EXAMPLE 1. Increase the value n and examine the convergence speed! 2. Solve system of linear equations by the 2D Laplacian matrix equations using Gauss-Seidel iteration! Compare the speed of convergence wit Jacobi-iteration
3. Relaxation method (SOR) The relaxation method accelerates the iteration further when relaxation parameter ω is cleverly chosen. The Gauss-Seidel iteration is derived from the use of relaxation. Let us start with the Gauss-Seidel iteration of the split . Transforming the right hand side by adding and subtracting term . Multiply the equation from left with the inverse of diagonal matrix
= This formula is equivalent to the Gauss-Seidel iteration and more evident that the steps (k + 1) -th is used in the previously calculated term . So = is the difference between the iteration steps. The correction or convergence acceleration idea is using a multiplier factor ω . It is obtained from the equation backwards to the previous derivation of the Gauss-Seidel iteration relaxation.
. Multiply the equation from left by matrix = The resulting formula of the successive over relaxation method (SOR), which is the original Gauss-Seidel-iteration when . Writing in form we get where in splitting is and 3.1. THEOREM. (Necessary condition of convergence for SOR ) It is necessary to fulfil the condition for the method of SOR. 3.2. THEOREM (Sufficient condition of convergence for SOR ) If the matrix is symmetric and positive definit (SPD) then the condition the convergence for method SOR. 73
is sufficient for