SISTEM PERSAMAAN LINIER DENGAN METODE DEKOMPOSISI LU

Introduction

This tutorial will guide you through solving linear equations using the LU decomposition method. LU decomposition is a powerful technique in linear algebra that allows for efficient solutions of systems of linear equations. This method is particularly useful for larger systems, making it an essential skill in various fields, including engineering, physics, and computer science.

Step 1: Understand the System of Linear Equations

• Identify the system of equations you want to solve. For example:

  2x + 3y = 5
  4x + 9y = 10
  

• Ensure that the system is represented in matrix form as AX = B, where:
  • A is the coefficient matrix,
  • X is the column matrix of variables,
  • B is the column matrix of constants.

Step 2: Perform LU Decomposition

• Factor the coefficient matrix A into two matrices: L (lower triangular matrix) and U (upper triangular matrix).
• The goal is to express A as:

  A = LU
  

• Follow these sub-steps to perform the decomposition:
  1. Set up the matrices L and U according to the dimensions of A.
  2. Use Gaussian elimination without row exchanges to fill the matrices:
     • For each pivot element in A, modify L and U accordingly.
     • Ensure all entries below the pivot in U become zero.

Step 3: Solve the Lower Triangular System

• Once you have L and U, solve the equation LY = B for Y:
  • Use forward substitution:
    1. Start with the first equation from LY = B.
    2. Substitute known values from previous equations to find new values of Y.

Step 4: Solve the Upper Triangular System

• Now, solve the equation UX = Y for X:
  • Use back substitution:
    1. Start from the last equation in UX = Y.
    2. Substitute values of Y to find corresponding values of X.

Step 5: Interpret the Results

• The final matrix X contains the values of the variables you were solving for in the original equations.
• For example, if X is:

  X = [x]
      [y]
  

  You can interpret these values as the solutions to your initial system.

Conclusion

By following the LU decomposition method, you can efficiently solve systems of linear equations. This technique streamlines the process, especially for larger systems, and is a valuable tool in many scientific and engineering applications. For further practice, try applying the LU decomposition method to different systems of equations to solidify your understanding.