Fibonacci Series

In continuation with the theme of the last lecture we define another infinite series --- the fibonacci series. Ofcourse all of you know the Fibonacci Series. It is defined by the linear recurrence relation $F_{i+2}=F_i+F_{i+1}$. We assume that $F_0=1$ and $F_1=1$ to begin with.

The defintion is straight forward; it is just a one liner, but we use this as an excuse to introduce two standard list functions

Ziping a list

The zip of the list $x_0,...,x_n$ and $y_0,\dots ,y_m$ is the list of tuples $(x_0,y_0),\dots ,(x_k,y_k)$ where $k$ is the minimum of $n$ and $m$.

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> zip :: [a] -> [b] -> [(a,b)]
> zip  []       _   = []
> zip   _      []   = []
> zip (x:xs) (y:ys) = (x,y) : zip xs ys

The function zipWith is a general form of ziping which instead of tupling combines the values with an input functions.

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> zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
> zipWith f xs ys = map (uncurry f) $ zip xs ys

We can now give the code for fibonacci numbers.

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> fib = 1:1:zipWith (+) fib (tail fib)

Notice that the zip of fib and tail fib gives a set of tuples whose caluse are consecutive fibonacci numbers. Therefore, once the intial values are set all that is required is zipping through with a (+) operation.