Exercise 9.2.1 Design a tail-recursive version of length.
Ah, tail recursion again. Re-iterating that in a tail recursion, the last line of the function has to be a pure call to itself. To achieve this, an extra parameter needs to be passed to the function that keeps track of some relevant state.
We are getting good at this now, arent we?
Exercise 9.1.1 Provide an implementation of the missing function insert.
Insert will also be recursive, and its pretty easy to lay down the flow. Do not forget to concatenate the head after insert is applied on reduced versions of the list since we will need to return the whole inserted list and not a subset.
Exercise 7.2.2 Complete the following implementations of function contains and insert for IntTree’s.
This is similar to the IntSet implementations in Chapter 6.
Exercise 6.0.3 Write an implementation Integer of integer numbers The implementation should support all operations of class Nat while adding two methods def isPositive: Boolean def negate: Integer
This will be revisited later.
Exercise 6.0.2 Add a method def excl(x: Int) to return the given set without the element x. To accomplish this, it is useful to also implement a test method def isEmpty: Boolean for sets.
This will be revisited later.
Exercise 6.0.1 Write methods union and intersection to form the union and intersection between two sets.
This will be revisited later.
Exercise 5.3.1 Write a function for cube roots using fixedPoint and averageDamp.
Both fixedPoint and averageDamp functions remain the same. Instead of sqrt, now theres a cbrt function which calculates the cube root. The math library is used to avail of the exponential function.
Exercise 5.2.4 Can you write an even more general function which generalizes both sum and product?
This function abstracts out the operation which can be passed in as a sum or product function. One thing to be noted here is that an extra argument needs to be passed as the base case which would be 0 for a sum and 1 for a product.
Exercise 5.2.3 Write factorial in terms of product.
Exercise5.2.2 Write a function product that computes the product of the values of functions at points over a given range.
This is very similar to the sum function that is already worked out in the chapter.
The key is to understand that converting linear recursion to a tail-recursive version involves carrying over a computed value over to the next iteration. Also, note that the call to the iter function at the end need not take in a and b as parameters directly but can be computed values. Thats what facilitates the tail-recursive version. Here's my solution to the exercise.
A tail-recursive call is a method where the last step of the method is to call itself. The key point to remember is that the call should be a pure call and not an expression that involves a call to the method. If it is a pure call, the stack frame can be reused rather than allocating a new stack frame. The version of the factorial function seen in most places is a recursive one but its not tail-recursive since the last line of the factorial function usually calls n * factorial(n -1).
Here's the tail-recursive version of factorial. As you can see below, making a tail-recursive version involves adding an extra parameter. This parameter involves a computation that's different from just a reduction as in simple recursion.
