LU Decomposition Method - Lecture Notes

Introduction

• LU Decomposition is a method to solve a system of linear equations by splitting a matrix into its lower triangular form (L) and upper triangular form (U).
• The method consists of two main steps: creating the matrices L and U, and then using them to solve for variable values.

System of Linear Equations

• The problem presented involves solving for variables x1, x2, and x3.
• The system can be represented as:
  • Ax = B
  • Where A is the coefficient matrix, x is the variable matrix, and B is the constant matrix.
• Example coefficients for matrix A:
  • A = | 1 1 1 | | 4 3 -1 | | 3 0 1 |
• B values are provided as:
  • B = [1, 6, 4]

Creating L and U Matrices

• L (Lower Triangular Matrix):
  • Only the lower triangle has non-zero elements.
  • Example structure: | 1 0 0 | | L21 1 0 | | L31 L32 1 |
• U (Upper Triangular Matrix):
  • Only the upper triangle has non-zero elements.
  • Example structure: | U11 U12 U13 | | 0 U22 U23 | | 0 0 U33 |

Solving for Values

• Start by multiplying the respective elements:
  • Example equations derived from the multiplications:
    • U11 = 1
    • U12 = 1
    • U13 = 1
    • Further equations to find L and U values.
• Solve for L21, L31, U22, U23, U33 by substituting known values and comparing positions.

Finding Intermediate Variable Matrix Y

• Set Ux = Y, where Y is defined as Y1, Y2, Y3.
• The equation becomes:
  • Ly = B
• Solve for y1, y2, y3 using the L matrix:
  • Derived equations as follows:
    • y1 = 1
    • 4y1 + y2 = 6
    • 3y1 - 2y2 + y3 = 4

Final Value Calculation for x

• Substitute values of y into the equation Ux = Y to find x1, x2, x3:
• Resulting equations:
  • x1 + x2 + x3 = 1
  • -x2 - 5x3 = 2
  • -10x3 = 5
• Solve these to get:
  • x1 = 1, x2 = 1/2, x3 = -1/2.

Conclusion

• LU Decomposition is essential for solving linear equations when specifically requested, despite other potentially simpler methods.
• Understanding this method is crucial for handling matrix equations systematically.

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