- Haven't had a look at your 4.1 assignments yet.
- Your second exam is next week, Wednesday, covering material from
Chapters 3 and 4. We'll do some review Monday, after wrapping
up 4.3.
There is an assignment for section 4.3 which will serve as a take home component (again, 30%) of the exam. You may submit it by Friday of next week.
So 4.3 will not be explicitly covered on the in-class portion.
- A final 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
- Checked out #17 of section
4.1, to see how a two-point integration scheme might be derived on
a particular stencil. We were re-deriving this scheme in
section 4.2, using an alternate strategy, when we were rudely
interrupted by the end of the class session!
- We went over the "big idea" of section 4.2:
- seeing what class of polynomials we could get right;
- and designing schemes of linear combinations of function values
from a stencil by ensuring that we got a certain class of polynomials
exactly right:
"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.
- We went over the examples of using a centered stencil to construct
first and second derivative formulas for x=x_0.
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).
- Then, as noted, we were in the process of creating a formula for the integral when we found ourselves hanging from a cliff....
- Let's wrap up problem #5 from 4.2, and see how our work
compares to that of #17 from 4.1.
- Next up is 4.3: Error Analysis
- We'll finally figure out how well we're doing, using
Taylor series and getting estimates.
- On page 152, we encounter a general error term formula for
the first derivative evaluated at a node: it is the
heart of things for first derivatives:
\[
f'(x_{0})-P_{n}'(x_{0})=\frac{f^{(n+1)}(\xi)}{(n+1)!}\theta_{1}\theta_{2}\cdots\theta_{n}(-h)^{n}.\label{eq:firstDerivErrorSimplified}
\]
We note that the formula includes two important parts: a derivative, and a power of \(h\): the error is of the form $O(h^{n}f^{(k)}(\xi_{h}))$.
The power of \(h\) tells us the rate of convergence -- how fast the method converges as the step size \(h\) tends to 0; the derivative \(k\) tells us the degree of precision (the power of the polynomial that the method gets right). Polynomials of degree \(k-1\) vanish under \(k\) derivatives, so those polynomials will be evaluated without error using the scheme.
-
Let's use the general rule to get the error terms for the three-point
formulas for the derivative: left, right, and centered.
(Note: there's an error in one of the formulas in the tables at the end of the section. What do you have when you have an error in your error formula?:)
- Checking out the tables at the end of section 4.3, we can see how
to use error formulas to construct higher order methods:
Let's use two-point backward and forward difference formulas to derive the three-point centered difference formula. We do it by "balancing errors".
- Second derivative and higher formulas require a bit more work: we
need to use Taylor series to derive these methods, and their error
terms. As an example, let's discover the centered three-point second
derivative formula.
- Integral formulas can be obtained in similar fashion, and our
author spotlights Gaussian elimination as a technique to get
extraordinary precision (the scheme gets high degree polynomials
exactly right). The placement of the nodes is at the roots of special
polynomials, called Legendre polynomials.
While there is no motivation for this, the author shows that it works. Maybe it's just magic.... or a good project?
Let's see how to use Taylor series to derive a more specific error formula for the first formula of this class, the midpoint method: \[ I = \int_{x_0-h}^{x_0+h}f(x)dx \approx M = 2h f(x_0) \] (given as $O(h^{3}f^{''}(\xi_{h}))$ in the table).
- We'll finally figure out how well we're doing, using
Taylor series and getting estimates.
- Our Textbook: Tea Time Numerical Analysis (Leon Q. Brin)
- 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.
- 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....