GaussSeidli

GaussSeidli is an iterative method used for solving systems of linear equations. It is a variation of the Gauss-Seidel method, which itself is an improvement upon the Jacobi method. The core idea behind Gauss-Seidli, like its predecessors, is to repeatedly update the solution vector until convergence is achieved.

The Gauss-Seidli method distinguishes itself by utilizing the most recently computed values of the variables within the same iteration. When calculating a new approximation for a variable x_i, it uses the newly computed values for x_1, x_2, ..., x_{i-1} from the current iteration, and the old values for x_{i+1}, ..., x_n from the previous iteration. This differs from the Jacobi method, which uses only values from the previous iteration for all variables.

This feature of using updated values as they become available often leads to faster convergence compared to the Jacobi method, particularly for diagonally dominant matrices. However, convergence is not guaranteed for all systems of linear equations. Like the Gauss-Seidel method, Gauss-Seidli is generally applied to systems where the coefficient matrix is symmetric and positive-definite, or strictly diagonally dominant. The method involves rearranging each equation to solve for one variable in terms of the others and then iteratively substituting these expressions.