How To Solve Linear Equasions In Matlab Gauss Seidel

Walter Cronk

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30-04-2025

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The Gauss-Seidel method is an iterative numerical technique used to solve systems of linear equations. It's particularly useful when dealing with large systems where direct methods (like Gaussian elimination) become computationally expensive or impractical. This guide will walk you through implementing the Gauss-Seidel method in MATLAB.

Understanding the Gauss-Seidel Method

Before diving into the MATLAB code, let's briefly review the underlying principles. The Gauss-Seidel method iteratively refines an initial guess for the solution until it converges to a sufficiently accurate result. Each iteration updates the solution components one by one, using the most recently computed values.

The core idea is to rearrange the system of equations to isolate each variable:

xᵢ = (bᵢ - Σⱼ≠ᵢ aᵢⱼxⱼ) / aᵢᵢ 

where:

• xᵢ is the i-th unknown variable.
• bᵢ is the i-th element of the constant vector.
• aᵢⱼ is the element in the i-th row and j-th column of the coefficient matrix.

The iteration continues until the difference between successive approximations falls below a specified tolerance.

Implementing the Gauss-Seidel Method in MATLAB

Here's how you can implement the Gauss-Seidel method in MATLAB. This code includes error handling and a convergence check:

function [x, iterations] = gaussSeidel(A, b, x0, tolerance, maxIterations)
  % Solves a system of linear equations Ax = b using the Gauss-Seidel method.
  %
  % Args:
  %   A: The coefficient matrix.
  %   b: The constant vector.
  %   x0: The initial guess for the solution vector.
  %   tolerance: The convergence tolerance.
  %   maxIterations: The maximum number of iterations.
  %
  % Returns:
  %   x: The approximate solution vector.
  %   iterations: The number of iterations performed.

  n = length(b);
  x = x0;
  iterations = 0;

  %Check for diagonally dominant matrix
  if ~isDiagonallyDominant(A)
      warning('Matrix is not diagonally dominant. Convergence is not guaranteed.');
  end

  for i = 1:maxIterations
    x_old = x;
    for j = 1:n
      sum = 0;
      for k = 1:n
        if k ~= j
          sum = sum + A(j, k) * x(k);
        end
      end
      x(j) = (b(j) - sum) / A(j, j);
    end

    if norm(x - x_old) < tolerance
      iterations = i;
      return;
    end
  end

  warning('Gauss-Seidel method did not converge within the specified number of iterations.');
end

function result = isDiagonallyDominant(A)
  % Checks if a matrix is diagonally dominant.
  n = size(A,1);
  result = true;
  for i = 1:n
    sum = 0;
    for j = 1:n
      if i ~= j
        sum = sum + abs(A(i,j));
      end
    end
    if abs(A(i,i)) <= sum
      result = false;
      break;
    end
  end
end

How to Use:

1. Define your system: Specify your coefficient matrix A and constant vector b.
2. Provide an initial guess: Choose an initial guess x0 for the solution vector. A vector of zeros is often a reasonable starting point.
3. Set parameters: Define the tolerance (e.g., 1e-6) and maxIterations (e.g., 1000).
4. Call the function: Use [x, iterations] = gaussSeidel(A, b, x0, tolerance, maxIterations); to solve the system.

Example:

A = [10 -1 2; -1 11 -1; 2 -1 10];
b = [6; 25; -11];
x0 = [0; 0; 0];
tolerance = 1e-6;
maxIterations = 1000;

[x, iterations] = gaussSeidel(A, b, x0, tolerance, maxIterations);

disp('Solution:');
disp(x);
disp(['Iterations: ', num2str(iterations)]);

Remember that the Gauss-Seidel method's convergence is not guaranteed for all systems. It works best when the coefficient matrix is diagonally dominant (the absolute value of each diagonal element is greater than the sum of the absolute values of the other elements in its row). The code includes a check for diagonal dominance, issuing a warning if this condition isn't met. If convergence issues arise, consider alternative iterative methods or direct solvers.