a. A grad student wants to solve f(x) = 0 by fixed point iteration. They naively...

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a. A grad student wants to solve f(x) = 0 by fixed point iteration. They naively choose the following fixed point form: which inspires the iteration scheme x = = 3x- 2x + 4x - 2, xk+1 = 3x - 2x + 4xk - 2. Does this fixed point iteration converge for any initial guess xo (assuming xo is not exactly the root of f)? Perform a small calculation to justify your answer. b. A postdoc tries to fix the grad student's mistake by proposing the following fixed point form: x = -x + 2 3 2 + which inspires the iteration scheme 2 k+1=-2 + + 3' Does this fixed point iteration converge for all initial guesses sufficiently close to the root of f? Perform a small calculation to justify your answer. c. A professor tries to improve convergence of the postdoc's scheme by proposing the following fixed point form: which inspires the iteration scheme 2 x = x = x + 3 xk+1 = - xk+ Does this fixed point iteration converge faster than the postdoc's for all initial guesses suffi- ciently close to the root of f? Perform a small calculation to justify your answer. Hint: Use WolframAlpha to calculate and/or evaluate any derivatives you need to take. a. A grad student wants to solve f(x) = 0 by fixed point iteration. They naively choose the following fixed point form: which inspires the iteration scheme x = = 3x- 2x + 4x - 2, xk+1 = 3x - 2x + 4xk - 2. Does this fixed point iteration converge for any initial guess xo (assuming xo is not exactly the root of f)? Perform a small calculation to justify your answer. b. A postdoc tries to fix the grad student's mistake by proposing the following fixed point form: x = -x + 2 3 2 + which inspires the iteration scheme 2 k+1=-2 + + 3' Does this fixed point iteration converge for all initial guesses sufficiently close to the root of f? Perform a small calculation to justify your answer. c. A professor tries to improve convergence of the postdoc's scheme by proposing the following fixed point form: which inspires the iteration scheme 2 x = x = x + 3 xk+1 = - xk+ Does this fixed point iteration converge faster than the postdoc's for all initial guesses suffi- ciently close to the root of f? Perform a small calculation to justify your answer. Hint: Use WolframAlpha to calculate and/or evaluate any derivatives you need to take.