CS 312: Algorithm Analysis

1. This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License. CS 312: Algorithm Analysis Lecture #8: Non-Homogeneous Recurrence Relations Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick

2. Announcements • HW #5 Due Today • Questions about Homogeneous RR? • Project #2 • Questions about the project? • Early Day: Friday • Due Date: next Monday

3. 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

4. Example (cont.): Linear, HomogeneousRecurrence Relation

6. Finding the Particular Solution

7. Fibonacci in Closed Form!

8. Fundamental Theorem of Algebra • For every polynomial of degree n, there are exactly n roots. • They may not be unique.

9. Roots of Multiplicity j

10. Roots of Multiplicity j

11. Roots of Multiplicity j

12. Roots of Multiplicity j

13. Example

14. Non-Homogeneous, LinearRecurrence Relations

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

16. Example (Cont.): d=0

19. Initial Conditions

20. Example (cont.)

21. Towers of Hanoi Revisited

22. Assignment • Read: Recurrence Relations Notes, Parts III & IV • HW #6: Part II Exercises (Section 2.2)