Question

this is numerical analysis please do all the questions

1. A function g(x) is called a contraction on the interval (a,b) if g([a, b]) c [a, b] and moreover, there exists 0 <k < 1 such that væ, y € [a, b] we have 19(x) – 9(y)<k|x – yl. (a) Find d > 0 such that the function g(x) = cos x is a contraction on (0.5 - 4,0.5 + d). Justify fully. Hint: The cosine of 1 radian is about 0.54. (b) Let g(x),d be as above. Show that g(x) cannot be a contraction on every interval [0.5-6,0.5+ C (0.5-d,0.5+d). Hint: Suppose it was a contraction on every interval. What would that imply? (c) Write down an iterative scheme of the form 4n+1 = G(xn) that converges quadratically to a fixed point of g(x) = cos x for initial guess close enough to the fixed point.

Answer 1

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