Lecture Videos Lecture 20 – Solving Recurrence Equations

Lecture 20 – Solving Recurrence Equations

You may want to download the lecture slides that were used for these videos (PDF).

1. Review of Recurrence Equations

This video reviews recurrence equations and four examples of where they arise.  Can you determine how fast the solution to a recurrence equation grows, just by looking at the recurrence? (4:37)

2. Developing a General Framework (1 & 2)

We consider the family V of all functions which map the set Z of all integers to the set C of complex numbers. V is an infinite dimensional vector space over the field C of complex numbers, with (f + g)(n) = f(n) + g(n) and (α f)(n) = α(f(n)). (6:10)

3. Developing a General Framework (3 & 4)

We will first focus on homogeneous linear recurrence equations. (3:25)

4. Developing a General Framework (5)

We define the advancement operator A on the vector space V by the rule A f(n) = f(n+1). (2:27)

5. Developing a General Framework (6)

This video introduces the following theorem: The set S of all solutions to a homogeneous linear recurrence equation is a d-dimensional subspace of the vector space, V. (3:29)

6. The Base Case

This video introduces the following theorem: If r ≠ 0, the 1-dimensional space S of all solutions to the linear homogeneous equation (A – r)f(n) = 0 has the function rn as a basis. Equivalently, all solutions of (A – r)f(n) = 0 are of the form, f(n) = crn for some constant c.  (1:47)

7. The Base Case – Proof

In this video, we prove the theorem introduced in the previous video, that if r ≠ 0, the 1-dimensional space S of all solutions to the linear homogeneous equation (A – r)f(n) = 0 has the function rn as a basis. (6:08)

8. Towards the General Case (1)

In this video, we verify that the functions (-7)n and 5n are solutions to the equation (A2 +2A-35)f(n)=0.  This is because we can factor the expression, obtaining (A + 7)(A – 5)f(n) = 0. (5:25)

9. Towards the General Case (2)

Here we extend our results from the previous slide to the complex case. (2:46)

10. Towards the General Case (3)

Here we extend our results from the previous slide to the case with repeated roots. (3:30)

11. Towards the General Case (4 & 5)

Here we once again extend our results from the previous slide to the complex case.  Then, we introduce a set of functions that are solutions to the homogeneous equation with a root of multiplicity m. (1:54)

12. Towards the General Case (6 & 7)

This video explores two examples of homogeneous equations and their solutions, while drawing connections to linear algebra. (5:33)

13. Partial Fractions

In this video, we provide a brief review of partial fractions and make connections between partial fractions and linear algebra. (5:07)

14. Differential Equations

This video explores the connections between finite dimensional vector spaces, homgeneous differential equations, and homgeneous recurrance relations. (3:57)

15. The Non-Homogeneous Equation

In this video we discuss the significantly more difficult problem of finding a solution to the non-homogenous equation. (2:57)

16. Example of a Non-Homogeneous Equation

This video introduces an example of a non-homogenous equation, gives a particular solution, and the general solution. (4:16)

17. Another Example of a Non-Homogeneous Equation

This video explores another example of a non-homogenous equation which is more complicated, and discusses some of the mathematics we will explore in the following lecture. (2:56)