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## The Fibonacci Sequence

version 2.2 (1.03 MB) by

Efficient computation of Fibonacci and Lucas numbers

Updated

Often I see students asking for help on a tool to compute the Fibonacci numbers. Or, I'll find them asking for help on a Project Euler problem. Or, a student has been assigned the problem of computing the fibonacci numbers using a recursive implementation. After all, these numbers lend themselves splendidly to teaching a student to use recursion.
The problem is that a direct, simple, recursive scheme is a poor one for the Fibonacci numbers, unless the recursion is written very carefully.
This tool teaches you how to compute the Fibonacci numbers in a variety of ways, good, bad, ugly. I teach the concept of memoization, a vitally important tool for many recursive schemes, not only for Fibonacci numbers. (If you do teach a student recursion, use it as an excuse to also teach them about memoization!)
Of course, I also employ some additional tricks to compute the n'th Fibonacci number without needing to compute every lower order number in the sequence. Some useful identities are introduced to achieve that task.
Since these numbers get very large, very rapidly, I return them in my VPI class, but don't be mistaken, these tools are indeed efficient. For example, to compute both the 1000'th Fibonacci and Lucas numbers, the time required was only 0.013 seconds.
>> tic,[F,L] = fibonacci(1000);toc
Elapsed time is 0.013347 seconds.

These are big numbers, each having over 200 decimal digits.

>> F
F =
43466557686937456435688527675040625802564660517371780402481729089536
555417949051890403879840079255169295922593080322634775209689623239873322
471161642996440906533187938298969649928516003704476137795166849228875
>> L
L =
97194177735908175207981982079326473737797879155345685082728081084772
518818444815269080619149045968297679578305403209347401163036907660573971
740862463751801641201490284097309096322681531675707666695323797578127

Yair Altman

### Yair Altman (view profile)

An excellent tutorial on the pitfalls of recursion, the benefits of memoization, and the importance of utilizing known mathematical identities to obviate the need to use any of them.

One small issue: I content that the fibrecurmemo() function is not typical use of memoization: In typical memoization the previous function results are cached and reused in subsequent function calls. In your implementation, the recursion is simply reimplemented to use a single recursion but the results are not cached and so every call to fibrecurmemo() would need to recompute them. Classic memoization would entail using persistent variables to store the results.

Another suggestion is to use round() when directly computing the fib values using Binet's formula. You rightfully mention the FP-precision issue and using round() is a natural simple workaround for values small-enough not to require VPI.

Still, these are only minor points in an otherwise excellent submission.

 5 Feb 2017 2.2 Final version. 5 Feb 2017 2.1 Uploaded the wrong version as version 2.0. 2.1 is correct 5 Feb 2017 1 Repaired a bug in computing the mods of Fibonacci & Lucas numbers 4 May 2012 1.1 Allows efficient computation of the modulus of large Fibonacci/Lucas numbers.
##### MATLAB Release
MATLAB 7.12 (R2011a)