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To write a recursive function, one first defines the grounding condition, or base case, at which the recursion terminates. In the example of factorial, both 1! and 0! return 1 by definition, and no further computation is necessary. When the grounding condition occurs, the function should stop calling itself and simply return the value 1. The factorial for a larger number can be defined as the larger number times the factorial of the next smaller integer. So the factorial function can be thought of as a process: multiply the number times the factorial of the next smaller number. The process continues until the number being evaluated is 1. At that point, the function returns 1, which provides the factor for computing 2!, which answer provides the factor for computing 3!, which answer provides the factor for computing 4!, etc. The recursion unwinds providing the answer for the factorial of the larger number. In a functional language, all computation proceeds by means of evaluating functions. Assignment of values to variables in order to maintain state is not permitted, so we cannot use a variable like n to keep track of our progress in a loop. Looping must be accomplished by recursion. Here is a Scheme function to compute the factorial: (define factorial (lambda (n) (if (<= n 1) 1 (* n (factorial(- n 1))) ))) Here we also see the conditional execution function if. The if function in Scheme is followed by three expressions. The first is evaluated for its truth. If the first expression following if is true, the second expression is evaluated and returned (in this case, if n <= 1, return 1). If the first expression is false, the third expression is evaluated and returned (in this case, if n > 1, return the product of n and the factorial of n 1). We can elaborate on our simple summation function and illustrate some more ideas. Here is a version of sum that takes a list as an argument. This way, our sum function can compute the sum of any number of integers: (define listSum (lambda (n) (cond ((null n) 0) ( (null (cdr n)) (car n) ) (else (+ (car n) (listSum (cdr n)))) ))) The cond (condition) operator is like multiple if and else-if statements in C or Java. Following the function cond is a list of condition/action pairs. Cond tests the first condition and, if it s true, cond executes the associated action. If the first condition is false, cond checks the second condition. If the second condition is true, cond executes the action associated with the second condition. There can be any number of conditions, and cond will execute only the action associated with the first true condition. At the end can be an else condition that cond will execute if no other condition is true. The else condition is not required, however. The listSum function tests to see whether the list it was passed is null. If so, the function returns 0. If not, then it tests to see if the cdr of the list is null. That will be true if the list consists of a single element, and in that case the function will simply return the value of the first and only element. Otherwise, the function recursively adds the first element of the list to the sum of the elements in the back (cdr) of the list. When we evaluate listSum, we get the correct result: > (listSum (list 2 4 5)) 11 We can go one step further and make a function that behaves like the + function, which accepts any number of addends. Notice that next to lambda, n appears without parentheses. Scheme will accept any number of parameters; in this case, create a list of the parameters, and pass the list to the function giving the list the name n.
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