Final answer:
The Gauss-Seidel method is an iterative process used to solve linear equations. It updates each variable sequentially using the most recent values, and iteration continues until the solution converges within a certain tolerance. Diagonal dominance is often necessary for ensuring convergence.
Step-by-step explanation:
Performing Gauss-Seidel Iteration
The Gauss-Seidel method is an iterative technique used to solve a system of linear equations. It's an improvement over the Jacobi method and often converges more rapidly. Here is a general step-by-step guide to perform the Gauss-Seidel iteration:
Arrange the equations so that the largest coefficient of each variable is on the diagonal of the coefficient matrix.
Start with an initial guess for each of the variables. Often, zeros or the values from a previous estimate are used.
Update each variable sequentially by solving the corresponding equation for that variable while using the most recent values for the other variables. This can be written as:
where k is the current iteration step.
Continue iterating through the variables until the changes in the values between iterations are below a chosen tolerance level, or a predetermined number of iterations has been reached.
Ensure that the diagonal dominance or convergence conditions are satisfied. Otherwise, the method may not converge.
The Gauss-Seidel method stands out for its ability to converge to the solution quicker under the correct conditions as compared to other iteration methods.