Numerical Analysis Homework #7 Solved

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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?

The above eight questions weigh 5, 8, 5, 12, 10 points, respectively, totaling 40 points.