C Program for Gauss-Jacobi Method

The Gauss-Jacobi method is an iterative technique for solving systems of linear equations. Its implementation in C involves setting up the system, performing iterations to refine the solution, and checking for convergence.  Explanation: Initialization: The program takes the number of unknowns n and the augmented matrix a as input. Initial guesses for the unknowns x are set to zero. Iteration Loop: The program iterates a maximum number of times (MAX_ITERATIONS) or until convergence is achieved. Calculation of New Values: In each iteration, x_new[i] is calculated using the formula derived from the system of equations, where the values from the previous iteration (x[j]) are used for the other variables in the equation. This is the key difference from the Gauss-Seidel method, which uses the most recently calculated values within the same iteration. Convergence Check: The max_diff variable tracks the maximum absolute difference between the current and previous iteration's values for any unknown. If max_diff falls below TOLERANCE, the solution is considered converged. Output: The program prints the values of the unknowns at each iteration and the final converged solution. If convergence is not achieved within MAX_ITERATIONS, a message indicating this is displayed. Important considerations Diagonal Dominance: The Gauss-Jacobi method's convergence is guaranteed when the coefficient matrix is diagonally dominant. Rearrangement: Rearranging the rows (equations) might be necessary to achieve convergence if the matrix is not diagonally dominant. Initial Guess: A suitable initial guess can significantly reduce the number of iterations required.