CS3911 Reading List: Week 4

• Section 2.1, 2.3
• The Non-Linear Equations slides can be found in the common directory: NL-eqn.pdf.

• Problems P2.21
• Extra Problems:
  • We discussed in class that if -LOG₁₀(|x-x^*|/ |x^*|) = k and x is close to x^*, the approximation value x and the true value x^* have about k most significant digits identical. Prove this claim with a convincing argument. Do keep in mind that prove-by-example is NOT a proof. Prove-by-example means the use of an example to prove a proposition.
  • Write a small Fortran program to compute the variance of three values with the one-pass and two-pass methods. Then, run it with 9000000, 9000001 and 9000002 as input. Is the computed variance 1? Use hand-calculation without the help of a calculator to determine what went wrong in your program's output.
  • Use fixed-point iteration to find a root of f(x) = e^-x - x². Do the following:
    1. Use gnuplot to plot the graph of f(x) = e^-x - x² to locate a root.
    2. Transform the equation to a form of x = g(x).
    3. Use gnuplot to plot a graph of x = g(x) and y = x, and check the tangent line slop of x = g(x) to ensure fixed-point iteration can converge.
    4. Use your calculator to do a fixed-point iteration. Note that you should avoid all kinds of floating point problems (e.g., over- and under- flow, division-by-0, cancellation, etc) . Otherwise, you should try a different transformation and/or a new initial guess.

You do not have to turn in your paper. What I really expect you to do is using these problems to gauge your understanding of the subject. So, do the problems after finish reading the above sections.