In order for the lower triangular matrix D L to be invertible it is necessary and sufficient for aii 0. Convergence and experimental results are presented. We now generalize the Gauss-Seidel iteration procedure. We will leave, as an exercise for the student, the derivation, but the matrix equation for the Gauss-Seidel iteration method is as follows: xk (D L) 1Uxk 1 + (D L) 1b. used for the solution of linear systems (like GaussSeidel, SOR, Jacobi, and others) to the solution of systems of nonlinear algebraic and/or transcendental equations, as well as to unconstrained optimization of nonlinear functions. For each generate the components of from by Namely, Matrix form of Gauss-Seidel method. Application to solve a diagonal flow and to solve the Smith-Hutton problem. Derive iteration equations for the Jacobi method and Gauss-Seidel method to solve The Gauss-Seidel Method. Numerical solution to the Convection-Diffusion Equations in Fluid Mechanics. An iterative method to solve the linear system A x = b starts with an initial approximation p 0 to the solution x and generates a sequence of vectors \( \left\_0 = (2,1,0), \) then iteration will converge to the solution (3,2,1) much faster than under Jacobi method. from the first equation, its value is then used in the second equation to obtain the new and so on.