Day 21, Mat360: Numerical Analysis

Today:

• Announcements:
  • Your homework assignment for section 3.3 is returned.

    Here's my key, and I've put your octave files on-line if you want to try out others' work.

    Octave-online version.

  • You have an assignment due today (over 4.1).

  • A reminder: 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!)
    • Here's the link to sign up, in case you don't like QR codes.
    • Schedule:
      • 9:00: Hyperbolic crochet
      • 10:30: Origami
      • 11:30: Lunch
      • noon: Korean drumming

• Let's recall what we did last time:
  • We worked on the 4.1 problems 4-7, making use of the tools that we learned from the interpolation chapter.
  • Didn't quite get to 17, so I emailed that over the weekend.

• Today:
  • Let's begin by reviewing #17 of section 4.1
    • Draw the situation (the "stencil").
    • What is the upshot?

  • Then we'll take a look at Section 4.2 (no homework assigned from 4.2).
    • The big idea: our methods are all producing the same result: we're using linear combinations of a few known data points and simple rules (built on "stencils"!) to produce approximations to function values, derivatives, integrals, etc.

    • What if we just assume that we're going to be able to write our estimates that way, and then try to figure out what the coefficients should be for a given stencil?

      But how are we going to figure out those coefficients?

      By making sure that our methods are exact for polynomials of degree $n$ or lower,

      where $n$ is chosen because we're using $n+1$ data points (usually $x_0,x_1,\dots ,x_n)$ as the backbone of our stencil.

      "In particular, it must be exact for the polynomials $p_j(x)=(x-x_0{)}^j,j=0,1,\dots ,n$." (p 143)

      These polynomials for a basis for the space of all polynomials of degree $n$ or lower. Since all of the operations we're considering are linear, if we know the answers for these polynomials we can build the answer for an arbitrary polynomial of degree $n$ or lower.

    • "The Crumpet is crucial": on 143, Vandermonde Matrices

    • Consider the example of computing first and second derivatives for the standard stencil (and notice that $x_0$ is right in the middle...:),
      

      (pp. 143-144).

    • Let's walk through this example.

      We end up with a system of linear equations (so time to whip out that Gaussian elimination!:). But also sometimes symbolic manipulation, and so our author includes a little bit about Maxima in this chapter (Crumpet 29, p. 147).

    • And, in these two cases, solving the system is easy enough, with a little inspection.

  • Let's take a look at problem #5 from 4.2, and see how our work compares to that of #17.

  • Next up is 4.3: Error Analysis
    • We'll finally figure out how well we're doing, using Taylor series and getting estimates
    • At the end of section 4.3 is a table with a lot of the standard methods for doing numerical calculus, and their error formulas.

  • Then, next week, we'll have our second exam, which will cover through 4.3. I'll have you submit your 4.3 homework once again with the exam, as your "take home" portion. It will again focus on some numerical implementations of these methods.

  • On Monday next week we'll review.

• 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