MAT 2720 Discrete Mathematics Section 7 2 Solving

MAT 2720 Discrete Mathematics Section 7. 2 Solving Recurrence Relations http: //myhome. spu. edu/lauw

Goals l Recurrence Relations (RR) • Definitions and Examples • Second Order Linear Homogeneous RR with constant coefficients

*Additional Materials… l l l We will cover some additional materials that may not make 100% senses to some of you. They are here to add another layer of understanding. They are for educational purposes only, i. e. will not appear in the HW/Exam.

*Principle of Superposition (DE) l

2. 5 Example 3 Fibonacci Sequence is defined by

2. 5 Example 3 Fibonacci Sequence is an example of RR.

Recurrence Relations (RR)

Recurrence Relations (RR) RR appears a lot in sciences. In particular, they have many applications in CS (that’s why IEEE/ACM put this in the DM syllabus. )

Example 1: Population Model (1202) l Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits.

Example 1: Population Model (1202) l l Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on.

Example 1: Population Model (1202) l l l Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?

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Example 1: Population Model (1202)

Example 1 Fibonacci Sequence How to find an explicit formula?

Example 2 l Let us look at a simple example (Fibonacci Sequence is a bit more complicated at this point. ) l It will give us some ideas of how to find formulas for RR.

Example 2(a) l

Example 2(b) l * I do not like this solution, but it serves the purpose of this problem.

Example 2(c)* l RR is closed related to recursions / recursive algorithms

Example 2(c)* l Recursions are like a person call him/herself in a big cave and hear him/herself calling over and over again. . .

Example 1 Fibonacci Sequence How to find an explicit formula?

Definitions Second Order Linear Homogeneous RR with constant coefficients Note: FS belongs to this class of RR

Example 3 Solve

Recall Example 2 l A person invests $1000 at 12 percent interest compounded annually.

Example 3 Solve l

Example 3 Solve l

Expectations l l You are required to clearly show the system of equations are being solved. Points in HW and Exam are allocated to those algebraic steps.

Verifications l How do I check that my formula is (probably) correct?

Generalized Method l The above method can be generalized to more situations and by-pass some of the steps.

Theorem Second Order Linear Homogeneous RR with constant coefficients Characteristic Equation 1. Distinct real roots t 1, t 2 : 2. Repeated root t :

Example 4 Solve

HW Expectations l l l You are expected to use the Theorem above as demo in example 4. Do not model your HW after example 3. It meant to introduce you to theorem and ideas. So it has extra steps that are not necessary if we use the Theorem. Also, a lot of points goes to the details of the algebra. Please do not skip steps. If you are doing solo, . .

*The Theorem looks familiar? l Where have you seem a similar theorem?