We discussed pattern matching, the Maybe Monad, filter, map and head. Base = 0.477305071 Recursive = 517.544341882 Iterative = 491.569636915 So, the recursive factorial function is slightly slower than the iterative function. GCD was defined two ways. One way took an iterative approach while the second way, Euclid’s Algorithm, used a simple recursive method. Ok great! Write a function which takes in an array and returns the result of adding up every item in the array: In JavaScript: All solutions were written in Haskell but the algorithms easily translate to other languages. Factorial in iterative and functional style public long factorial(int n) { return LongStream .rangeClosed(1, n) .reduce((a, b) -> a * b) .getAsLong(); } factorial(5) // Output: 120 It’s worth repeating that by abstracting the how part we can write more maintainable and scalable software. There are quite a few cases where a recursive solution is worse than an iterative one. Even a pure functional language like Haskell supports iterative solutions in the form of list comprehension. Haskell can use tail call optimisation to turn a recursion into a loop under the hood. 3. For the two aforementioned examples that converge, this is readily seen: Write a factorial function with declarative style (Haskell): factorial n = product [1..n] factorial 5 -- 120. We discussed the Fibonacci sequence, LCM and GCD. Even if we don’t know what a factorial is, we can understand it by reading this simple code. The same kinds of techniques can also be used to encode behaviors more often associated with dependent types and polytypic programming, and are thus a topic of much recent interest in the Haskell community. These two hand crafted functions are both much slower than the built-in factorial because Base uses some lookup table magics. A fixed point of a function f is a value a such that f a == a.For example, 0 is a fixed point of the function (* 3) since 0 * 3 == 0.This is where the name of fix comes from: it finds the least-defined fixed point of a function. Note that an implementation isn't necessarily either iterative or recursive. (We'll come to what "least defined" means in a minute.) Tail Calls Consider the factorial function below: When we make the call fac(3), two recursive calls are made: fac(2, 3) and fac(1, 6). Factorial in Haskell factorial :: Integer -> Integer factorial 0 = 1 ... Iterative computation • An iterative computation is one whose execution stack is bounded by a constant, independent of the length of the computation • Iterative computation starts with an initial state S 0 Iterative solution. The last call returns 6, then fac(2, 3) returns 6, and finally the original call returns 6. The code shown here is based on an account by Thomas Hallgren (see ), extended to include factorial. For example, here are three different definitions of the factorial function in the language Haskell: 2. ( acc * n ) Note that we have used accumulator with strict evaluation in order to suppress the default laziness of Haskell computations - this code really computes new n and acc on every recursion step. factorial 0 acc = acc factorial n acc = factorial (n-1) $! factorial n = fac n 1 Where fac n acc = if n < 2 then acc else fac (n-1) (acc*n) fix and fixed points []. An implementation of the factorial function can be either iterative or recursive, but the function itself isn't inherently either. 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