Question

Please answer all questions

Q2 2015

a) show that the function f(x) = pi/2-x-sin(x)

has at least one root x* in the interval [0,pi/2]

b)in a fixed-point formulation of the root-finding problem, the equation f(x) = 0 is rewritten in the equivalent form x = g(x). thus the root x* satisfies the equation x* = g(x*), and then the numerical iteration scheme takes the form x(n+1) = g(x(n))

prove that the iterations converge to the root, provided that the starting guess x0 id in some interval around the root x* in which the condition |g'(x)| <K<1 holds true for every value of x in that interval.

c)by writing the root-finding problem in part a? in the form x=pi/2-sinx

show that a fixed point iteration scheme should converge for any starting guess x0 in the interval 0<x0<pi/2

2. (a) Show that the function f(x) -x-sin x 2 has at least one root x* in the interval 0, [5 points (b) In a fixed-point formulation of the root-finding problem, the equation f(x) 0 is re-written in the equivalent form x = g(x). Thus the root x* satisfies the equation x*=g(x*), and then the numerical iteration scheme takes the form 1 = g (x,) Prove that the iterations converge to the root, provided that the starting guess x is in some interval around the root x* in which the condition g (xK<1 x in that interval holds true for every value of [10 points] (c) By writing the root-finding problem in part (a) in the form x= -sinx 2 show that a fixed-point iteration scheme should converge for any starting guess in the interval 0<xo <7. 5 points]

Answer 1

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