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Understanding Non-Homogeneous Recurrence Relations in Algorithm Analysis

In Lecture #8 of CS 312, presented by Eric Ringger with input from Mike Jones, Eric Mercer, and Sean Warnick, we dive into the complexities of non-homogeneous recurrence relations with constant coefficients. Key topics include defining roots of multiplicity, solving non-homogeneous linear relations using geometric forcing functions, and deriving specific solutions from initial conditions. Notably, the discussion reiterates the Fundamental Theorem of Algebra concerning polynomial roots. This lecture is vital for mastering advanced algorithm analysis concepts.

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Understanding Non-Homogeneous Recurrence Relations in Algorithm Analysis

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  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

  5. Finding the Particular Solution

  6. Fibonacci in Closed Form!

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

  8. Roots of Multiplicity j

  9. Roots of Multiplicity j

  10. Roots of Multiplicity j

  11. Roots of Multiplicity j

  12. Example

  13. Non-Homogeneous, LinearRecurrence Relations

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

  15. Example (Cont.): d=0

  16. Initial Conditions

  17. Example (cont.)

  18. Towers of Hanoi Revisited

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

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