} & \lim _{n\rightarrow \infty }q_k^{(n)}=\lambda _k(A^{(0)}),\quad k=1,2,\dots ,m,\\ & \lim _{n\rightarrow \infty }e_k^{(n)}=0,\quad k=1x,2,\dots ,m-1. \end{aligned}$$

2.2 Nonautonomous Discrete Hungry Integrable Systems

We now consider the LR transformations of upper Hessenberg matrices. Introducing shifts into the LR transformations is useful for accelerating convergence. A shifted version of (3.9) is given by

$$\begin{aligned} \left\{ \begin{array}{l} A^{(n)}-s^{(n)}I=\hat{L}_0^{(n)}\hat{R}^{(n)},\\ \hat{R}^{(n)}\hat{L}_0^{(n)}+s^{(n)}I=A^{(n+M)}, \end{array}\right. \end{aligned}$$

(3.12)

where $\hat{L}_0^{(n)}$ is a lower bidiagonal matrix, $\hat{R}^{(n)}$ is an upper triangle matrix, and $s^{(n)}$ denotes the shift parameter. Since $A^{(n+M)}=(\hat{L}_0^{(n)})^{-1}{A}^{(n)}\hat{L}_0^{(n)}$ in (3.12), the eigenvalues of $A^{(n+M)}$ are identical to those of $A^{(n)}$. Such a shift, which does not change the eigenvalues, is called an “implicit” shift. In contrast, an “explicit” shift changes the eigenvalues. We can verify that the implicit-shift LR transformation (3.12) is generated by a pair of LR and LL transformations, such that

$$\begin{aligned} \left\{ \begin{array}{l} \hat{L}_{\ell +1}^{(n)} R^{(n+M+\ell )}=R^{(n+\ell )}\hat{L}_{\ell }^{(n)},\quad \ell =0,1,\dots ,M-1,\\ \hat{L}_0^{(n,0)}L^{(n+M)}=L^{(n)}\hat{L}_M^{(n)}, \end{array}\right. \end{aligned}$$

(3.13)

where $\hat{L}_\ell ^{(n)}$ are bidiagonal matrices involving new variables $\hat{e}_{\ell ,k}^{(n)}$ and are given by

$$\begin{aligned} \hat{L}_{\ell }^{(n)}=\left[ \begin{array}{cccc} 1 &{} &{} &{} \\ \hat{e}_{\ell ,1}^{(n)} &{} 1 &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} \hat{e}_{\ell ,m-1}^{(n)} &{} 1 \end{array}\right] . \end{aligned}$$

Using the entry identities and boundary conditions in (3.13), we can derive a recursion formula involving an arbitrary $s^{(n)}$:

$$\begin{aligned} \left\{ \begin{array}{l} q_k^{(n+M+\ell )}+\hat{e}_{\ell +1,k-1}^{(n)}=q_k^{(n+\ell )}+\hat{e}_{\ell ,k}^{(n)}, \quad k=1,2,\dots ,m,\quad \ell =0,1,\dots ,M-1,\\ q_k^{(n+M+\ell )}\hat{e}_{\ell +1,k}^{(n)}=q_{k+1}^{(n+\ell )}\hat{e}_{\ell ,k}^{(n)}, \quad k=1,2,\dots ,m,\quad \ell =0,1,\dots ,M-1,\\ e_k^{(n+M)}+\hat{e}_{0,k}^{(n)}=e_k^{(n)}+\hat{e}_{M,k}^{(n)},\quad k=1,2,\dots ,m-1,\\ \hat{e}_{0,1}^{(n)}=\dfrac{q_1^{(n)}q_1^{(n+1)}\cdots q_1^{(n+M-1)}}{q_1^{(n)}q_1^{(n+1)}\cdots q_1^{(n+M-1)}-s^{(n)}}e_1^{(n)},\\ e_k^{(n+M)}\hat{e}_{0,k+1}^{(n)}=e_{k+1}^{(n)}\hat{e}_{M,k}^{(n)},\quad k=1,2,\dots ,m-1\\ \hat{e}_{\ell ,0}^{(n)}:=0,\quad \hat{e}_{\ell ,m}^{(n)}:=0,\quad \ell =0,1,\dots ,M-1,\\ e_0^{(n)}:=0,\quad e_m^{(n)}:=0. \end{array}\right. \end{aligned}$$

(3.14)

Since (3.14) with $s^{(n)}=0$ simplifies to the dhToda-I equation, we can regard (3.14) as a nonautonomous dhToda-I equation with an arbitrary parameter. As $n\rightarrow \infty$, the nonautonomous dhToda-I equation exhibits the same asymptotic convergence to the upper Hessenberg eigenvalues as (3.6).

Theorem 3.3

(Fukuda et al. [10]) Suppose that $A^{(0)}$ is TN, that is, $q_k^{(0)}>0$ and $e_k^{(j)}>0$ for $j=0,1,\dots ,M-1$. If $s^{(nM)}<\lambda _m(A^{(0)})$ for $n=0,1,\dots$, then the following holds:

$$\begin{aligned} & \lim _{\ell \rightarrow \infty }q_k^{(\ell M)}q_k^{(\ell M+1)}\cdots q_k^{(\ell M+M-1)}=\lambda _k(A^{(0)}), \quad k=1,2,\dots ,m,\\ & \lim _{\ell \rightarrow \infty }e_k^{(\ell M)}=0,\quad k=1,2,\dots ,,m-1. \end{aligned}$$

The shift strategy and its effect on the computation of eigenvalues can be found in [11].

Similar to the dhToda-I equation, considering the implicit-shift LR transformation of the lower Hessenberg matrices yields the nonautonomous dhToda-II equation with an arbitrary $s^{(n)}$ [28]:

$$\begin{aligned} \left\{ \begin{array}{l} e_{\ell ,k-1}^{(n+1)}+\hat{q}_{\ell +1,k}^{(n)}=e_{\ell ,k}^{(n)}+\hat{q}_{\ell ,k}^{(n)}, \quad k=1,2,\dots ,m,\quad \ell =0,1,\dots ,M-1,\\ e_{\ell ,k}^{(n+1)}\hat{q}_{\ell +1,k}^{(n)}=e_{\ell ,k}^{(n)}\hat{q}_{\ell ,k+1}^{(n)}, \quad k=1,2,\dots ,m-1,\quad \ell =0,1,\dots ,M-1,\\ q_k^{(n+1)}+\hat{q}_{0,k+1}^{(n)}=q_{k+1}^{(n)}+\hat{q}_{M,k}^{(n)}, \quad k=1,2,\dots ,m,\\ \hat{q}_{0,1}^{(n)}=q_{1}^{(n)}-s^{(n)},\\ q_k^{(n+1)}\hat{q}_{0,k}^{(n)}=q_k^{(n)}\hat{q}_{M,k}^{(n)},\quad k=1,2,\dots ,m,\\ e_{\ell ,0}^{(n)}:=0,\quad e_{\ell ,m}^{(n)}:=0. \end{array}\right. \end{aligned}$$

(3.15)

We can rewrite the discrete-time evolutions generated by (3.15) using the LR and RR transformations as

$$\begin{aligned} & \left\{ \begin{array}{l} L_{\ell }^{(n+1)}\hat{R}_{\ell +1}^{(n)}=\hat{R}_{\ell }^{(n)}L_{\ell }^{(n)},\quad \ell =0,1,\dots ,M-1,\\ R^{(n+1)}\hat{R}_0^{(n)}=\hat{R}_M^{(n)}R^{(n)}, \end{array}\right. \\ & L_{\ell }^{(n)}:=\left[ \begin{array}{cccc} 1 &{} &{} &{} \\ e_{\ell ,1}^{(n)} &{} 1 &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} e_{\ell ,m-1}^{(n)} &{} 1 \end{array}\right] ,\quad \hat{R}_{\ell }^{(n)}:=\left[ \begin{array}{cccc} \hat{q}_{\ell ,1}^{(n)} &{} 1 &{} &{} \\ &{} \hat{q}_{\ell ,2}^{(n)} &{} \ddots &{} \\ &{} &{} \ddots &{} 1 \\ &{} &{} &{} \hat{q}_{\ell ,m}^{(n)} \end{array}\right] .\nonumber \end{aligned}$$

(3.16)

Moreover, a combination of LR and RR transformations leads to the implicit-shift lower Hessenberg LR transformation from $A^{(n)}$ to $A^{(n+1)}$:

$$\begin{aligned} \left\{ \begin{array}{l} A^{(n)}-s^{(n)}I=\hat{L}^{(n)}\hat{R}_0^{(n)},\\ \hat{R}_0^{(n)}\hat{L}^{(n)}+s^{(n)}I=A^{(n+M)}, \end{array}\right. \end{aligned}$$

where $\hat{L}^{(n)}$ is a lower triangular matrix and $\hat{R}_0^{(n)}$ is an upper bidiagonal matrix. Expressing the solution to the nonautonomous dhToda-II equation using determinants, and examining their asymptotic expansions as $n\rightarrow \infty$, we can clarify the asymptotic convergence for the nonautonomous dhToda-II equation as $n\rightarrow \infty$ without restricting $A^{(0)}$ to being TN.

Theorem 3.4

(Shinjo et al. [28]) If $q_k^{(0)}\ne 0$ and $e_{\ell ,k}^{(0)}\ne 0$ for $\ell =0,1,\dots ,M-1$, then

$$\begin{aligned} & \lim _{n\rightarrow \infty }q_k^{(n)}=\lambda _k(A^{(0)}),\quad k=1,2,\dots ,m,\\ & \lim _{n\rightarrow \infty }e_{\ell ,k}^{(n)}=0,\quad k=1,2,\dots ,m-1,\quad \ell =0,1,\dots ,M-1,\\ & \lim _{n\rightarrow \infty }\hat{q}_{\ell ,k}^{(n)}=\lambda _k(A^{(0)})-s^{*},\quad k=1,2,\dots ,m,\quad \ell =0,1,\dots ,M, \end{aligned}$$

where $s^{*}$ is a constant such that $s^{(n)}\rightarrow s^{*}$ as $n\rightarrow \infty$.

For the nonautonomous dhToda-I equation, we can also obtain a convergence theorem similar to Theorem 3.4 by analyzing the determinant solution.

We now introduce new variables $u_k^{(n)}$ that comprise the nonautonomous dhToda-II variables:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle q_k^{(n)}=u_{(M+1)(k-1)+1}^{(n)}\prod _{j=1}^M(1+\delta ^{(n)} u_{(M+1)(k-1)+1-j}^{(n)}),\\ \displaystyle e_{\ell -1,k}^{(n)}=u_{(M+1)(k-1)+\ell +1}^{(n)}\prod _{j=1}^M(1+\delta ^{(n)} u_{(M+1)(k-1)+j+1-j}^{(n)}),\\ \displaystyle \hat{q}_{\ell ,k}^{(n)}=\frac{1}{\delta ^{(n)}}\prod _{j=0}^M (1+\delta ^{(n)}u_{(M+1)(k-1)+j+1-j}^{(n)}). \end{array}\right. \end{aligned}$$

(3.17)

Applying (3.17) to (3.15), we derive a dhLV system:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle u_k^{(n+1)}\prod _{j=1}^{M}(1+\delta ^{(n+1)}u_{k-j}^{(n+1)}) =u_k^{(n)}\prod _{j=1}^M(1+\delta ^{(n)}u_{k+j}^{(n)}),\\ \quad k=1,2,\dots ,(M+1)m-M,\\ u_{\ell -M}^{(n)}:=0,\quad u_{(M+1)m+\ell -M}^{(n)}:=0,\quad \ell =1,2,\dots ,M,\quad n=0,1,\dots . \end{array}\right. \end{aligned}$$

(3.18)

Equation (3.18) is the Bäcklund transformation between the nonautonomous dhToda-II equation and dhLV system. We can systematically find this Bäcklund transformation from the perspective of the LR transformation. Details of Bäcklund transformations between the nonautonomous dhToda-I equation and dhLV systems can be found in [9, 26]. Combining these Bäcklund transformations with Theorems 3.3 and 3.4, we can also obtain convergence theorems for dhLV systems.

3 Discrete Relativistic Toda Equation

The relativistic Toda equation is a Poincaré-invariant generalization of the Toda equation. Since its introduction by Ruijsenaars [23], it has received considerable attention, and various studies have been conducted on its integrability [23], special solutions [3, 22], integrable discretization [29] and its solutions [18, 19], Lax representations [29], and application to numerical algorithms [19]. In this section, we consider the following nonautonomous discrete relativistic Toda (drToda) equation [20, 36]:

$$\begin{aligned} \left\{ \begin{array}{l} a_k^{(n+1)}+s^{(n+1)}(1+b_k^{(n+1)})=a_k^{(n)}\dfrac{1+b_{k+1}^{(n)}}{1+b_k^{(n)}}+s^{(n)}(1+b_{k+1}^{(n)}),\quad k=0,1,\dots , K-1,\\ a_{k-1}^{(n+1)}b_k^{(n+1)}=a_k^{(n)}b_k^{(n)}\dfrac{1+b_{k+1}^{(n)}}{1+b_k^{(n)}},\quad k=1,2,\dots , K-1,\\ b_0^{(n)}:=0,\quad n=0,1,\dots , \end{array}\right. \end{aligned}$$

(3.19)

where $a_k^{(n)}$ and $b_k^{(n)}$ are variables at the kth grid point at discrete-time n and $s^{(n)}$ is a scalar parameter that can be chosen arbitrarily for each n. We consider this equation from the perspective of a shifted LR transformation so that we can discuss its derivation, Lax representation, relationship with other discrete integrable systems, and conserved quantities in an elementary and unified manner.

3.1 Derivation from the Perspective of a Shifted LR Transformation

We derive the nonautonomous drToda equation from the autonomous drToda equation, which is given by

$$\begin{aligned} \left\{ \begin{array}{l} u_k^{(n)}+v_k^{(n+1)}=u_k^{(n+1)}+v_{k+1}^{(n)},\quad k=0,1,\dots , K-1,\\ u_k^{(n+1)}v_k^{(n)}=u_{k-1}^{(n)}v_k^{(n+1)},\quad k=1,2,\dots , K-1,\\ v_0^{(n)}:=0,\quad n=0,1,\dots . \end{array}\right. \end{aligned}$$

This resembles the discrete Toda equation, except for the differences in the indices. Its Lax representation is given by

$$\begin{aligned} R^{(n)}(L^{(n)})^{-1}=(L^{(n+1)})^{-1}R^{(n+1)},\quad n=0,1,\dots , \end{aligned}$$

(3.20)

where

$$\begin{aligned} & R^{(n)}=\left[ \begin{array}{cccc} u_0^{(n)} &{} 1 &{} &{} \\ &{} u_1^{(n)} &{} \ddots &{} \\ &{} &{} \ddots &{} 1 \\ &{} &{} &{} u_{K-1}^{(n)} \end{array}\right] ,\quad L^{(n)}=\left[ \begin{array}{cccc} 1 &{} &{} &{} \\ v_1^{(n)} &{} 1 &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} v_{K-1}^{(n)} &{} 1 \end{array}\right] . \end{aligned}$$

(3.21)

Equation (3.20) can be viewed as an LR transformation of the structured matrix $A^{(n)}:=\left( L^{(n)}\right) ^{-1}R^{(n)}$.

We now introduce a shift into the LR transformation (3.20). To simplify the notation, we denote the variables at discrete-time n without an asterisk, and those at discrete-time $n+1$ with an asterisk. The shifted LR transformation can then be written as follows:

$$\begin{aligned} & A-s I=\breve{L}\breve{R}, \end{aligned}$$

(3.22)

$$\begin{aligned} & \breve{R}\breve{L}+s I=A^*. \end{aligned}$$

(3.23)

Here, (3.22) decomposes the shifted matrix $A-s I$ into a product of L and R factors, and (3.23) forms the matrix $A^*$ at the next time step by reversing the order of the product and re-adding the shift. Interestingly, we can show that if the matrix A can be written as $A=L^{-1}R$, then $A^*$ can also be written as $A^*=(L^*)^{-1}R^*$, where $L^*$ and $R^*$ are of the form of (3.21). Furthermore, the structures of $\breve{L}$ and $\breve{R}$ can be identified as follows.

Lemma 3.1

(Yamamoto et al. [36]) Let $L=I+L^{\prime }$ and $R=D+J$, where J is an upper bidiagonal matrix with zeros on the diagonal and ones on the upper sub-diagonal. Then, the matrices $\breve{L}$ and $\breve{R}$ defined by (3.22) with $A=L^{-1}R$ can be expressed as

$$\begin{aligned} & \breve{R}=\breve{D}+J,\end{aligned}$$

(3.24)

$$\begin{aligned} & \breve{L}=(I+L^{\prime })^{-1}(I-s L^{\prime }\breve{D}^{-1}), \end{aligned}$$

(3.25)

where $\breve{D}$ is the solution to the matrix equation:

$$\begin{aligned} \breve{D}=D-s(I-L^{\prime }\breve{D}^{-1}J). \end{aligned}$$

(3.26)

Now, we can write the two steps of the shifted LR transformation (3.22) and (3.23) as

$$\begin{aligned} & L^{-1}R-s I=\breve{L}\breve{R}, \end{aligned}$$

(3.27)

$$\begin{aligned} & \breve{R}\breve{L}+s I=(L^*)^{-1}R^*, \end{aligned}$$

(3.28)

$$\begin{aligned} & (L^*)^{-1}R^*-s^*I=\breve{L}^*\breve{R}^*, \end{aligned}$$

(3.29)

$$\begin{aligned} & \breve{R}^*\breve{L}^*+s^*I=(L^{**})^{-1}R^{**}, \end{aligned}$$

(3.30)

where the symbols with double asterisks denote the variables at discrete-time $n+2$. Then, we can regard $\breve{L}$ and $\breve{R}$ as basic variables and consider (3.28) and (3.29) as the equations that govern their discrete-time evolutions. Equations (3.28) and (3.29) can be rewritten as

$$\begin{aligned} \breve{R}\breve{L}-(s^*-s)I=\breve{L}^*\breve{R}^*. \end{aligned}$$

(3.31)

This can be viewed as an explicitly shifted LR transformation with the shift $s^*-s$. By changing the variables from $L^{\prime }$ to $\tilde{L}^{\prime }:=-L^{\prime }\breve{D}^{-1}$, we can rewrite (3.25) as

$$\begin{aligned} \breve{L}=(I-\tilde{L}^{\prime }\breve{D})^{-1}(I+s\tilde{L}^{\prime }). \end{aligned}$$

(3.32)

By substituting (3.24) and (3.32) into (3.31), we can express the explicitly shifted LR transformation in terms of $\breve{D}$ and $\tilde{L}^{\prime }$:

$$\begin{aligned} & (\breve{D}+J)(I-\tilde{L}^{\prime }\breve{D})^{-1}(I+s \tilde{L}^{\prime })+s I \nonumber \\ & \quad =(I-\tilde{L}^{*\prime }\breve{D}^*)^{-1}(I+s^*\tilde{L}^{*\prime })(\breve{D}^*+J)+s^*I. \end{aligned}$$

(3.33)

Although this appears to be a complicated matrix equality, we can show that if the equalities between the diagonal entries and between the lower sub-diagonal entries hold, then equality automatically holds for the entire matrix. This leads to the following lemma.

Lemma 3.2

(Yamamoto et al. [36]) Equation (3.33) is a combination of the following two equations:

$$\begin{aligned} & (I+J\tilde{L}^{\prime })(\breve{D}+s I) =(\breve{D}^*+s^* I)+\tilde{L}^{*\prime }(\breve{D}^*+s^* I)J, \end{aligned}$$

(3.34)

$$\begin{aligned} & (I+J\tilde{L}^{\prime })(\breve{D}\tilde{L}^{\prime })=(\tilde{L}^{*\prime }\breve{D}^*)(I+J\tilde{L}^{\prime }). \end{aligned}$$

(3.35)

Here, (3.34) and (3.35) represent the equality between the diagonal entries and between the lower sub-diagonal entries, respectively.

To rewrite (3.34) and (3.35) entry-by-entry, let

$$\begin{aligned} & \breve{D}=\left[ \begin{array}{cccc} a_0 &{} &{} &{} \\ &{} a_1 &{} &{} \\ &{} &{} \ddots &{} \\ &{} &{} &{} a_{K-1} \end{array} \right] ,\quad \tilde{L}^{\prime }=\left[ \begin{array}{cccc} 0 &{} &{} &{} \\ b_1 &{} 0 &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} b_{K-1} &{} 0 \end{array} \right] ,\nonumber \\ & b_0=b_K=0, \end{aligned}$$

(3.36)

and assume the same expressions for the variables with asterisks. Then, (3.34) and (3.35) can be rewritten as

$$\begin{aligned} & (1+b_{k+1})(a_k+s)=(a_k^*+s^*)+b_k^*(a_{k-1}^*+s^*),\quad k=0,1,\dots ,K-1,\end{aligned}$$

(3.37)

$$\begin{aligned} & (1+b_{k+1})a_kb_k=a_{k-1}^*b_k^*(1+b_k),\quad k=1,2,\dots ,K-1. \end{aligned}$$

(3.38)

Equation (3.38) is the second equation of (3.19). Rewriting (3.37) and using (3.38), we obtain

$$\begin{aligned} a_k^*+s^*(1+b_k^*) &=a_k(1+b_{k+1})-a_{k-1}^* b_k^*+s(1+b_{k+1}) \nonumber \\ &=a_k(1+b_{k+1})-a_k b_k\,\dfrac{1+b_{k+1}}{1+b_k}+s(1+b_{k+1}) \nonumber \\ &=a_k\dfrac{1+b_{k+1}}{1+b_k}+s(1+b_{k+1}),\quad k=0,1,\dots ,K-1, \end{aligned}$$

(3.39)

which is the first equation of (3.19). We summarize this observation as the main theorem of this section.

Theorem 3.5

([36]) One step of the nonautonomous drToda equation is equivalent to the explicitly shifted LR transformation (3.31), where $\breve{L}$ and $\breve{R}$ are given by (3.32) and (3.24), respectively, and $\breve{D}$ and $\tilde{L}^{\prime }$ are given by (3.36).

This theorem enables us to derive a condition on s to ensure the absence of a breakdown in the discrete-time evolution and the positivity of the variables. Further details can be found in [36].

3.2 Relationship with the Discrete Hungry Lotka–Volterra System

From the discussion in Sect. 3.3.1, we know that the nonautonomous drToda equation can be expressed by two equivalent systems. The first is to use $\breve{L}$ and $\breve{R}$ (or $\breve{D}$ and $\tilde{L}^{\prime }$) as the basic variables and (3.28) and (3.29) (or (3.38) and (3.39)) as the equations for discrete-time evolution. The second is to use L and R as the basic variables and use (3.27) and (3.28) for the discrete-time evolution. These two systems can be interchanged using (3.27) as the change-of-variables formula.

Using this fact, we can construct a Bäcklund transformation between the drToda equation (3.19) and dhLV system (3.18). To illustrate this, note that the dhLV system can be expressed as the following implicitly shifted LR transformation [6]:

$$\begin{aligned} \left\{ \begin{array}{l} L_1L_2\cdots L_M R-s I=\breve{L}\breve{R},\\ \breve{R}\breve{L}+s I=L_1^*L_2^*\cdots L_M^*R^*, \end{array}\right. \end{aligned}$$

where $L_i$ for $i=1,2,\dots ,M$ and R are given by

$$\begin{aligned} L_i=\left[ \begin{array}{cccc} 1 &{} &{} &{} \\ e_{i,1} &{} 1 &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} e_{i,m-1} &{} 1 \end{array}\right] ,\quad R=\left[ \begin{array}{cccc} q_1 &{} 1 &{} &{} \\ &{} q_2 &{} \ddots &{} \\ &{} &{} \ddots &{} 1\\ &{} &{} &{} q_m \end{array}\right] . \end{aligned}$$

Consider the special case where $m=K$, $M=K-1$, and only one lower sub-diagonal entry, $e_{i,K-i}$, is nonzero in each $L_i$. Thus, we have

$$\begin{aligned} & L_1L_2\cdots L_M =\left[ \begin{array}{ccccc} 1 &{} &{} &{} &{} \\ -e_{K-1,1} &{} 1 &{} &{} &{} \\ &{} \ddots &{} \ddots &{} &{} \\ &{} &{} -e_{2,K-2} &{} 1 &{} \\ &{} &{} &{} -e_{1,K-1} &{} 1 \end{array}\right] ^{-1}. \end{aligned}$$

For this particular case, the dhLV system has the same form as that of (3.27) and (3.28). The relationship between the dhLV and drToda variables (expressed in terms of L and R) is

$$\begin{aligned} & q_k=u_{k-1},\quad k=1,2,\dots ,K,\end{aligned}$$

(3.40)

$$\begin{aligned} & e_{K-k,k}=-v_k,\quad k=1,2,\dots ,K-1. \end{aligned}$$

(3.41)

The dhLV variables can also be related to the drToda variables expressed in terms of $\breve{D}$ and $\tilde{L}^{\prime }$. To this end, we note the following relationship between the drToda variables:

$$\begin{aligned} & u_k=a_k+s(1+b_k),\quad n=0,1,\dots ,K-1, \end{aligned}$$

(3.42)

$$\begin{aligned} & v_k=-a_{k-1}b_k,\quad k=1,2,\dots ,K-1. \end{aligned}$$

(3.43)

Equation (3.42) is an entry-wise representation of (3.26), while (3.43) follows from $\tilde{L}^{\prime }:=-L^{\prime }\breve{D}^{-1}$. Substituting these equations into (3.40) and (3.41) yields the Bäcklund transformation between the dhLV variables and drToda variables in (3.19):

$$\begin{aligned} & q_k=a_{k-1}+s(1+b_{k-1}),\quad k=1,2,\dots ,K, \end{aligned}$$

(3.44)

$$\begin{aligned} & e_{K-k,k}=a_{k-1}b_k,\quad k=1,2,\dots ,K-1, \end{aligned}$$

(3.45)

$$\begin{aligned} & e_{i,k}=0,\quad i\ne K-k,\,\,k=1,2,\dots ,K-1. \end{aligned}$$

(3.46)

Finally, as an application of this Bäcklund transformation, we present the conserved quantities of the drToda equation. Because the conserved quantities of the dhLV system are known as functions of q and e variables [15], we only need to rewrite them in terms of the drToda variables using (3.45) and (3.46). The result is given by the following theorem.

Theorem 3.6

Let $U_1,U_2,\dots ,U_{2K-1}$ be defined by

$$\begin{aligned} \left\{ \begin{array}{l} U_{2i-1}=a_{i-1}+s(1+b_{i-1}),\quad i=1,2,\dots ,K,\\ U_{2i}=a_{i-1}b_i,\quad i=1,2,\dots ,K-1, \end{array}\right. \end{aligned}$$

and define the relational operator $\ll$ as

$$\begin{aligned} i\ll j\;\;\Leftrightarrow \;\; i+\textrm{mod}(i+1,2)+2\le j. \end{aligned}$$

Then,

$$\begin{aligned} C_p=\sum _{1\le i_1\ll i_2 \ll \cdots \ll i_p\le 2K-1}U_{i_1}U_{i_2}\cdots U_{i_p},\quad p=1,2,\dots ,K \end{aligned}$$

are conserved quantities of the drToda equation (3.19).

Viewing a discrete integrable system from the perspective of a shifted LR transformation makes determining its relationship with other discrete integrable systems and exploiting the results obtained for the latter easier. The Bäcklund transformation and conserved quantities derived in this subsection illustrate the effectiveness of this approach.

4 Ultradiscrete Toda Equation

In this section, we demonstrate that the eigenvalues of matrices can be obtained using the discrete-time evolution of the udToda equation (3.1). The discussion here does not consider eigenvalues over ordinary linear algebra, but over min-plus algebra. In other words, we clarify a min-plus analogue of the qd algorithm for linear eigenvalue computation.

4.1 Min-Plus Algebra

Min-plus algebra is a commutative and idempotent semiring and is closely related to weighted directed graphs constructed with sets of nodes and directed edges with weights. Let $\mathbb {R}_{\min }:=\mathbb {R}\cup \{+\infty \}$. For $a,b\in \mathbb {R}_{\min }$, min-plus algebra has two binary operations:

$$\begin{aligned} \left\{ \begin{array}{l} a\oplus b=\min \{a,b\},\\ a\otimes b=a+b. \end{array}\right. \end{aligned}$$

Here, $\oplus$ and $\otimes$ are both associative and commutative; $\otimes$ is distributive with respect to $\oplus$; and $\varepsilon :=+\infty$ and $e:=0$ are the identities with respect to $\oplus$ and $\otimes$, respectively. Moreover, let $\oslash$ be the inverse of $\otimes$ such that $a\otimes b\oslash b=a$.

The set of all m-by-n min-plus matrices is denoted as $\mathbb {R}_{\min }^{m\times n}$. For $A=(a_{ij}), B=(b_{ij})\in \mathbb {R}_{\min }^{m\times m}$, the matrix sum $A\oplus B =([A\oplus B]_{ij})$ and product $A\otimes B=([A\otimes B]_{ij})$ are respectively given by

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle [A\oplus B]_{ij}=a_{ij}\oplus b_{ij}=\min \{a_{ij},b_{ij}\},\\[3pt] \displaystyle [A\otimes B]_{ij}=\bigoplus _{\ell =1}^{m}(a_{i\ell }\otimes b_{\ell j}) =\min _{\ell =1,2,\dots ,m}\{a_{i\ell }+b_{\ell j}\}. \end{array}\right. \end{aligned}$$

The following definition provides reasonable min-plus analogues for eigenvalues and eigenvectors.

Definition 3.1

For $A \in \mathbb {R}_{\min }^{m\times m}$, if $\lambda \in \mathbb {R}_{\min }$ and the vector $\boldsymbol{x}\in \mathbb {R}_{\min }^m$, which is not equal to $(\varepsilon ,\varepsilon ,\dots ,\varepsilon )^{\top }$, satisfy

$$\begin{aligned} A\otimes \boldsymbol{x}=\lambda \otimes \boldsymbol{x}, \end{aligned}$$

then $\lambda$ and $\boldsymbol{x}$ are an eigenvalue of A and its corresponding eigenvector, respectively.

The following definition provides the min-plus analogue of determinants.

Definition 3.2

The tropical determinant $\textrm{tropdet}(A)$ of $A\in \mathbb {R}_{\min }^{m\times m}$ is defined as

$$\begin{aligned} \textrm{tropdet}(A):=\bigoplus _{\sigma \in S_m}a_{1\sigma (1)}\otimes a_{2\sigma (2)}\otimes \cdots \otimes a_{m\sigma (m)}, \end{aligned}$$

where $S_m$ denotes the symmetric group of permutations of $\{1,2,\dots ,m\}$. Moreover, the characteristic polynomial $f_A(\lambda )$ for $A\in \mathbb {R}_{\min }^{m\times m}$ is given by

$$\begin{aligned} f_A(\lambda ):=\textrm{tropdet}(A\oplus \lambda \otimes I), \end{aligned}$$

where I denotes the m-by-m identity matrix, the (i, j) entry of which is 0 if $i=j$ or $\varepsilon$ otherwise.

Proposition 3.1

(Maclagan-Sturmfels [17]) The characteristic polynomial $f_{A}(\lambda )$ of $A\in \mathbb {R}_{\min }^{m\times m}$ is factorized as

$$\begin{aligned} f_A(\lambda )\equiv & (\lambda \oplus p_1)^{q_1}\otimes (\lambda \oplus p_2)^{q_2} \otimes \cdots \otimes (\lambda \oplus p_k)^{q_k}, \end{aligned}$$

where $p_1< p_2 <\cdots <p_k$, $q_1+q_2+\cdots +q_k=m$, and the symbol “$\equiv$” indicates that the graphs of the functions are equal. The minimum root $p_1$ of $f_A(\lambda )$ coincides with the eigenvalue of A.

For a weighted digraph G involving m vertices, the entries of the m-by-m weighted adjacency matrix $A(G)=(a_{ij})\in \mathbb {R}_{\min }^{m\times m}$ are given by

$$\begin{aligned} a_{ij}=\left\{ \begin{array}{ll} w(\textrm{e}), &{} \quad \textrm{if}\ \textrm{e}=(\textrm{v}_i,\textrm{v}_j)\in E,\\ \varepsilon , &{} \quad \textrm{otherwise}. \end{array}\right. \end{aligned}$$

Conversely, for any $A\in \mathbb {R}_{\min }^{m\times m}$, there exists a weighted digraph G(A) whose weighted adjacency matrix is A. In a circuit C on a weighted digraph G(A), the number of edges and sum of edge weights are referred to as the length $\ell (C)$ and weight w(C) of C, respectively. The average weight $w_\textrm{ave}(C)$ is the ratio of the weight w(C) to the length $\ell (C)$:

$$\begin{aligned} w_\textrm{ave}(C)=\frac{w(C)}{\ell (C)}. \end{aligned}$$

The following theorem provides an interesting relationship between the eigenvalue of $A\in {\mathbb R}_{\min }^{m\times m}$ and the average weight of the circuits on the weighted digraph G(A).

Theorem 3.7

(Baccelli et al. [1]) Let the weighted digraph G(A) be strongly connected. Then, the weighted adjacency matrix A(G) only has one eigenvalue. Moreover, the minimum value of the average weights of the circuits in G(A) coincides with the eigenvalue of A(G).

4.2 Relationship with Min-Plus Eigenvalue

The ultradiscrete Toda (udToda) equation (3.1) is obtained by applying variable transformations $q_k^{(n)}=e^{-Q^{(n)}/\varepsilon }$ and $e_k^{(n)}=e^{-E^{(n)}/\varepsilon }$ to the dToda equation (3.2), taking the logarithm, multiplying both sides by $\varepsilon$, and taking the limit $\varepsilon \rightarrow +0$. Using the notation of min-plus algebra, that is, $\oplus$ and $\otimes$, the udToda equation can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle Q_k^{(n+1)}=\bigotimes _{j=1}^k Q_j^{(n)}\oslash \bigotimes _{j=1}^{k-1}Q_j^{(n+1)}\oplus E_k^{(n)},\quad k=1,2,\dots ,m,\\ E_k^{(n+1)}=Q_{k+1}^{(n)}\otimes E_k^{(n)}\oslash Q_k^{(n+1)},\quad k=1,2,\dots ,m-1,\\ E_0^{(n)}:=\varepsilon ,\quad E_m^{(n)}:=\varepsilon ,\quad n=0,1,\dots . \end{array}\right. \end{aligned}$$

Let ${\mathcal L}^{(n)}$ and ${\mathcal R}^{(n)}$ in $\mathbb {R}_\mathrm{{min}}^{m\times m}$ be ultradiscrete analogues of $(R^{(n)})^{\top }$ and $(L^{(n)})^{\top }$ in (3.3), respectively.

$$\begin{aligned} {\mathcal L}^{(n)}=\left[ \begin{array}{ccccc} Q_1^{(n)} &{} \varepsilon &{} \cdots &{} \cdots &{} \varepsilon \\ e &{} Q_2^{(n)} &{} \ddots &{} &{} \vdots \\ \varepsilon &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \varepsilon \\ \varepsilon &{} \cdots &{} \varepsilon &{} e &{} Q_m^{(n)} \end{array}\right] , \quad {\mathcal R}^{(n)}=\left[ \begin{array}{ccccc} e &{} E_1^{(n)} &{} \varepsilon &{} \cdots &{} \varepsilon \\ \varepsilon &{} e &{} E_2^{(n)} &{} \ddots &{} \vdots \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \varepsilon \\ \vdots &{} &{} \ddots &{} \ddots &{} E_{m-1}^{(n)} \\ \varepsilon &{} \cdots &{} \cdots &{} \varepsilon &{} e \end{array}\right] . \end{aligned}$$

(3.47)

Similar to the discrete case, we define the tridiagonal matrix ${\mathcal A}^{(n)}\in \mathbb {R}_\mathrm{{min}}^{m\times m}$ as the product of ${\mathcal L}^{(n)}$ and ${\mathcal R}^{(n)}$:

$$\begin{aligned} {\mathcal A}^{(n)}&:={\mathcal L}^{(n)}\otimes {\mathcal R}^{(n)}\\ &=\left[ \begin{array}{ccccc} Q_1^{(n)} &{} Q_1^{(n)}\otimes E_1^{(n)} &{} \varepsilon &{} \cdots &{} \varepsilon \\ e &{} Q_2^{(n)}\oplus E_1^{(n)} &{} Q_2^{(n)}\otimes E_2^{(n)} &{} \ddots &{} \vdots \\ \varepsilon &{} \ddots &{} \ddots &{} \ddots &{} \varepsilon \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} Q_{m-1}^{(n)}\otimes E_{m-1}^{(n)} \\ \varepsilon &{} \cdots &{} \varepsilon &{} e &{} Q_m^{(n)}\oplus E_{m-1}^{(n)} \end{array}\right] . \end{aligned}$$

To characterize the eigenvalues of ${\mathcal A}^{(n)}$ without the corresponding weighted digraph, we consider the characteristic polynomials of the min-plus tridiagonal matrices ${\mathcal A}^{(n)}$.

Theorem 3.8

(Watanabe et al. [34]) The characteristic polynomials $f_{{\mathcal A}^{(n)}}(\lambda )$ of the tridiagonal matrices ${\mathcal A}^{(n)}\in \mathbb {R}_\mathrm{{min}}^{m\times m}$ can be factorized as

$$\begin{aligned} f_{{\mathcal A}^{(n)}}(\lambda )=(\lambda \oplus Q_1^{(n)})\otimes (\lambda \oplus Q_2^{(n)}\oplus E_1^{(n)}) \otimes \cdots \otimes (\lambda \oplus Q_m^{(n)}\oplus E_{m-1}^{(n)}). \end{aligned}$$

Theorem 3.8 suggests that $Q_1^{(n)}$, $Q_2^{(n)}\oplus E_1^{(n)}$, $\dots$, $Q_m^{(n)}\oplus E_{m-1}^{(n)}$ are the roots of the characteristic polynomial $f_{{\mathcal A}^{(n)}}(\lambda )$. From Proposition 3.1, the minimum root of $f_{{\mathcal A}^{(n)}}(\lambda )$ is an eigenvalue of ${\mathcal A}^{(n)}$. Thus, we can express the eigenvalues $\lambda ^{(n)}$ using udToda variables:

$$\begin{aligned} \lambda ^{(n)}=\bigoplus _{k=1}^m Q_k^{(n)}\oplus \bigoplus _{k=1}^{m-1}E_k^{(n)}. \end{aligned}$$

(3.48)

The right-hand side of (3.48) coincides with the conserved quantities of the udToda equation [32]. This means that $\lambda ^{(n)}$ is a constant, that is, it is independent of discrete-time n. Thus, the eigenvalue $\lambda =\lambda ^{(0)}$ of the tridiagonal ${\mathbb R}_{\min }^{m\times m}$ matrix ${\mathcal A}^{(0)}$ is given by the right-hand side of (3.48) for any discrete-time n. In the BBS, the interchange of ball groups terminates at a sufficiently large discrete-time n. Simultaneously, the number of empty boxes between two neighboring ball groups monotonically increases as n increases. Recall that $Q_k^{(n)}$ corresponds to the number of successive balls in the kth ball group. A property of the values of $Q_k^{(n)}$ and $E_k^{(n)}$ [32] is that there exists a discrete-time N such that, for any $n\ge N$, the following holds:

$$\begin{aligned} & Q_1^{(n)}\le E_k^{(n)},\quad k=1,2,\dots ,m-1,\end{aligned}$$

(3.49)

$$\begin{aligned} & Q_1^{(n)}\le Q_2^{(n)}\le \cdots \le Q_m^{(n)}. \end{aligned}$$

(3.50)

This implies that for a sufficiently large n, $Q_1^{(n)}$ is equal to the minimum value of all udToda variables. Combining (3.49) and (3.50) with (3.48) leads to the following theorem.

Theorem 3.9

(Watanabe et al. [34]) For the eigenvalue $\lambda$ of ${\mathcal A}^{(0)}\in {\mathbb R}_{\min }^{m\times m}$, the following holds for $n\ge N$:

$$\begin{aligned} \lambda =Q_1^{(n)}. \end{aligned}$$

Under the initial condition that $Q_1^{(0)}$, $Q_2^{(0)}$, $\dots$, $Q_m^{(0)}$ and $E_1^{(0)}$, $E_2^{(0)}$, $\dots$, $E_{m-1}^{(0)}$ are provided by the tridiagonal matrices ${\mathcal A}^{(0)}\in {\mathbb R}_{\min }^{m\times m}$, the udToda equation generates matrix transformations of ${\mathcal A}^{(0)}$ without changing an eigenvalue, and then makes the values of $Q_1^{(n)}$ equal to the eigenvalue after sufficient discrete-time evolution. Furthermore, in the weighted digraph $G({\mathcal A}^{(0)})$, the value of $Q_1^{(n)}$ for a sufficiently large n is equal to the minimum value of the average weights of all circuits.

The ultradiscrete hungry Toda-I (udhToda-I) equation, which is a hungry extension of the ultradiscrete Toda equation, can also be related to eigenvalues over the min-plus algebra in a similar manner as for the udToda equation. The udhToda-I equation is obtained by the ultradiscretization of the dhToda-I equation (3.6) as follows:

$$\begin{aligned} \left\{ \begin{array}{l} Q_k^{(n+M)}=\displaystyle \bigotimes _{j=1}^k Q_j^{(n)}\oslash \bigotimes _{j=1}^{k-1}Q_j^{(n+M)}\oplus E_k^{(n)},\quad k=1,2,\dots ,m,\\ E_k^{(n+1)}=Q_{k+1}^{(n)}\otimes E_k^{(n)}\oslash Q_k^{(n+M)},\quad k=1,2,\dots ,m-1,\\ E_0^{(n)}:=\varepsilon ,\quad E_m^{(n)}:=\varepsilon ,\quad n=0,1,\dots . \end{array}\right. \end{aligned}$$

(3.51)

We define the lower Hessenberg matrix ${\mathcal A}^{(n)}\in \mathbb {R}_\mathrm{{min}}^{m\times m}$ as the product of ${\mathcal L}^{(n)}$ and ${\mathcal R}^{(n)}$ in (3.47):

$$\begin{aligned} {\mathcal A}^{(n)}:={\mathcal L}^{(n)}\otimes {\mathcal L}^{(n+1)}\otimes \cdots \otimes {\mathcal L}^{(n+M-1)}\otimes {\mathcal R}^{(n)}. \end{aligned}$$

There exist some circuits for the weighted digraph $G({\mathcal A}^{(n)})$ for the min-plus matrix ${\mathcal A}^{(n)}$. The digraph is strongly connected; from Theorem 3.7, the eigenvalue of ${\mathcal A}^{(n)}$ coincides with the minimum average weight of all circuits of $G({\mathcal A}^{(n)})$. Considering the udhToda-I equation, specifically the asymptotic behavior of the corresponding nBBS and its conserved quantities, we obtain the following theorem.

Theorem 3.10

(Kan et al. [16]) For the eigenvalue $\lambda$ of ${\mathcal A}^{(0)}\in {\mathbb R}_{\min }^{m\times m}$, the following holds for $n\ge N$:

$$\begin{aligned} \lambda =Q_1^{(n)}\otimes Q_1^{(n+1)}\otimes \cdots \otimes Q_1^{(n+M-1)}. \end{aligned}$$

Further details can be found in [16]. The ultradiscretization of the dLV system (3.5) can also be related to min-plus eigenvalues. Further details on this are provided in [8].

5 Numbered Box and Ball System

According to [32], the udhToda-I equation (3.51) governs the nBBS, where every ball is numbered. Based on the ultradiscretization of the dhToda-II equation (3.10), found in the study of Hessenberg LR transformations, we can design an nBBS where every box (rather than ball) is numbered [36]. To distinguish between the two types of nBBSs, we refer to the former and latter as nBBS-I and nBBS-II, respectively. In this section, we describe the numbering rules for the boxes, moving rules for the balls, and soliton behavior of the ball groups for the nBBS-II.

At discrete-time n, the nBBS-II with M types of boxes enforces that all boxes and balls satisfy the following rules:

• (a) Only boxes between the leftmost and rightmost ball groups are numbered.

• (b) Each box is numbered by one of the following integers: $n,n+1,\dots ,n+M-1$.

• (c) Each box array, which is an array of successive empty boxes, contains at least one box with the number $n+i$ for $i=0,1,\dots ,M-1$.

• (d) The boxes in each box array are lined up in the increasing order of their assigned numbers.

The discrete-time evolution from n to $n+1$ allows the identification numbers of the boxes to change. The balls move according to the following rules:

• (i) Starting from the leftmost ball, each ball moves, one by one, to the nearest empty box to its right with the number n or without a number.

• (ii) Remove the number n from the boxes that have been newly filled with a ball. Assign the number n to the boxes that have become empty, except for the boxes to the left of the leftmost ball.

• (iii) Replace the n assigned to the box with $n+M$.

• (iv) Reorder the boxes between adjacent ball groups in ascending order of their assigned numbers.

This discrete-time evolution is mathematically formulated by the ultradiscretization of dhToda-II (udhToda-II) equation:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle Q_k^{(n+1)}=\min \left( \sum _{i=1}^kQ_i^{(n)}-\sum _{i=1}^{k-1}Q_i^{(n+1)},E_k^{(n)}\right) ,\quad k=1,2,\dots ,m,\\ Q_k^{(n+1)}+E_k^{(n+M)}=Q_{k+1}^{(n)}+E_k^{(n)},\quad k=1,2,\dots ,m-1,\\ E_0^{(n)}:= +\infty ,\quad E_m^{(n)}:=+\infty . \end{array}\right. \end{aligned}$$

(3.52)

Note that, if rules (a)–(d) hold at discrete-time $n=0$, then they always hold under the discrete-time evolution generated by rules (i)–(iv). Figure 3.2 show an example of discrete-time evolution from $n=0$ to $n=6$ for the nBBS-II with $M=3$.

Fig. 3.2

An example of discrete-time evolution from $n=0$ to $n=6$ for the nBBS-II with $M=3$

To demonstrate the soliton behavior in the nBBS-II, we need to clarify the conserved quantities of the udhToda-II equation. Thus, we need to determine the conserved quantities of the dhToda-II equation (3.10) such that their ultradiscretization becomes a conserved quantity in the udhToda-II equation (3.52). The target quantities can be derived by reconsidering the similarity transformation of a Hessenberg matrix generated by the dhToda-II equation (3.10). Although these conserved quantities are equal to the coefficients of the Hessenberg eigenpolynomial, seeking them directly is not easy. Recall that the Hessenberg similarity transformation can be decomposed into multiple bidiagonal LR transformations. Combining this with the Watkins’ theorem [35], we can then obtain the target quantities by computing the coefficients of the eigenpolynomial of the sparse band matrix instead of the dense Hessenberg matrix. Details of this process can be found in [15]. Using the resulting conserved quantities, we obtain a theorem concerning soliton behavior in the nBBS-II.

Theorem 3.11

(Yamamoto et al. [36]) Suppose that m ball groups contain $L_1$, $L_2$, $\dots$, $L_m$ successive balls, which are sufficiently far from each other, at discrete-time $n=0$. Moreover, let $\sigma$ be a permutation of $\{1,2,\dots ,m\}$ such that $L_{\sigma (1)}\le L_{\sigma (2)}\le \cdots \le L_{\sigma (m)}$, and let $n^{*}$ be the discrete time immediately after the completion of all interactions. Then, the following holds for $n=n^{*},n^{*}+1,\dots$:

$$\begin{aligned} Q_1^{(n)}=L_{\sigma (1)},\quad Q_2^{(n)}=L_{\sigma (2)},\quad \dots ,\quad Q_m^{(n)}=L_{\sigma (m)}. \end{aligned}$$

6 Concluding Remarks

In our chapter, we first demonstrated that various discrete integrable systems can be related to matrix LR transformations. We also showed that their ultradiscretization of such systems can generate matrix transformations without changing an eigenvalue, and that these transformations can be used to compute the eigenvalue of matrices over min-plus algebra. Finally, we detailed an nBBS with numbered boxes, which arose from the LR perspective of the discrete hungry integrable systems.

Hessenberg matrices including tridiagonal matrices, frequently encountered in this chapter, have a special structure that can be decomposed into products of bidiagonal matrices. This suggests that we can find new discrete integrable systems, and BBSs based on them, by considering other decompositions of Hessenberg matrices from the LR perspective. However, we do not need to only consider Hessenberg matrices in this way. We also do not need to only consider meticulous integrable systems where neighbors are always assumed to affect each other in some way. For example, not all drivers pay continuous attention to their environment. Therefore, tolerant integrable systems may be considered to capture realistic phenomena. We hope that the accumulation of studies on integrable systems and their related LR transformations presented in this chapter improves our ability to represent mobility phenomena.