Jacobi and Gauss-Seidel Iteration methods are numerical techniques used to solve systems of linear equations. They involve repeatedly updating the values of the unknown variables until a solution is reached.
The main difference between these two methods is the way in which they update the values of the unknown variables. In Jacobi Iteration, all variables are updated simultaneously using the values from the previous iteration. In Gauss-Seidel Iteration, each variable is updated as soon as a new value is computed, using the most recent values of the other variables.
Both methods are effective for solving systems of linear equations, but Jacobi Iteration is generally faster and more accurate when the system is diagonally dominant (when the absolute value of the diagonal element in each row is larger than the sum of the absolute values of the other elements in the same row). Gauss-Seidel Iteration is better suited for systems that are not diagonally dominant.
These methods are useful because they can be applied to any size system of linear equations and do not require the equations to be in any specific form. They also converge to a solution faster than other iterative methods, such as the Gauss-Jordan elimination method.
One limitation is that these methods may not converge to a solution if the system is not diagonally dominant or if the initial values chosen for the unknown variables are not close enough to the actual solution. In addition, they may not be efficient for large, sparse systems of linear equations.