Cross rules and non-Abelian lattice equations for the discrete and confluent non-scalar ε-algorithms

Abstract

In this paper, we give the cross rules of the discrete and confluent vector, topological and matrix ε-algorithms. Then, from the rules of these confluent algorithms, we derive non-Abelian lattice equations, in particular some extensions of the Lotka–Volterra system, in the style of the equation related to the confluent form of the scalar ε-algorithm.

Non-isospectral extension of the Volterra lattice hierarchy, and Hankel determinants

“…Via the Miura transformation (cf. [8,9] in the present context) u n (t) = −M n−1 (t) M n (t) , M n (t) =γ n (t)/g n (t) , n = 1, 2, . .…”

Confluent Form of the Multistep ɛ‐Algorithm, and the Relevant Integrable System

“…We have (the variable t has been suppressed for simplicity) (5) f (6) + f f f (7) − f f (4) f (6) − f f f (8) f (5) f (7) − f (4) f (8) .…”

“…It is known that, when m = 1 in (4), the corresponding confluent form based on the ε-algorithm can be transformed into the Lotka-Volterra equation (see, for example, [8]). In this Section, we will show that, for any value of m ≥ 1,equation (4) can be similarly transformed into an integrable system under a proper variable transformation.…”

“…For example, in [23], Wynn derived the confluent ε-algorithm and the confluent ρ-algorithm from the discrete scalar ε-algorithm and ρ-algorithm, respectively; more details can be found in [1,25]. Since many convergence acceleration algorithms have close connection with discrete integrable systems (see [9-11, 16, 26]), it is natural to study the relations between their confluent forms and integrable systems [8].…”

“…When m = 1, if we set M k (t) = ε k,1 (t) and N k (t) = M k (t)M k+1 (t), then we obtain the Lotka-Volterra equation (see, for example, [8])…”