The Toda Molecule Equation and the Ε-algorithm

One of the well-known convergence acceleration methods, the ε-algorithm is investigated from the viewpoint of the Toda molecule equation. It is shown that the error caused by the algorithm is evaluated by means of solutions for the equation. The acceleration algorithm based on the discrete Toda molecule equation is also presented. Discrete integrable systems play important roles in the field of numerical analysis. Matrix eigenvalue algorithms ([11], [19], [20]) and convergence acceleration methods ([1], [14], [15]) are typical examples. We here focus our attention mainly on the convergence acceleration methods and investigate both qualitatively and quantitatively their features from the viewpoint of discrete integrable systems. In particular, we show that there is a strong relation between the ε-algorithm and the discrete Toda molecule equation. Let us first consider the equation of motion given by dQ1 dt = −e−(Q2−Q1), dQn dt = −e−(Qn+1−Qn) + e−(Qn−Qn−1) (n = 1, 2, . . . , N − 1), (1) dQN dt = e−(QN−QN−1). Equation (1) is obtained by imposing the formal boundary condition Q0 = −∞, QN+1 =∞ (2) in the Toda lattice equation [21] and is also called the Toda molecule equation. Owing to its boundary condition (2), each particle moves freely and the distance between two neighboring particles becomes infinite as t → ∞. Under Flaschka’s change of variables [7], an = 1 2 exp ( −n+1 −Qn 2 ) , bn = 1 2 Q̇n = 1 2 dQ dt , (3) equation (1) is rewritten in the following matrix differential equation, dX dt = [X, B], (4) Received by the editor May 20, 1996 and, in revised form, November 5, 1996 and February 13, 1997. 1991 Mathematics Subject Classification. Primary 58F07, 65B10.