Day 19, Mat360: Numerical Analysis

Today:

• Announcements:
  • Your homework over Lagrange interpolation is returned.

    Let's have a look at some of it, using my Mathematica file (and here's a pdf).

    Octave on-line: Danny's code

  • You have a homework assignment for section 3.3 due today.

  • You have a new assignment, due next Monday.

  • Reminder that we agreed to move our second exam back by a week, to 4/3.

  • Also a reminder that I've assigned your first project.

    At the time that you submit that one (due Friday, 4/5), you should submit your proposal for your second project, which will be of your own design. Your proposal should be a page or two: enough for me to gauge what you've got in mind, and hopefully make some useful suggestions.

    That project will be due the last day of class, when you will present your projects.

  • You're invited: 2024 Math/Art Pop-Up Conference We'd love to have you:
    • Fun activities
    • Meet other math majors from other schools
    • Free lunch! (Yes, there is!)

• Let's recall what we did last time:
  • Another exciting conclusion to another cliff-hanger! (The board work for that story is here.)

  • Then we discussed splines, and Bezier polynomials in particular, as we prepare to create our "sigmacs" (project 1).
  • We got acquainted with Bezier polynomials, which you'll be using in your project.

    We were just at the point of looking into these two pieces of software:

    • Bezier in Desmos.
    • Mathematica

• Today:

  We begin chapter 4, "Numerical Calculus". And I have to tell you about a warning that Prof. Holden gave me....

  "Beware of stencils!"

  • So today we'll work our way through the "motivating example": and I'll use some Mathematica code to help us.

  • Talk about these stencils:
    • The author argues that they're an organizing tool
    • They're a little abstract for the motivating example of numerical calculus, but I think that I see where our author is going (and where he will ultimately end up):

      As an example, let's think about the three standard rules for calculating the derivative, and their related "stencils":

      • backwards difference
      • centered difference
      • forward difference
      Let's try to appreciate stencils, without getting hung up on stencils. That's the challenge....

  • And then let's begin the homework with some stencil work!:)

• Links:
  • Our Textbook: Tea Time Numerical Analysis (Leon Q. Brin)
  • An example of cubic splines: how do we intepolate two points and their slopes using a cubic? We can derive the so-called "Hermite spline" as a variant of Lagrange interpolation, featuring "get out of the way" functions!

    Here's another picture of how we might build one of these.

    More on my implementation of this in Mathematica.

  • 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
  • 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