Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 5.8 Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University How to Find an Explicit Formula? Iteration is a basic technique, and thus limited. A second-order linear homogeneous recurrence relation with constant coefficients is a recurrence relation of the form for all integer some fixed integer, where A and B are fixed real numbers with 2

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Terms A second-order linear homogeneous recurrence relation with constant coefficients? Second-order refers to the fact that the expression for contains the two previous terms Linear to the fact that and appear in separate terms (Non-linear: ) Homogeneous to the fact that the total degree of each term is the same Constant coefficients to the fact that A and B are fixed real numbers which do not depend on k 3

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University A Simple Test 4 Non second- order Non linear Non homogeneous Non constant coefficients

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Logic (from Chapter 2) How to show two statements are equivalent? Suppose we want to prove “ A is true if and only if B is true.” First, show B is true given that A is true: Second, show A is true given that B is true: Then, A and B are equivalent! Examples 3 k = 6 if and only if k = 2 4 k = 8 if and only if ( k -2)( k -3) = 0 5

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Lemma and Characteristic Equation Let A and B be real numbers. A recurrence relation of the form (a second-order linear homogenous recurrence relation with constant coefficients) is satisfied by the sequence where t is a nonzero real number, if and only if t satisfies the equation 6 : the characteristic equation of the relation.

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Lemma and Characteristic Equation Suppose is satisfied by the sequence where t is a nonzero real number. Then, we have 7

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Lemma and Characteristic Equation If a non-zero real number t satisfies the equation then we have Now, suppose. Then, we have That is satisfies the recurrence relation. 8

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Example Suppose we have Then, the characteristic equation of this relation is Here we have two possible sequences: Possible explicit formulas are 9 Which one is correct? => Depend on the initial cond.

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Lemma If and are sequences that satisfy the same second-order linear homogeneous recurrence relation with constant coefficients, and if C and D are any numbers, then the sequence defined by the formula also satisfies the same recurrence relation. 10

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Example Suppose we have Then, the characteristic equation of this relation is None of are right. Set We have 11

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Theorem Distinct-Roots Theorem Suppose a sequence satisfies a recurrence relation for some real numbers A and B with and all integers. If the characteristic equation has two distance roots r and s, then is given by the explicit formula Where C and D are the numbers whose value are determined by the values and. 12

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Distinct-Roots Theorem Example The Fibonacci sequence Characteristic equation: Note that Set 13

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Lemma The Single-Root Case Let A and B be real numbers and suppose the characteristic equation has a single root r. Then the sequences and both satisfy the recurrence relation for all integer since and if we assume 14

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Theorem Single-Roots Theorem Suppose a sequence satisfies a recurrence relation for some real numbers A and B with and all integers. If the characteristic equation has a single (real) root r, then is given by the explicit formula Where C and D are the real numbers whose values are determined by the values of and any other known value of the sequence. 15

Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Example Suppose a sequence satisfies a recurrence relation with initial conditions. Find an explicit formula for. 16