Note on Miller's recurrence algorithm

An extension of Miller's algorithm

“…From this more general standpoint the algorithm can be recast into a new form which enables the correct number of recurrence steps to be determined automatically without appeal to an asymptotic or other analytical formula. In this respect it resembles an algorithm proposed recently by SHINTANI [4].…”

A continued fraction algorithm

“…3) has already been used in connection with linear recurrence relations [3,5]. Gautschi calls it the "dual" recursion.…”

On Recursive Computation

“…Conditions on the solutions of the equa¬ tion which will guarantee the convergence of the algorithm have been given by problems as the calculation of successive deriva/tives of , where f Gautschi (1961b) and by Olver (1964). Methods of increasing the efficiency of the algorithm by using the adjoint equation to generate auxiliary sequences have been given by Shintani (1965). (Our Theorem 2.7 is a generalization of one of this author's results.…”