Understanding LU Decomposition Method

Sep 23, 2024

Notes on LU Decomposition Method

Introduction

  • Topic: Solving a system of linear equations using LU decomposition method.
  • LU decomposition involves splitting a matrix into:
    • Lower triangular matrix (L)
    • Upper triangular matrix (U)

System of Linear Equations

  • Given a system with variables x1, x2, x3.
  • Format: A * x = B
    • A: Coefficient matrix
    • x: Variables to solve for
    • B: Result matrix

Coefficient Matrix (A)

  • Example of matrix A:
    • A =
      | 1 1 1 |
      | 4 3 -1 |
      | 3 2 0 |
  • Ensure the order of coefficients is correct; if missing, use 0.

Definitions of L and U

  • L: Lower triangular matrix (Upper triangle elements = 0)

    • Example structure:
      | 1 0 0 |
      | L21 1 0 |
      | L31 L32 1 |

  • U: Upper triangular matrix (Lower triangle elements = 0)

    • Example structure:
      | U11 U12 U13 |
      | 0 U22 U23 |
      | 0 0 U33 |

Steps to Solve for L and U Values

  1. Identify elements for L and U using the equations derived from their definitions.
  2. Multiply corresponding elements to fill in the matrices based on given values.
  3. Solve for each variable (L and U) by comparing positions.

Example Calculation:

  • For U11 = 1, L21 = 4, etc.
  • L21 * U11 = 4
  • Update U and L as:
    • L21 = 4
    • U22 = 3 - 4 = -1
    • U33 = 3 - (-2) = 10

Forming the Equation

  • After obtaining L and U, substitute A with LU:
    • Let Ux = Y
    • L * Y = B

Finding Values of Y

  • Solve LY = B:
    • Example equations derived:
      • y1 = 1
      • 4y1 + y2 = 6
      • 3y1 - 2y2 + y3 = 4

Solving for Y

  • y1 = 1
  • y2 = 2
  • y3 = 5

Back Substitution to Find X

  • Solve Ux = Y:
    • Example equations derived:
      • x1 + x2 + x3 = 1
      • -x2 - 5x3 = 2
      • -10x3 = 5

Final Values

  • x3 = 1/2
  • x2 = 1
  • x1 = 1

Conclusion

  • The values obtained from decomposition are x1 = 1, x2 = 1/2, x3 = -1/2.
  • Importance of following specific methods as directed by problem statements.

Closing Remarks

  • Encouragement to engage with the content (like, share, subscribe).
  • Acknowledgment of the monotony during explanations and a promise for more engaging content in future videos.