Notes on Gauss-Seidel Method in MATLAB

Introduction

• Course on scientific computing using MATLAB
• Previous lecture covered iterative methods for solving linear systems (Jacobi method)
• Today's focus: Gauss-Seidel method

Convergence Conditions

• Gauss-Seidel method is convergent if the spectral radius (( \rho(H) )) of the convergence matrix ( H ) is less than 1.
• Spectral radius defined as:
  [ \rho(H) = \max { |\lambda| : \lambda \text{ is an eigenvalue of } H } ]
• Convergence matrix is defined as:
  [ H = -L + D^{-1}U ]
  where
  • ( L ) = lower triangular part of matrix
  • ( U ) = upper triangular part of matrix
  • ( D ) = diagonal part of matrix
• Rate of convergence is given by:
  [ -\log(\rho(H)) ]

MATLAB Code Development

• Function definition for Gauss-Seidel method:

  function x = gauss_seidel(A, b, x0, tolerance)  
  

  • Parameters:
    • A: Coefficient matrix
    • b: Right-hand side vector
    • x0: Initial guess
    • tolerance: Desired accuracy

Code Structure

1. Initialization

   • Determine matrix size ( n )
   • Initialize solution vector ( x ) with zeros
   • Start error at 1 for the while loop

2. Iteration Loop

   • While error > tolerance and iterations < 100:
     • Update solution using the formula:
     • Break down the matrix into lower, upper, and diagonal components
     • Numerical calculations for ( x_n )

3. Error Calculation

   • Find the error as the norm of the difference between the new and old estimates

4. Display Results

   • Print out the results for each iteration
   • Show eigenvalues and spectral radius

Example Runs

• Tested various initial conditions and matrix setups:
  • 3x3 systems: convergence and divergence cases showcased.
  • Rate of convergence related to the spectral radius values observed.
  • Different initial approximations yield similar convergence behavior.
• 4x4 systems demonstrated similar behavior with variations in iterations based on initial conditions.

Conclusions

• Gauss-Seidel method is effective for solving linear systems but requires diagonal dominance for guaranteed convergence.
• Importance of spectral radius in determining convergence speed and behavior.
• The next lecture will address more problems related to iterative methods.