Guessing Handbook
نویسنده
چکیده
Introduction This package provides functions for discoverying linear recurrence equations with polynomial coefficients satisfied by the entries of a given array. It extends the functionality of existing packages like Mallinger's GeneratingFunctions.m or Salvy and Zimmermann's gfun in two directions: (1) it is more general in that, e.g., multivariate recurrences, or recurrences of a particular shape can be searched for. (2) it is more efficient than the corresponding functions of existing packages, by using fine tuned linear system solvers based on modular techniques. The basic principle of all functions provided in this package is the same: they take an array of numbers as input and return as output a set of recurrence equations these numbers satisfy. For example, one way to rediscover the Fibonacci recurrence is to type In[2]:= GuessMinRE@81, 1, 2, 3, 5, 8, 13, 21, 34<, f@nDD Out[2]= -f@nD f@1 + nD + f@2 + nD This is a recurrence with constant coefficients, and it can also be found by Mathematica's builtin function FindLinearRecurrence (available since Mathematica 7). The builtin function, however, is not able to discover recurrence equations with polynomial coefficients, like the recurrence for the harmonic numbers, which our package can find without any problem:
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