How to use Fibonacci numbers to teach recursive programming

There are too many examples and programming guides (which, e.g., an internet search for "recursive procedure Fibonacci" will turn up) to count that use Fibonacci numbers as an example to illustrate recursive programming. The motivation for this article is to show why the naive way of doing this is a bad idea, as it is horrendously inefficient. We will exhibit much more efficient ways of computing Fibonacci numbers, both iterative and recursive, and analyze and compare worst case running times and memory usages. Using some mathematical properties of Fibonacci numbers leads to the most efficient method for their computation. For illustration and benchmarking, we will use Maple and its programming language, however, similar behaviour can be demonstrated in almost any other programming language. This exposition combines and explores the mathematical properties of Fibonacci numbers, notions of algorithmic complexity, and efficient Maple programming and profiling techniques, and may be used as an introduction to any of these three subjects. The techniques described can be readily generalized to more general types of linear recurrences with constant coefficients.

Decision problems for linear recurrences involving arbitrary real numbers

We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show that for each problem there exists a correct partial algorithm which halts for all problem instances for which the answer is locally constant, thus establishing that all three problems are as close to decidable as one can expect them to be in this setting. We further show that the algorithms for the Positivity Problem and the Ultimate Positivity Problem halt on almost every instance with respect to the usual Lebesgue measure on Euclidean space. In comparison, the analogous problems for exact rational or real algebraic coefficients are known to be decidable only for linear recurrences of fairly low order.