(a) Write a program for GaussSeidel iteration. (b) Apply the program A(t)x = b, to starting from...

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(a) Write a program for Gauss–Seidel iteration.

(b) Apply the program A(t)x = b, to starting from [0    0    0]T, where

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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) Show that by adding and subtracting x(m) on the right, formula (6) can be written

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Anticipation of further corrections motivates the introduction of an over relaxation factor ω > 1 to get the SOR formula for Gauss–Seidel

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intended to give more rapid convergence. A recommended value is ω = 2/(1 + √1 - ρ) where ρ 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 improvement of convergence. (Spectacular gains are made with larger systems.)

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