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"Given data x_n + 1 = x_n - f(x_n)/f'(x_0) the rate convergence of an Interactive method is ""p"" it e_n + 1 alpha e^p_n c e_n + 1 is the error in 'n + 1' iteration and 'e_n' is the error in 'n' the iteration f c is constant Let 'alpha' be the root of f(x) = 0 rightarrow f(alpha) = 0 the n^th approximation to the root is 'x_n' the err orison then x_n e_n = alpha rightarrow x_n = alpha + e_n .. (1) similarly to n + 1 approximation x_n + 1 = alpha = e_n + 1 .. (2) By Newton Ruphson method x_n + 1 = x_n = f(x_n)/f'(x_n) rightarrow x_n + 1 = x_n f(x_n)/f'(x_n) rightarrow alpha + e_n + 1 alpha + e_n - f(alpha e_n)/f'(alpha + e_n) rightarrow e_n - f(alpha e_n)/f'(alpha + e_n) \r\n By Taylor series expansion to f(e + alpha) + f'(alpha + e_n) e_n + 1 = e_n - f(alpha) = f'(alpha) e_n + f''(alpha) + .. /f'(alpha) + e_n + f''"
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