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CS 312: Algorithm Analysis
Lecture #8: Non-Homogeneous
Recurrence Relations
Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick
Announcements
 HW #5 Due Today
 Questions about Homogeneous RR?
 Project #2
 Questions about the project?
 Early Day: Wednesday
 Due Date: next Friday!
Objectives
 Find specific solutions using initial
conditions
 Understand how to solve
non-homogeneous, linear, recurrence
relations with constant coefficients
 Geometric forcing functions
 Define roots of multiplicity j
Example (cont.): Linear, Homogeneous
Recurrence Relation
Finding the Specific Solution
Linear Combinations
Fibonacci in Closed Form!
Fundamental Theorem of Algebra
 For every polynomial of degree n, there
are exactly n roots.
 They may not be unique.
Roots of Multiplicity j
Roots of Multiplicity j
Roots of Multiplicity j
Roots of Multiplicity j
Example
Non-Homogeneous, Linear
Recurrence Relations
Non-Homogeneous Example
What do you notice about the problem now?
Example (Cont.)
Initial Conditions
Initial Conditions
Example (cont.)
Example (cont.)
Towers of Hanoi Revisited
Assignment
 Read: Recurrence Relations Notes, Parts
III & IV
 HW #6:
 Part II Exercises (Section 2.2)
 Towers of Hanoi using method of recurrence
relations.