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CS 312: Algorithm Analysis

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. Announcements. HW #5 Due Today

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CS 312: 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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