Question

Consider the system of equations 2x + 3y = 7, x − y = 1. (i). Show that a Jacobi scheme based on writing x = (7 − 3y)/2, y = x −1 fails to converge. (ii). Reformulate the problem in such a way that a Jacobi scheme does converge.
(iii). Formulate a convergent Gauss-Seidel method for the system (*), showing that the method is convergent. Compute two iterations of the method, starting from x = 0, y = 0. (iv). Use the method obtained in (iii) to write down an S.O.R. scheme for (*) and determine the value of the acceleration parameter that gives the fastest convergence.​

Answer 1

Step-by-step explanation:

Consider the system of equations 2x + 3y = 7, x − y = 1. (i). Show that a Jacobi scheme based on writing x = (7 − 3y)/2, y = x −1 fails to converge. (ii). Reformulate the problem in such a way that a Jacobi scheme does converge.

(iii). Formulate a convergent Gauss-Seidel method for the system (*), showing that the method is convergent. Compute two iterations of the method, starting from x = 0, y = 0. (iv). Use the method obtained in (iii) to write down an S.O.R. scheme for (*) and determine the value of the acceleration parameter that gives the fastest convergence.