LU Decomposition

Theory :

LU Decomposition is a method of factorizing a matrix into the product of two simpler matrices:

A=LUA = LUA=LU

Where:

• A is the original square matrix
  
  

• L is a lower triangular matrix (all entries above the diagonal are 0)
  
  

• U is an upper triangular matrix (all entries below the diagonal are 0)

Application Problem :

In a distributed computing system , 3 servers process tasks with varying execution times. Server 1 takes 2 seconds for task A , 4 seconds for task B and 1 second for task C

Server 2 takes 3 seconds for task A , 1 seconds for task B and 2 seconds for task C.

Server 3 takes 1 seconds for task A, 2 seconds for task B and 3 seconds for task C.

The system must follow the constraints :

1. Task A must take exactly 6 seconds across all servers.

2. Task B must take exactly 8 seconds across all servers.

3. Task C must take exactly 7 seconds across all servers.

Using LU decomposition, determine how much time each server should run to satisfy all these constraints.

Reflection Questions :

1. What is LU decomposition, and what conditions must a matrix satisfy for LU decomposition to be performed without pivoting?
LU decomposition factors a matrix AAA into the product of a lower triangular matrix LLL and an upper triangular matrix UUU, such that A=LUA = LUA=LU.
For LU decomposition to be performed without pivoting, the matrix must be nonsingular, and all leading principal minors (determinants of top-left k×kk \times kk×k submatrices) must be non-zero. This ensures numerical stability and prevents division by zero during the process.

2. How does LU decomposition improve computational efficiency when solving multiple systems with the same coefficient matrix?
When solving systems of the form AX=BAX = BAX=B for multiple right-hand sides BBB but the same coefficient matrix AAA, LU decomposition allows you to:

• Factor AAA once as LULULU

• Then solve LY=BLY = BLY=B using forward substitution, and UX=YUX = YUX=Y using backward substitution
  This avoids refactoring AAA every time, making the process significantly faster for multiple systems.

3. How do forward and backward substitution simplify the process of solving linear systems compared to Gaussian elimination?
Once LU decomposition is done, forward and backward substitution are much more efficient than Gaussian elimination.

• Forward substitution solves LY=BLY = BLY=B step by step since LLL is lower triangular.

• Backward substitution solves UX=YUX = YUX=Y step by step since UUU is upper triangular.
  These methods involve fewer operations and are easier to implement, especially for triangular systems.

4. How does LU decomposition compare to matrix inversion and Gaussian elimination in terms of computational efficiency?

• LU decomposition is typically faster and more stable than matrix inversion for solving linear systems.

• Matrix inversion is computationally expensive and not preferred when solving equations.

• Compared to Gaussian elimination, LU decomposition avoids repeating the elimination steps when solving multiple systems with the same AAA, leading to better performance.

5. What are some real-world applications of LU decomposition, and why is it beneficial in those fields?
LU decomposition is widely used in:

• Engineering simulations (e.g., structural analysis, electrical circuits)

• Computer graphics (for solving systems related to transformations)

• Economics and finance (for solving large systems in models)

• Scientific computing (like finite element analysis or climate models)
  It is beneficial because it allows for efficient, stable, and reusable solutions, especially when solving multiple systems with the same coefficient matrix.