I. 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?
II. What are the condition numbers of the followingfunctions? Where are they large?
• (x − 1)α,
• lnx,
• ex,
• arccosx.
III. The last Exercise in Section 1.3.5 in the notes.
IV. Consider the function f(x) = 1 − e−x for x ∈ [0,1].
• Show that condf(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 condA(x) for x ∈ [0,1].
• Use C++ to plot condf(x) and condA(x) as a function of x on [0,1]. Discuss your results.
V. The math problem of root finding for a polynomial
can be considered as a vector function f : Rn →C: r = f(a0,a1,...,an−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?
