Chapter Seven - Fibonacci on the Fast Track

Abstract

This chapter discusses the computational mathematics and programming involving Fibonacci numbers. The sequence of Fibonacci numbers is easily defined in the terms of fn and these definitions can be programmed directly into Mathematica through Fibonacci .m. This method of computing Fibonacci numbers is very inefficient. This inefficiency can be overcome by employing dynamic programming. The disadvantage of this method is that all intermediate Fibonacci numbers are stored as rules; entering these rules into the system and subsequent lookup is slow and uses a lot of storage space. As fn is of size n, the measure to use is n; the loop that computes fibc[n] is executed n times. Thereafter, using a closed formula for this purpose though slow in symbolic expansion is still a fast way of computing fn; to get all the digits right, the computation with a precision equal to the size of fn must be performed. Another way to compute fnis using the matrix method; the improvements derived by finding better formulae for matrix multiplications are specific to the computing of Fibonacci numbers. Finally, there is a range of variance in between the computational power of the above methods; the fastest algorithm being one involving the compute fn asymptotically.