(We'll come to what "least defined" means in a minute.) We discussed pattern matching, the Maybe Monad, filter, map and head. 2. An implementation of the factorial function can be either iterative or recursive, but the function itself isn't inherently either. We discussed the Fibonacci sequence, LCM and GCD. One way took an iterative approach while the second way, Euclid’s Algorithm, used a simple recursive method. All solutions were written in Haskell but the algorithms easily translate to other languages. Write a factorial function with declarative style (Haskell): factorial n = product [1..n] factorial 5 -- 120. fix and fixed points []. Base = 0.477305071 Recursive = 517.544341882 Iterative = 491.569636915 So, the recursive factorial function is slightly slower than the iterative function. 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. factorial n = fac n 1 Where fac n acc = if n < 2 then acc else fac (n-1) (acc*n) Note that an implementation isn't necessarily either iterative or recursive. These two hand crafted functions are both much slower than the built-in factorial because Base uses some lookup table magics. 3. 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. Haskell can use tail call optimisation to turn a recursion into a loop under the hood. 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 factorial 0 acc = acc factorial n acc = factorial (n-1) $! 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. For the two aforementioned examples that converge, this is readily seen: Iterative solution. The last call returns 6, then fac(2, 3) returns 6, and finally the original call returns 6. GCD was defined two ways. Even if we don’t know what a factorial is, we can understand it by reading this simple code. Ok great! The code shown here is based on an account by Thomas Hallgren (see ), extended to include factorial. 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. Write a function which takes in an array and returns the result of adding up every item in the array: In JavaScript: ( 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. 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). For example, here are three different definitions of the factorial function in the language Haskell: Haskell uses a lazy evaluation system which allows you define as many terms as you like, safe in the knowledge that the compiler will only allocate the ones you use in an expression. 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