Ultradiscrete hungry Toda equation and eigenvalues over min-plus algebra

The recursion formula of the quotient difference algorithm for computing matrix eigenvalues corresponds with the discrete Toda equation, which is well-known in discrete integrable systems. Previous studies have revealed that the ultradiscrete Toda equation computes eigenvalues of tridiagonal matrices over the min-plus algebra. The min-plus algebra is a commutative semiring in which minimum and plus operations are introduced into the union of the set of real numbers and positive infinity. The discrete hungry Toda equation, which is a generalization of the discrete Toda equation, can compute the eigenvalues of lower Hessenberg banded matrices. This study focuses on the ultradiscrete hungry Toda equation and show that the time evolution of the equation yields the eigenvalues of lower Hessenberg banded matrices over the min-plus algebra.


Introduction
Several numerical algorithms with intrinsic mathematical structures of integrable systems have been discovered.For example, the discrete Toda equation which is obtained through a discretization of the well-known continuous Toda equation in integrable systems, is identical to the recursion formula of the quotient difference (qd) algorithm for computing eigenvalues of tridiagonal matrices A (n) = L (n) R (n) defined by The qd algorithm coincides with the LR algorithm for tridiagonal matrices A (n) and the discrete Toda equation ( 1) can be derived through the diagonal and subdiagonal entries of the LR transformation [12].Regarding the discrete Toda equation ( 1) as a discrete time dynamical system, lim n→∞ q (n) k , k = 1, 2, . . ., m coincide with the eigenvalues of the given tridiagonal matrix A (0) .Examples of other relationships between numerical algorithms and integrable systems include the QR algorithm [14], singular value computation [7,8], eigenvalue computation for banded matrices [2,3], and convergence acceleration algorithms [11].
The box and ball system is a discrete dynamical system in which a finite number of balls are placed in a row of boxes and the balls are moved according to certain rules.The box and ball system was discovered by Takahashi and Satsuma in [15].The equation of motion of the box and ball system is written as Equation ( 3) is called the ultradiscrete Toda equation, which is an ultradiscrete version of the discrete Toda equation (1).Here, the ultradiscretization [16] is a limiting procedure which transforms a difference equation into a piecewise linear equation.The variable is the number of kth consecutive row of balls (soliton) and E (n)  k is the number of empty boxes between the kth and (k + 1)st solitons.Solitons are shown to be ordered in ascending order by time evolution.Moreover, the modified version of the evolution equation for convergence acceleration corresponds with the bubble sort [10].The ultradiscrete Toda equation is an equation over the min-plus algebra which is a commutative and idempotent semiring whose addition ⊕ is min and multiplication ⊗ is +.Min-plus algebra (or its isomorphic max-plus algebra) and their generalized algebraic structure are studied in [4,9].Eigenvalues of linear systems described by the min-plus algebra appear in various applications, such as scheduling problems, railway systems, control theory, discrete event systems [5,6,13].In [18], an ultradiscrete analog of the relationship between the discrete Toda equation and the qd algorithm is considered, in which it is shown that eigenvalues over the min-plus algebra can be obtained by time evolution of the ultradiscrete Toda equation (3).
The discrete hungry Toda (dhToda) equation [17] is known as a generalization of the discrete Toda equation (1).The dhToda algorithm, an extension of the qd algorithm, for computing eigenvalues has been designed using the dhToda equation ( 4) [3].The qd algorithm can compute the eigenvalues of tridiagonal matrices L (n) R (n) , whereas the dhToda algorithm can compute eigenvalues of totally nonnegative lower Hessenberg banded matrices n) which is defined using bidiagonal matrices L (n) and R (n) in (2).Here, the lower Hessenberg banded matrix is a square matrix which has zero entries above the first superdiagonal.If M is set to 1, the dhToda algorithm is reduced to the qd algorithm.
The ultradiscrete hungry Toda equation [17], which is obtained through an ultradiscretization of the dhToda equation ( 4), is the equation of motion for a numbered box and ball system, which is an extension of the box and ball system.In [17], conserved quantities and solitonical nature for the numbered box and ball systems are discussed; however, the relationship between the ultradiscrete hungry Toda equation and eigenvalues over the min-plus algebra is not discussed.In this study, we show that the ultradiscrete hungry Toda equation can be applied to eigenvalue computation over the min-plus algebra.The resulting algorithm is used for lower Hessenberg banded matrices over the min-plus algebra and is a min-plus analog of the dhToda algorithm.
The remainder of this paper is organized as follows.Section 2 introduces basic definitions, notations, and basic properties of the min-plus algebra; further, describes a graph theoretical property of matrix products in min-plus algebra, which is crucial for presenting the results of this paper.Section 3 reviews the existing results on the ultradiscrete hungry Toda equation in terms of conserved quantities and solitonical nature and discussed the relaxation of the sorting condition as a by-product of the process of deriving the main results.Section 4 shows that the ultradiscrete hungry Toda equation computes eigenvalues of the lower Hessenberg banded matrices in a finite number of time evolution.Finally, Section 5 presents the concluding remarks.

Min-plus algebra
This section introduces basic results and definitions related to the min-plus algebra.Let R min = R ∪ {+∞} be the set of real numbers R and +∞.The binary operations ⊕ and ⊗ are defined as follows.
a ⊕ b = min(a, b) and a ⊗ b = a + b, for any a, b ∈ R min .Such an R min with binary operations ⊕ and ⊗ is called the min-plus algebra.Addition ⊕ is commutative, associative and has ε = +∞ as the zero element.Multiplication ⊗ is also commutative, associative, and has e = 0 as the identity element.
Symbol ⊗ is omitted if no particular confusion exists.There is no inverse element with respect to ⊕, whereas that of ⊗ exists except for ε.The inverse operation of ⊗ is defined as = −.
Let us introduce the set of all m × n min-plus matrices as R m×n min .In the case of n = 1, R m×1 min = R m min is a set of all vectors of order m.For two min-plus matrices A = (a i,j ), B = (b i,j ) ∈ R m×n min , their sum The product A ⊗ B = ([A ⊗ B] i,j ) ∈ R m×n min for two min-plus matrices A = (a i,j ) ∈ R m× min and B = (b i,j ) ∈ R ×n min is defined as follows.
For α ∈ R min , the scalar multiplication of A = (a i,j ) ∈ R m×n min is defined as Next, we define the eigenvalues and eigenvectors of the min-plus matrix.

Definition 2.1 (Eigenvalues and eigenvectors):
For a given min-plus matrix A ∈ R m×m min , the scalar λ ∈ R min is called the eigenvalue of A if there exists x = [ε, ε, . . ., ε] ∈ R m min satisfying: The vector x is called the eigenvector of A corresponding to λ.Some basic definitions and properties of graphs are introduced.Let G = (V, E) be a directed graph with the set of vertices V and the set of directed edges E. A directed edge from i ∈ V to j ∈ V is expressed by e i,j ∈ E. We assume that all edges e i,j in G have weights w i,j ∈ R. The adjacency matrix A = (a i,j ) ∈ R |V|×|V| min is defined as follows.
Conversely, if a square matrix A ∈ R m×m min is given, there exists a weighted directed graph G(A) whose adjacency matrix is A. For i 1 , i 2 , . . ., i ∈ V and e i 1 ,i 2 , e i 2 ,i 3 , . . ., e i −1 ,i ∈ E, . The length of the circuit C is the number of edges − 1 expressed as |C|.The weight of the circuit C is defined as Here, the average weight ave(C) of the circuit C is defined as follows.
A directed graph G = (V, E) is strongly connected if there exists at least one path from i to j, for all the vertices i, j ∈ V.
Considering the corresponding graphs is helpful in matrix analysis over min-plus algebra.Let us consider the multiplication of min-plus matrices and its corresponding explanation over the graphs.For example, for , we consider their multiplication Figure 1 displays the weighted directed graph G(A 1 ) and G(A 2 ).The weight w i,j of edges of graphs G(A 1 ) and G(A 2 ) is denoted by w A 1 i,j and w A 2 i,j , respectively.Then the following holds.
In general, it is easy to show that a similar result holds for a multiplication of p matrices of arbitrary size m × m.

Proposition 2.2: For p matrices A
The eigenvalues of min-plus matrices are related to circuits of weighted directed graphs.

Conserved quantities and asymptotic behaviour of the numbered box and ball system
The numbered box and ball system (NBBS) [17] is a generalization of the original box and ball system such that several types of balls can be treated by assigning numbers to each ball.
Here we assume M types of balls, where each ball is assigned a number as 1, 2, . . ., M. The time evolution rule for the NBBS at discrete time n is given as follows.Here, a continuous row of balls, regardless of the ball number, is called a soliton.At discrete time n, Q (n)  k represents the number of balls with number (n mod M) + 1 in the kth soliton and E (n)  k represents the number of empty boxes between the kth and (k + 1)st solitons.Then, the dynamics of NBBS can be expressed by the ultradiscrete hungry Toda (udhToda) equation as follows.
Let us substitute the variables of the udhToda equation ( 5) (6) in this order, that is Then, the conserved quantities of the udhToda equation ( 5) is given as the following proposition.

Proposition 3.1 (T. Tokihiro et al. [17]):
For the ultradiscrete hungry Toda equation (5), the conserved quantities, which are independent of the discrete time evolution, are given as follows.
Let P (n) k be a length of the kth soliton at discrete time n, namely, . The following proposition is known for the behaviour of the udhToda equation ( 5) after sufficient time evolution.

Proposition 3.2 (T. Tokihiro et al. [17]):
In the ultradiscrete hungry Toda equation (5), there exists a discrete time N, such that the following holds for n ≥ N.
From Proposition 3.2, for sufficiently large discrete times n, the solitons in the NBBS are arranged from left to right in ascending order of length, and the length of solitons does not change with discrete time evolution for arbitrary initial values.However, the set of solitons ordered in ascending order after time evolution does not necessarily coincide with the set of solitons given in the initial states.In [17], the following theorem is obtained concerning the solitonical nature of the NBBS.Theorem 3.3 (T.Tokihiro et al. [17]): Suppose that P (0)  k , k = 1, 2, . . ., m are arranged in descending order; that is and that each E (0) k , k = 1, 2, . . ., m − 1 is sufficiently large to make no contribution in taking the minimum of equation (7).Then for n ≥ N with sufficiently large time N, we obtain Solitons arranged in descending order at initial states are arranged in ascending order by time evolution.Moreover, for the length of solitons P (n) k , the following lemma is obtained.

Lemma 3.4 (T. Tokihiro et al. [17]
): There exists a discrete time N such that Q Here we put M−1 s=M Q Lemma 3.4 suggests that (8) holds for arbitrary initial values after sufficient time evolution.Let us consider that the initial states of the udhToda equation ( 5) holds (8), namely, This means that, for the conserved quantities (7), the term which does not contain k is selected as the minimum value at n = 0. Therefore, under the condition (9), the conserved quantities of the udhToda equation ( 5) yields . . .
Thus, the length of m solitons P (0) k , k = 1, 2, . . ., m are m independent conserved quantities at the initial state.On the other hand, from Lemma 3.4, (8) holds for n ≥ N and arbitrary initial states.Then the conserved quantities are also expressed as follows.
Namely, from ( 10) and ( 11), if the initial states of the NBBS satisfy the condition (9), the set of the length of solitons {P (0) This means that even if the solitons in the initial states are not arranged in descending order as is in Theorem 3.3, the length of the solitons can be sorted in ascending order by time evolution if the condition (9) holds.
Let us share a numerical example in Tables 1 and 2. Setting m = 5, M = 3, and the initial states as in Table 1 for n = 0, 1, 2, these initial values satisfy the condition (9).Table 1 displays the values of Q (n)  k and E (n) k obtained by the time evolution of the udhToda Equation ( 5) from n = 3 to 27.Table 2 displays the time evolution of the length of solitons P (n) k from n = 0 to 25.The subsequent time after n = 24, the length of solitons P (n)  k does not change.Evidently, from Table 2 it is observed that even if solitons given as initial values are not arranged in descending order, they are sorted in ascending order at n = 24.

Ultradiscrete hungry toda equation and eigenvalue computation over min-plus algebra
This section shows that the time evolution of the udhToda equation ( 5) yields the eigenvalues of lower Hessenberg banded matrices over the min-plus algebra, which is an analog of the dhToda algorithm.
The two bidiagonal matrices L (n) , R (n) ∈ R m×m min are defined using the variables k of the udhToda equation ( 5) as follows.
k with time evolution of the udhToda equation (5).where e = 0 and ε = +∞.We define the matrix A (n) ∈ R m×m min as follows.
where the matrix A (n) is a lower Hessenberg matrix with lower band width min(M + 1, m) over the min-plus algebra.Moreover, we define , where L i,j denotes the (i, j)-entry of L.Then, we obtain The directed graph G(L (n) ) whose adjacency matrix is L (n) is displayed in Figure 2. Using Proposition 2.2 and Figure 2, for i = 1, 2, . . ., m, the ith diagonal entries L i,i of L are given as follows.
), (14) where if p = 0, we put Hereafter, the same treatment will be applied to the boundary of p in the summation .For i = 3, 4, . . ., m, the (i, i − 2) entries of L are Therefore, for 1 ≤ j < i ≤ m, the (i, i − j) entries of L are expressed as Let k and be the minimum index of a vertex that composes a circuit, and the length of the circuit respectively.For given k and , the circuit in the directed graph G(A (n) ) is uniquely determined.This circuit is denoted by C k, .To investigate the eigenvalues of the matrix A (n) , the following proposition is introduced.
Proposition 4.1: For a directed graph G(A (n) ) whose adjacency matrix is A (n) , the following inequality holds for any k and .
Proof: For any k and , the weight w(C k, ) of a circuit C k, is expressed as follows Here, let us define On the other hand, the right-hand side of ( 16) can be written as Here, (18) contains the following terms From ( 17) and ( 19), for any k and , if the following holds, then the inequality ( 16) holds.
Proposition 4.1 suggests that the sum of the weight of the self-loop gives the minimum weight.Since the directed graph G(A (n) ) is strongly connected, the eigenvalue of A (n) is unique and equal to the minimum average weight of the circuit in G(A (n) ) from Proposition 2.3.From Proposition 4.1, the average weight of the circuit of arbitrary length satisfies the following inequality.
Therefore, we have the following theorem with respect to the eigenvalue of A (n) .Theorem 4.2: The eigenvalue λ(A (n) ) of A (n) is given as From ( 12), the right-hand side of (23) can be expressed as ).
Here, the variables are renamed as in (6), and a new variable is introduced as Therefore, the eigenvalue of A (n) can be written as follows.
The value on the right-hand side of (25) is the conserved quantity of the udhToda equation (5).Therefore, since the eigenvalue of A (n) is invariant with the time evolution of the udhToda equation ( 5), it follows that Note that (8) can be written, using W k , as then, for any n ≥ N, the right-hand side of (25) yields From Proposition 3.2, it holds that Therefore, the following theorem is obtained as the main result of this study.

Theorem 4.3:
For any n ≥ N, the eigenvalue λ(A (0) ) of A (0) is given as In the following, it is confirmed that the eigenvalue can be obtained as in Theorem 4.3 by a numerical example.We use the example in Table 1 again.From the initial values Q k , k = 1, 2, 3, 4, given in Table 1, matrices L (0) , L (1) , L (2) , and R (0) are given as  A lower Hessenberg banded matrix that give the decomposition A (0) = L (0) ⊗ L (1) ⊗ L (2) ⊗ R (0) are whose corresponding directed graph G(A (0) ) are illustrated in Figure 3. Proposition 2.3 indicate that the eigenvalue of A (0) is the minimum average weight of all circuit in G(A (0) ).
The circuit with minimum average weight in G(A (0) ) is the self-loop at vertex 3 and its average weight is 11, which gives that λ(A (0) ) = 11.In order to find the eigenvalue using the udhToda equation ( 5), we first give the initial values Q k , k = 1, 2, 3, 4 as in Table 1.Then compute the time evolution using the udhToda equation (5).After n = 24, P (n) = 11 does not change and coincides with the eigenvalue λ(A (0) ), which indicate that the time evolution finishes in a finite time.

Concluding remarks
In this paper, we show that the eigenvalues of lower Hessenberg banded matrices over the min-plus algebra can be computed by the time evolution of the ultradiscrete hungry Toda equation.The minimum soliton in the numbered box and ball system is obtained as an eigenvalue of banded matrices over the min-plus algebra after sufficiently large discrete time.The method of computing eigenvalues obtained in this study can be said to be an ultradiscrete analog of the dhToda algorithm for computing eigenvalues of banded matrices in the sense of linear algebra formulated based on the discrete hungry Toda equation.It is shown that the eigenvalues of banded matrices are the conserved quantities of the ultradiscrete hungry Toda equation, as in the case of the dhToda algorithm.Moreover, the sorting condition of the numbered box and ball system is slightly generalized into the case that even if the initial array of solitons are not arranged in descending order.
It has been shown that the eigenvalues can be obtained with a finite number of time evolutions, but a concrete estimate of the computational complexity is not given, which is a subject for future works.The matrices intended in this paper were lower Hessenberg matrices with half-bandwidth M. It has been shown that another extended type of the discrete Toda equation computes eigenvalues for matrices with upper and lower bands.Its ultradiscrete analog is a subject for future works.
(i) Move all balls with number (n mod M) + 1 only once.(ii) Move the leftmost ball, with number (n mod M) + 1, to the nearest empty box on the right.(iii) Move the leftmost ball among the remaining balls, with number (n mod M) + 1, to the nearest empty box on the right.(iv) Repeat operation (iii) to move all balls with number (n mod M) + 1.

Table 2 .
Values of P (5)ith the time evolution of the udhToda equation(5).