Question

What are the basic steps involved in solving a system of equations with the Gauss-Seidel method?

Short answer

Answer: The key steps involved in solving a system of linear equations using the Gauss-Seidel method are: 1. Write the system in matrix form Ax = b. 2. Decompose matrix A into L, D, and U matrices. 3. Choose an initial approximation for the solution vector x^(0). 4. Define the Gauss-Seidel iterative formula (x^(k+1) = (D+L)^(-1) * (b - U*x^(k))). 5. Compute the next approximation x^(k+1) sequentially by updating x^(k) as you go. 6. Check for convergence using an error tolerance. If not converged, repeat Steps 5 and 6. 7. When converged, the most recent approximation x^(k+1) is the solution to the system of equations.

Step-by-step solution

Matrix form of the system

Write the given system of linear equations in the matrix form Ax = b, where A is the matrix of coefficients, x is the vector of unknowns, and b is the vector of constants.

Define the matrices L, D, and U

Decompose matrix A into three matrices: L (lower triangular matrix), D (diagonal matrix), and U (upper triangular matrix). That is, A = L + D + U.

Starting values for x

Choose an initial approximation for the solution vector x^(0), typically a vector of zeros or a vector based on initial guesses.

Iterative formula

Define the Gauss-Seidel iterative formula as follows: x^(k+1) = (D+L)^(-1) * (b - U*x^(k)) where k is the iteration number.

Compute the next approximation

Using the iterative formula above, compute the next approximation x^(k+1) for each unknown in the system sequentially by updating x^(k) as you go. Continue updating the rest of the unknowns following the order defined in the linear system.

Check for convergence

Once the next approximation x^(k+1) is calculated, check for convergence. The convergence criterion can be found using one of the methods such as the absolute error |x^(k+1) - x^(k)| of unknowns or the relative error |x^(k+1) - x^(k)| / |x^(k+1)|. If the error is less than a given tolerance, move to the next step. Otherwise, iterate the process with Step 5 and 6.

Output the solution

When the convergence criterion is satisfied, the most recent approximation x^(k+1) represents the solution to the system of linear equations. Thus, x^(k+1) is the iterative solution obtained using the Gauss-Seidel method.