Section 2.4: Recursion

Abstract:

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

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.
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):

Solving Recurrence Relations

Vocabulary:

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

Example:

  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.

displaymath150

``Expand, guess, verify''!



LONG ANDREW E
Tue Feb 12 19:32:48 EST 2002