CS 312: Algorithm Analysis

Lecture #8: Non-Homogeneous Recurrence Relations

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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: Friday Due Date: next Monday

Objectives

Define “roots of multiplicity j” Understand how to solvenon-homogeneous, linear, recurrence relations with constant coefficients Geometric forcing function

Find the specific solution from initial conditions

Example (cont.): Linear, HomogeneousRecurrence Relation

Finding the Particular Solution

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, LinearRecurrence Relations

Non-Homogeneous Example

What do you notice about the problem now?

Example (Cont.): d=0

Initial Conditions

Example (cont.)

Towers of Hanoi Revisited

Assignment

Read: Recurrence Relations Notes, Parts III & IV

HW #6: Part II Exercises (Section 2.2)