Day 30, Mat360: Numerical Analysis

Today:

• Announcements:
  • It's project day! Time for you to show off some numerical analysis of your own.

    I hope that you've found something interesting to share, but, even if you don't win the Nobel Prize for numerical analysis (which does not exist, of course!) for your work, hopefully you've discovered something interesting to share.

  • Danny did succeed in generating Bezier quadratics for each person in the world. You draw your signature, convert it to svg, and then you've got Bezier Quadratics.

    For "my project", I converted those coordinates (using lisp), and then had my lisp code write the Mathematica commands to draw Danny's signature.

    I'm interested in figuring out how to best approximate a slew of quadratic Bezier with set of Bezier cubics. But I'm not done yet!

• Last time:
  • We summarized the RK stuff
  • We reviewed.

• Today: projects!
  • We'll draw for the order of presentation, although Danny asked to go last, in case he can use a little more time.
    1. Yimin Word doc
    2. Madison's powerpoint.
    3. Stefanie's Octave code is here, along with a powerpoint.
    4. Danny's Octave code is here, along with a report (pdf).

• Links:
  • Our Textbook: Tea Time Numerical Analysis (Leon Q. Brin)

  • Some old lisp routines illustrating the connection between adaptive quadrature and adaptive plotting:
    • Here's my lisp version of the plotting routine, then turned into an adaptive quadrature scheme:
    • Here's a similar lisp version of a local adaptive quadrature routine, which is then turned into an adaptive plotting scheme:

      It's not quite what you're asked to do in this algorithm, however, since you're to keep track of the errors, and keep poking at the largest:

    • Here's the global adaptive quadrature routine, which is then turned into an adaptive plotting scheme:

  • A paper my dad and I wrote about integration techniques: including lots of details!:)
  • Some notes
  • An interesting alternative derivation technique. Here, for example, is a four-point forward difference formula for the derivative, obtained using the Newton polynomial interpolator (notice, however, that the \(x\) values in the numerator should actually be \(y\) values -- function values):
    

    But the idea is really swell, and I am delighted by this method!

  • 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)
    • 3.2 (Mathematica)
    • 3.3 (Mathematica)
    • 4.1 (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