Lecture Notes: Method of Characteristic Root for Linear Recurrence Relations

Introduction

• Topic: Method of Characteristic Root to solve linear recurrence relations with constant coefficients.
• Prerequisites: Understanding of what recurrence and linear recurrence relations are.
• References: Video 1 of the playlist for foundational concepts.
• Categories of Recurrence Relations:
  • Homogeneous Recurrence Relation
  • Non-homogeneous Recurrence Relation

Homogeneous Recurrence Relation

• Definition: A recurrence relation is homogeneous if the function fn = 0.
• Steps to Solve:
  1. Assume (a_n = r^n), (a_{n-1} = r^{n-1}), etc.
  2. Substitute in the recurrence relation to derive the characteristic equation.
  3. Solve the characteristic equation to find characteristic roots._

Characteristic Roots

• Distinct Roots: If roots are distinct (e.g., (r_1, r_2, r_3)), the general solution (a_n = B_1r_1^n + B_2r_2^n + B_3r_3^n).
• Repeated Roots:
  • If a root repeats twice (e.g., (r_1, r_1)), the solution is ((B_1 + nB_2)r_1^n).
  • If a root repeats thrice (e.g., (r_1, r_1, r_1)), the solution is ((B_1 + nB_2 + n^2B_3)r_1^n).

Solving Recurrence Relation

• Example 1: Solve (a_n = 4a_{n-1} - a_{n-2})

  • Initial Conditions: (a_0 = 0), (a_1 = 1).
  • Derive the characteristic equation: (r^2 - 4r + 1 = 0).
  • Roots: (r_1 = 2, r_2 = -1).
  • General Solution: (a_n = B_1 , 2^n + B_2 , (-1)^n).
  • Solve for constants using initial conditions.

• Example 2: Solve (a_n = 4a_{n-1} - a_{n-2}) with initial conditions (a_0 = 0) and (a_1 = 1).

  • Repeat steps to find the general solution.

Advanced Example

• Example 3: (a_n = a_{n-1} + 2a_{n-2} + a_{n-3})
  • Characteristic Equation: (r^3 - r^2 - 2r - 1 = 0).
  • Roots: (r_1 = -1, r_2 = 2, r_3 = -2).
  • Solution: (a_n = B_1(-1)^n + B_2(2)^n + B_3(-2)^n).
  • Use given initial conditions to solve for constants._

Practice Problems

• Two unsolved questions provided for practice.
• Answers included for self-assessment.

Conclusion

• Encouragement to practice and apply learned methods.
• Upcoming video topics: Methods for non-homogeneous recurrence relations, covering different cases of fn.

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