Interactive Methods (Numeric Methods) Easy and Unique Notes of Mathematics, Study notes of Mathematics

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1.2.3 Iterative Methods As discussed earlier, iterative methods are based on the idea of successive approximations. We start with an initial approximation to the solution vector x = x,, to solve the system of equations Ax = b, and obtain a sequence of approximate vectors Xg, Xj, -.., X,, ---- Which in the limit as k > ~, converges to the exact solution vector x = Ab. A general linear iterative method for the solution of the system of equations Ax = b, can be written in matrix form as x*+*) = Hx”) +e, k=0,1,2,... (1.43) where x'*+)) and x‘*) are the approximations for x at the (k + 1)th and kth iterations respec- tively. H is called the iteration matrix, which depends on A and ¢ is a column vector, which depends on A and b. When to stop the iteration We stop the iteration procedure when the magnitudes of the differences between the two successive iterates of all the variables are smaller than a given accuracy or error tolerance or an error bound ¢, that is, (k+1) _ i | x “