[SOLVED] Numerical Analysis-Homework 7

Description

1. If the bisection method is used in single preci-sion FPNs of IEEE 754 starting with the interval [128,129], can we compute the root with absolute accuracy < 10⁻⁶? Why?
2. What are the condition numbers of the followingfunctions? Where are they large?
   • (x − 1)^α,
   • lnx,
   • e^x,
   • arccosx.

• The last Exercise in Section 1.3.5 in the notes.

1. Consider the function f(x) = 1 − e^−x for x ∈ [0,1].
   • Show that cond_f(x) ≤ 1 for x ∈ [0,1].
   • Let A be the algorithm that evaluates f(x) for the machine number x ∈ F. Assume that the exponential function is computed with relative error within machine roundoff. Estimate cond_A(x) for x ∈ [0,1].
   • Use C++ to plot cond_f(x) and cond_A(x) as a function of x on [0,1]. Discuss your results.
2. The math problem of root finding for a polynomial

can be considered as a vector function f : Rⁿ →C: r = f(a₀,a₁,…,a_n−1).

Derive the componentwise condition number of f based on the 1-norm. For the Wilkinson example, compute your condition number, and compare your result with that in the Wilkinson Example. What does the comparison tell you?