CS 312: Algorithm Analysis Lecture #8: Non-Homogeneous Recurrence Relations This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.Creative Commons Attribution-Share Alike 3.0 Unported License 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

Finding the Particular 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

Example

Non-Homogeneous, Linear Recurrence Relations

Non-Homogeneous Example What do you notice about the problem now?

Example (Cont.)

Possible Update  Point out existence of homog. RR for every non-homog. RR.  Notation: Use y(k) (homog.) instead of z(k) (non-homog.) to emphasize the difference.

Initial Conditions

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.