4 HW questions on Numerical Analysis

1.Let g(x) = π + 0.5 cos(x/ 2) . (a) Using the theorem from the course, show that g(x) has a unique fixed point on the interval [0, 2π]. (b) For p0 = π/2, compute p1. Is the fixed point iteration convergent? Justify your answer. (c) How many iterations are necessary to achieve the accuracy 10^−2 if fixed-point iteration is used to approximate the fixed point? 2.Let f(x) = xe^(1−x ) − 1. (a) p = 1 is a zero of f. Apply Newton’s iteration to f. Using an arbitrary p_0 close to 1, give the formula for calculating p_(n+1) from p_n for this function. Are there any restrictions on applying the iteration step? (b) What can you say about the rate of convergence of Newton’s method in this case? Explain your answer. (c) To improve the rate of convergence, please give another iteration formula based on this function f 3.Consider the data obtained by evaluating f(x) = 2^x x f(x) 0 1 1 2 2 4 3 8 (a) Give the Lagrange polynomial that interpolates the above data. (b) Verify the result obtained in (a) using Newton’s Divided-Difference Formula. (c) Use the theorem from the course to find an error bound for the above approximation. (d) If using piecewise linear polynomial to approximate f(x) = 2^x on [0,3], how many points (including endpoints 0 and 3) are required to achieve the accuracy 10^(−4)? 4.True (T) or False (F) (a) For a function whose derivatives are difficult to compute, the bisection method is the only method for finding the roots. (b) If there are three roots of a continuous function in an interval [a, b] and f(a) f(b) ≤ 0, then the bisection method starting with [a, b] will converge to one of the roots in the interval. (c) Given six distinct data values xi , i = 0, …, 5, there exits a unique polynomial of degree 6 that interpolates the data. (d) Let f(x) be an affine function with non-zero slope. Newtons method will converge to the root of f(x) = 0 for any initial guess x0. (e) If f(x0) and f(x1) are such that f(x0)f(x1) < 0, then the secant method will always converge to at least one root in [x0, x1]. Requirements: As clear as possible