- Haven't even looked at you homework
assignment for section 3.3! Not proud of it, but it's a fact...:)
- You have an assignment due
next Monday (over 4.1); but we're going to do about half of it today,
together, so you'll be off to a great start!
- 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!
- No cliff-hangers; we can't have a cliff-hanger every day, you know!
- We got started on "Numerical Calculus", keeping in mind a warning
that Prof. Holden gave me....
"Beware of stencils!"
- We worked our way through the "motivating examples", the basic
idea being to use interpolating polynomials to approximate
functions, and then doing calculus on those polynomials
(easy!).
(Here is some Mathematica code to check and try to understand the motivating examples).
- We talked a little about those stencils:
- The author argues that they're an organizing tool
- Maybe a little abstract, but they're supposed to
be abstract in some sense: there's a pattern to
the data locations, which, if repeated in some
other problem, will provide the solution for
that problem as well.
As examples, consider rules for differentiation, which use different stencils for the evaluation of \(f'(x_0)\):
- backwards difference \[ \frac{f(x_0)-f(x_0-h)}{h} \]
- centered difference \[ \frac{f(x_0+h)-f(x_0-h)}{2h} \]
- forward difference \[ \frac{f(x_0+h)-f(x_0)}{h} \]
- Let's begin the
homework with some stencil work!:)
If we're really lucky, we'll get through 4-7, and 17;
- 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 \(x_0,x_1,\ldots,x_n)\) as the backbone of our stencil.
"In particular, it must be exact for the polynomials $p_{j}(x)=(x-x_{0})^{j}, j=0,1,\ldots,n$." (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.
- 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).
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).
- 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.
- 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.
- 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....