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Question: numerical analysis suppose that x gx has unique solution...

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(Numerical Analysis)

Suppose that x = g(x) has unique solution ξ with g′(ξ) = 0. Also assume that g is twice differentiable on some closed interval I containing ξ and x0, and that g′′(x) is continuous on I.

(a) Show that if g′(ξ) = 0, then the simple iteration given by xk+1 = g(xk), with x0 given above, converges quadratically to ξ. (hint: Perform a Taylor expansion of g(xk) at x = ξ)

(b) Show that with g(x) = x − f(x) , x = g(x ) yields Newton’s method.f′(x) k+1 k

Suppose that x-g(z) has unique solution ξ with g(E)-0. Also assume that g is twice differentiable on some closed interval I containing ξ and zo, and that g(x) is continuous on I. (a) Show that if g(£)0, then the simple iteration given by k+1- g(xk), with zo given above, converges quadratically to ξ. (hint: Perform a Taylor expansion of g(x) at x-ξ) b) Show that with 9(2)-X- M-, Xk±1-9(2k) yelds Newtons method.

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