Day 12, Mat360: Numerical Analysis

Today:

• Announcements:
  • Our first exam is Wednesday, and will cover everything through Section 2.6.
  • Section 2.4 homework is returned, as is every other homework I believe.
  • Your homework for section 2.6 will make up the "take-home" portion of your exam. Please work on this on your own, although you can ask me questions.

    Homework pp. 93--, #1-5, 6, 8, 11, 13, 14, 16, 22 (use your Muller code), 23

• Questions on anything?

• Let's recall what we did last time:
  • We revisited section 2.6, and reviewed how we could use synthetic division to find both \(p(t)\) and \(p'(t)\) for a polynomial very easily.

    This allowed us to compute a step of Newton's method.

  • We went over my code, to see how we could put all of this together, to compute Newton's with deflation.

  • We discussed Muller's method, which generalizes the secant method (two points and a straight line to probe the x-axis for a better estimate of the root): Muller's uses three points, a quadratic, and a parabola to probe for a root.

• Today:
  • Let's revisit my Mathematica code, to see how to carry out a few of the exercises in section 2.6.

  • Then it's on to your questions, in preparation for the exam:
    • Danny requested that we look at 2.3, #13

  • If we have time, I'll show you my implementation of deflating quadratics in Mathematica, described here.

    I haven't yet implemented the suggestion that Danny made, to use the reduced polynomial discovered at the time when we decide that we have approximated a root to the desired accuracy. But every millisecond counts, so I would like to do so.

• Links:
  • My Octave Project for 2.6 homework; and the answers are collected in this pdf.
  • Octave project for 2.5
  • My Mathematica code for Section 2.6. Muller code, too.
  • Divided Differences. I use these in my Muller code, and they will prove useful when it comes to interpolating polynomials.
  • Author's octave code for these complex fractal convergence diagrams
  • Our Textbook: Tea Time Numerical Analysis (Leon Q. Brin)
  • Newton's method in Mathematica
  • fixed point with Aitken's acceleration in Mathematica
  • Some nice notes (and code) snippets on fixed point iteration, including these acceleration techniques.
  • fixed point in Mathematica
  • Check out how my bisection works, and how it is applied to arcsine....
  • Various sins....
    • Sins in Mathematica
    • Sins in Octave
    • ArcSins in Mathematica