Day 13, Mat360: Numerical Analysis

Today:

• Announcements:
  • Our first exam is Wednesday,
    • It will cover everything through Section 2.6.
    • You may use your hand-written notes. You might be advised to consolidate formulas and things you're pretty sure you'll need onto a sheet or two.

    • You may have the equivalent of a graphing calculator.

  • Section 2.4 homework is returned, as is every other homework that I have, I believe.

    My "solutions" for the homework are in the links below, but be forewarned that they're designed to give me the answers, rather than to lay out the solutions cleanly; so you may have to ask questions (which I'm happy to answer).

  • 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.

    Let's make that 30% of your grade on the exam, and the in-class portion 70%.

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

    Let's make it due Friday at 5pm.

• 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 quickly revisit My Mathematica code for Section 2.6., to see how to carry out a few of the exercises in section 2.6.

    This makes sense, since you'll be doing your own versions for the take home part of the exam.

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

      Issues:

      • How do we know if we really have a complex pair of roots, and not a real root with a spurious imaginary part?

        We know that complex pairs may have really small imaginary parts: how do we proceed?

    • 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!

  • A few things that you might expect:
    • A little bit from each section -- and that's about 8 sections, so I can't get too deep!
    • Be able to manipulate Taylor series, and bound the error term.
    • "Faking functions": we always make errors, it's just how we minimize them....

      		
       Understand alpha (speed of convergence), and
      			understand the relationship between significant digits
      			and alpha
      

    • Know conditions under which a solution will exist.
    • geometrical demonstrations of methods -- bisection, cobwebbing for fixed point, straight lines for Newton's and Secant method, quadratics for Muller.
    • Be able to do a step of methods by hand: Secant and other root finders, Aitkens, Steffensen's, etc.
    • A snippet of code, poorly constructed....

  • Then it's on to your questions, in preparation for the exam:
    • Let's look at #13, hw 2.4; it often pays to look at graphs of functions; we might also look at #11....
    • Danny requested that we look at 2.3, #13

• Links:
  • Homework solutions:
    • 1.1 (Mathematica)
    • 1.2 (Mathematica)
    • 1.3 (Mathematica)
    • 2.1 (Mathematica)
    • 2.2 (Mathematica)
    • 2.3 (Mathematica)
    • 2.4 (Mathematica)

  • My Mathematica code for Section 2.6. Muller code, too.
  • My Octave Project for 2.6 homework.
  • Octave project for 2.5
  • 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