CAS PROJECT. Gauss-Seidel Iteration. ta) Write a program for Gauss-Seidel iteration. (b)

Chapter 20, Problem 20.1.47

(choose chapter or problem)

CAS PROJECT. Gauss-Seidel Iteration. ta) Write a program for Gauss-Seidel iteration. (b) Apply the program to A(t)x = b, starting from [0 0 O]T. where A(t) = r: t J b = rJ For t = 0.2.0.5.0.8.0.9 determine the number of steps to obtain the exact solution to 6S and the corresponding spectral radius of C. Graph the number of steps and the spectral radius as functions of t and comment. (c) Successive overrelaxation (SOR). Show that by adding and subtracting xCm) on the right, formula (6) can be written XCm+l) = XCm) + b - LxCm + 1 ) - (U + l)xCm) Anticipation of further corrections motivates the introduction of an overrelaxation factor w > I to get the SOR formula for Gauss-Seidel XCm + ll = XCm) + web - LxCm + ll (14) (ajj = 1) intended to give more rapid convergence. A recommended value is w = 2/(1 + ,h - p). where p is the spectral radius of C in (7). Apply SOR to the matrix in (b) for t = 0.5 and 0.8 and notice the 851 improvement of convergence. (Spectacular gains are made with larger systems.)