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Algorithm Analysis: Solving Non-Homogeneous Recurrence Relations

Understand how to solve non-homogeneous, linear recurrence relations with constant coefficients using geometric forcing functions. Learn to find specific solutions with initial conditions and explore roots of multiplicity. Includes examples and fundamental theorem of algebra.

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Algorithm Analysis: Solving Non-Homogeneous Recurrence Relations

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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: Wednesday • Due Date: next Friday!

  3. Objectives • Find specific solutions using initial conditions • Understand how to solvenon-homogeneous, linear, recurrence relations with constant coefficients • Geometric forcing functions • Define roots of multiplicityj

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

  5. Finding the Specific Solution

  6. Linear Combinations

  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.)

  17. Initial Conditions

  18. Initial Conditions

  19. Example (cont.)

  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) • Towers of Hanoi usingmethod of recurrence relations.

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