Numerical Methods Question:

2. (a) The nonlinear equation x) = I] with {(2} = .12 5$+ 6.1476 has two positive zeros .11 and .12 with 12 13 iitl'ig and EigiB. {i} To approximate the zeros on and 4:22 we consider the fixedpoint method using the following iteration functions: 2 zL'fg and dull}: 53'5'1476' Using the information provided before, determine if 0:1 and {12 are fixed points of 9'51 (3} duh-j and %{I} and if these iteration functions are suitable to compute them. Clearly justify your answers. [12] (ii) A computer code that implements the two proposed fixedpoint methods con- verged in 85 iterations using is] {2:} and in 94 iterations using (152(3) starting the iterations from sin} = 2.5 and using tolerance 1D"3 for the stopping criterion of the increment. Similar results were obtained for other choices of 3'1"}. Why is the number of iterations so high? [10] [b] A function gs') has a zero in the interval [0, 4}. The following table reports the values of the function g at suitable points in (G, 4]: Use these values to perform three steps of the bisection method to compute the zero of 9(1) assuming that the initial interval is [0,4]. What is the obtained approximation of the zero? [3]