Section 2.4: Recursion and Recurrence Relations

Abstract:

In this section we examine the definition and multiple applications of recursion, and encounter many examples. We also solve one type of linear recurrence relation to give a general closed-form solution.

Recursion

A recursive definition is one in which

1. A basis case (or cases) is given, and

2. an inductive or recursive step describes how to generate additional cases from known ones.

Example: the Factorial function sequence:

1. F(0)=1, and

2. F(n)= nF(n-1).

Note: This method of defining the Factorial function obviates the need to ``explain'' the fact that F(0)=0!=1. For that reason, it's better than defining the Factorial function as ``the product of the first n positive integers,'' which it is from n=1 on....

In this section we encounter examples of several different objects which are defined recursively (See Table 2.5, p. 131):

• sequences (e.g. Fibonacci numbers - Practice 12, p. 122 - history, #29/32, p. 140/142)

  Note: Examples #31 and #32 illustrate why you want to stop and think before you attempt the proof!

• sets (e.g. palindromic strings - Practice 16 and 17, p. 124/125)

• operations (e.g. string concatenation - Practice 18, p. 125/126)

• algorithms (e.g. BinarySearch - Practice 20, p. 131; check out Example #41, p. 130, for the definition of ``middle''.)

Solving Recurrence Relations

Vocabulary:

• linear recurrence relation: S(n) depends linearly on previous S(r), r<n:

  

  The relation is called homogeneous if g(n)=0. (Both Fibonacci and factorial are examples of homogeneous linear recurrence relations.)

• first-order: S(n) depends only on S(n-1), and not previous terms. (Factorial is first-order, while Fibonacci is second-order, depending on the two previous terms.)

• constant coefficient: In the linear recurrence relation, when the coefficients of previous terms are constants. (Fibonacci is constant coefficient; factorial is not.)

• closed-form solution: S(n) is given by a formula which is simply a function of n, rather than a recursive definition of itself. (Both Fibonacci and factorial have closed-form solutions.)

The author suggests an ``expand, guess, verify'' method for solving recurrence relations.

Examples:

1. Practice 11, p. 121

2. Practice 19, p. 128

3. Practice 21, p. 133

Example: general linear first-order recurrence relations with constant coefficients.

``Expand, guess, verify'' (then prove by induction!):