Numerical Analysis Suppose that we want to use the fixed poi

Numerical Analysis

Suppose that we want to use the fixed point iteration method for g(x) = x, starting at xO. Consider the following two functions Explain why the fixed point iteration for both functions must converge if xO is sufficiently close to the fixed point. Do you anticipate a difference in the convergence rate of the iterations? Explain. Do you expect the iterations to be globally convergent? Why or why not?

Solution

(a) The error in the calculation of the value of x

error after kth iteration = xk - x*, where x* denotest the final value

g(x*) = x* will be the final solution, using the taylor theorem e_(k+1) = g\'(x*)e(k), therefore if the modulus value of |g\'(x*)| < 1, then the error will always be less than the final error, hence if we choose initial variable very close then the fixed point iteration will be going to converge

b) The difference in the convergence rate of iterations will be depending upon the value of g\'(x), since the value of g\'(x) = -sin(x) for both the function, hence there will be no difference in the convergence rate of the iterations

c) The iterations must be locally convergent, the iterations can be globally convergent is not true, since we can expect the iterations to provide a valid solution