MATH 245 Ch16 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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DISCRETE MATHEMATICS W W L CHEN c ± W W L Chen, 1992, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 16 RECURRENCE RELATIONS 16.1. Introduction Any equation involving several terms of a sequence is called a recurrence relation. We shall think of the integer n as the independent variable, and restrict our attention to real sequences, so that the sequence a n is considered as a function of the type f : N ∪ { 0 } → R : n 7→ a n . A recurrence relation is then an equation of the type F ( n,a n ,a n +1 ,...,a n + k ) = 0 , where k N is fixed. Example 16.1.1. a n +1 = 5 a n is a recurrence relation of order 1. Example 16.1.2. a 4 n +1 + a 5 n = n is a recurrence relation of order 1. Example 16.1.3. a n +3 + 5 a n +2 + 4 a n +1 + a n = cos n is a recurrence relation of order 3. Example 16.1.4. a n +2 + 5( a 2 n +1 + a n ) 1 / 3 = 0 is a recurrence relation of order 2. We now define the order of a recurrence relation. Definition. The order of a recurrence relation is the difference between the greatest and lowest sub- scripts of the terms of the sequence in the equation. Definition. A recurrence relation of order k is said to be linear if it is linear in a n ,a n +1 ,...,a n + k . Otherwise, the recurrence relation is said to be non-linear. Chapter 16 : Recurrence Relations page 1 of 17
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Discrete Mathematics c ± W W L Chen, 1992, 2008 Example 16.1.5. The recurrence relations in Examples 16.1.1 and 16.1.3 are linear, while those in Examples 16.1.2 and 16.1.4 are non-linear. Example 16.1.6. a n +1 a n +2 = 5 a n is a non-linear recurrence relation of order 2. Remark. The recurrence relation a n +3 + 5 a n +2 + 4 a n +1 + a n = cos n can also be written in the form a n +2 +5 a n +1 +4 a n + a n - 1 = cos( n - 1). There is no reason why the term of the sequence in the equation with the lowest subscript should always have subscript n . For the sake of uniformity and convenience, we shall in this chapter always follow the convention that the term of the sequence in the equation with the lowest subscript has subscript n . 16.2. How Recurrence Relations Arise We shall first of all consider a few examples. Do not worry about the details. Example 16.2.1. Consider the equation a n = A ( n !), where A is a constant. Replacing n by ( n + 1) in the equation, we obtain a n +1 = A (( n +1)!). Combining the two equations and eliminating A , we obtain the first-order recurrence relation a n +1 = ( n + 1) a n .
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This note was uploaded on 12/20/2009 for the course MATH 245 taught by Professor ramaswamy during the Spring '08 term at San Diego State.

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MATH 245 Ch16 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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