Computing Digit Selection Regions for Digit Recurrences

Digit selection is often the most difficult part of evaluating digit recurrence equations. Knowing the correct bounds on the digit selection regions is important for a number of reasons: to ensure that the recurrences converge, to know how to initialize the computation, to know the maximal convergence range of input values, and to compute the comparison values along with the maximum margins of error used to select digits. Digit selection regions are often a complicated function of the iteration index for the first few iterations, but then settle asymptotically to constants. Current methods often only estimate these bounds-and since the estimates must be conservative to guarantee convergence, they result in underestimating the input convergence range, and overestimating the required precision needed for comparisons. This paper demonstrates a general method for exactly computing digit selection regions for general digit recurrences, and is meant to help algorithmists to better visualize the exact shape and behavior of their digit selection regions.