Yes. The primitive recursive solution takes a lot of time. The reason for this is that for each number calculated, it needs to calculate all the previous numbers more than once. Take a look at the following image. It represents calculating Fibonacci(5) with your function. As you can see, it computes the value of Fibonacci(2) … Read more
Since OP has asked about matrix solution not involving any floating point computations, here it is. We can achieve O(logn) complexity this way, assuming numeric operations have O(1) complexity. Let’s take 2×2 matrix A having following structure 1 1 1 0 Now consider vector (8, 5), storing two consecutive fibonacci numbers. If you multiply it … Read more
I’ll give a bit of an explanation of how it works internally. First, you must realise that Haskell uses a thing called a thunk for its values. A thunk is basically a value that has not yet been computed yet — think of it as a function of 0 arguments. Whenever Haskell wants to, it … Read more
How to write a generator class? You’re almost there, writing an Iterator class (I show a Generator at the end of the answer), but __next__ gets called every time you call the object with next, returning a generator object. Instead, to make your code work with the least changes, and the fewest lines of code, … Read more
A very nice test is that N is a Fibonacci number if and only if 5 N^2 + 4 or 5N^2 – 4 is a square number. For ideas on how to efficiently test that a number is square refer to the SO discussion.
You can use the matrix exponentiation method (linear recurrence method). You can find detailed explanation and procedure in this or this blog. Run time is O(log n). I don’t think there is a better way of doing this.
Following from Pillsy’s reference to matrix exponentiation, such that for the matrix M = [1 1] [1 0] then fib(n) = Mn1,2 Raising matrices to powers using repeated multiplication is not very efficient. Two approaches to matrix exponentiation are divide and conquer which yields Mn in O(ln n) steps, or eigenvalue decomposition which is constant … Read more
The Fibonacci numbers have all sorts of really nice mathematical properties that make them excellent in computer science. Here’s a few: They grow exponentially fast. One interesting data structure in which the Fibonacci series comes up is the AVL tree, a form of self-balancing binary tree. The intuition behind this tree is that each node … Read more
You’re defining a function in terms of itself. In general, fibonnaci(n) = fibonnaci(n – 2) + fibonnaci(n – 1). We’re just representing this relationship in code. So, for fibonnaci(7) we can observe: fibonacci(7) is equal to fibonacci(6) + fibonacci(5) fibonacci(6) is equal to fibonacci(5) + fibonacci(4) fibonacci(5) is equal to fibonacci(4) + fibonacci(3) fibonacci(4) is … Read more
The evaluation mechanism in Haskell is by-need: when a value is needed, it is calculated, and kept ready in case it is asked for again. If we define some list, xs=[0..] and later ask for its 100th element, xs!!99, the 100th slot in the list gets “fleshed out”, holding the number 99 now, ready for … Read more