- Exam 1: Danny has compiled a collection of software which worked
as required, and I have put his files on-line which you can access and try.
You can run "danny.m" to verify that some of the commands are working.
- Your homework over Lagrange interpolation is due today.
- You have a homework
assignment for section 3.3, which will be due Monday.
I ask that you give it special attention: you are to write some octave code, and then use it to work several exercises. I need to see that you can actually code. It's a huge part of numerical analysis....
- I want to move our second exam back by a week -- are there any problems with that? It will now be on 4/3, if that's okay. I want to get a little more material from chapter four in prior to the exam.
- And I've assigned your first project.
- The exciting conclusion of "the Lagrange/Neville coincidence".
The board work for that story are here.
- Now we're moving on to Newton's form of the interpolating
polynomial, and we want to begin by noting the differences:
- While Lagrange's form \(L_{n}(x)\)is a sum of \(n+1\) \(n^{th}\) degree
polynomials, Newton's method builds: it is the sum of a
constant, a linear term, a quadratic term, etc., all the way up
to the \(n^{th}\) degree term.
- We began looking into Newton's form of the interpolating polynomial, and we were using divided differences to compute the coefficients of this version.
- While Lagrange's form \(L_{n}(x)\)is a sum of \(n+1\) \(n^{th}\) degree
polynomials, Newton's method builds: it is the sum of a
constant, a linear term, a quadratic term, etc., all the way up
to the \(n^{th}\) degree term.
- And there was another cliff-hanger! Goodness, this class is hard on the heart!
- The exciting conclusion!
The board work for that story are here.
- Newton's form can be easily "Hornerized": \[ \begin{array}{c} {N_{n}(x)= a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})(x-x_{1})+\cdots+a_{n}(x-x_{0})\cdots(x-x_{n-1})}\cr {=a_{0}+(x-x_{0})(a_{1}+(x-x_{1})(a_{2}+\cdots+(x-x_{n-2})(a_{n-1}+a_{n}(x-x_{n-1}))\ldots)} \end{array} \]
- Then we should talk about splines, and Bezier polynomials in
particular, as we prepare to create our "sigmacs".
- What's our objective? A signature. Here's Zach Bezold's:
Zach's signature 
Zach's signature, with the control points used to "guide" the set of Bezier curves which produce it. - What are these Bezier polynomials?
- Let's check out a few highlights from our text,
sections 5.1 and 5.2 first.
- "The Taylor polynomials of Section 1.2 and interpolating polynomials of Chapter 3 represent opposite extremes in the spectrum of osculating polynomials. Taylor polynomials require the value of the polynomial at a single point while interpolating polynomials require the value of the polynomial at, generally anyway, multiple points. Taylor polynomials require the values of, generally anyway, multiple derivative s while interpolating polynomials do not allow derivative specification."
- We begin with a Hermite cubic polynomial: \[ p(t)=\frac{t-t_{1}}{t_{0}-t_{1}}y_{0}+\frac{t-t_{0}}{t_{1}-t_{0}}y_{1}+\frac{(t-t_{1})^{2}(t-t_{0})}{(t_{0}-t_{1})^{2}}(\dot{y}_{0}-m)+\frac{(t-t_{0})^{2}(t-t_{1})}{(t_{1}-t_{0})^{2}}(\dot{y}_{1}-m) \] which passes through \((t_0,y_0)\) and \((t_1,y_1)\), and has the appropriate slopes there (notice that we've changed to \(\dot{y}\) notation -- we're going to be thinking of \(t\) as time, and often we represent time derivatives this way.
- Why are we getting away from \(x\) as independent variable? Because we're going to be fitting points in the plane, simultaneously requiring \(x(t)\) and \(y(t)\); and fitting each.
- Notice that on page 185 we switch to vector notation: we're creating what's called a "parametric curve" in the plane.
- The picture on page 187 is enlightening, and will be animated in the software below.
- Crumpet 34 on p. 188 puts several of these togethers, and illustrates the conditions necessary to assure slope continuity at the joins....
- I really only want to point out one thing
from section 5.2, on page 195: the calculation
of constraints on a spline:
A cubic spline required to interpolate $n+1$ points has $n-1$ joints and $n$ pieces. It follows that the set of cubics has $4n$ coefficients. Requiring each cubic to pass through $2$ points gives $2n$ conditions on the coefficients. Requiring first derivative matching at the joints gives $n-1$ more conditions. Requiring second derivative matching at the joints gives an additional $n-1$ conditions for a grand total of $4n-2$ conditions. That leaves $2$ "free" coefficients. Mathematically speaking, we have a family of splines with two degrees of freedom. To find any specific spline, we need to enforce two more conditions on the coefficients. These conditions may include the first, second, or third derivative at two of the nodes, both the first and second derivative at a single node, or some other combination of two derivative requirements.
- Bezier in Desmos.
- Mathematica
- Let's check out a few highlights from our text,
sections 5.1 and 5.2 first.
- What's our objective? A signature. Here's Zach Bezold's:
- 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....