- Your 4.1 assignments are
returned. We did 4-7 together in class, and 17 as well, so I just
focused on your implementation of these methods in the problems #8-12.
- Danny's Octave implementation
This was a great opportunity to use Octave! Those calculations by hand are painful. Save yourself all that pain!
- Here are my solutions
(using Mathematica)
- There was trouble in 11/12, because some of you used
\(x_0=0\) in your formulas.
What should we use?
- I went on a good deal longer in 11/12 than required, but what do you think would happen if we let \(h\) get smaller and smaller and smaller?
- Danny's Octave implementation
- Your second exam is Wednesday, covering material from
Chapters 3 and 4. We'll do some review today, 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.
So 4.3 will not be explicitly covered on the in-class portion.
- This pushed your first project deadline back a week, to Friday of
next week (the 12th).
- Nice to see 50% of our class at the Math/Art PopUp! Thanks for coming out...
- We finished up a problem (#5) from section 4.2, to compare it to the problem #17 of section 4.1.
- Then we began 4.3: Error Analysis. There was a neat trick
for getting the form of the error for the first
derivative over any stencil. We encountered 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 noted 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.
-
We used the general rule to get the error terms for the three-point
formulas for the derivative: left, right, and centered.
(Note: there is 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 saw how
to use error formulas to construct higher order methods, by trying to cancel the error terms via a smart use of coefficients:
- We also discussed using the error terms of competing schemes to
create better schemes (by cancelling errors).
We used the two-point backward and forward difference formulas to derive the three-point centered difference formula, by "balancing errors".
- More from 4.3: Error Analysis
- 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).
For the derivation of the error term, we rely on the weighted mean value theorem for integrals (which is found on page 158 of the text).
- 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.
- Now: what do we need to review? Just four sections!
- Section 3.2: Lagrange Polynomials
- Section 3.3: Newton Polynomials
- Section 4.1: Rudiments of Numerical Calculus
- Section 4.2: Undetermined coefficients
- Our Textbook: Tea Time Numerical Analysis (Leon Q. Brin)
- 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.
- 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....