- Your 2nd project (comparison of root finding methods) is due
today. I told Lauren that you could have until MIDNIGHT TONIGHT!
- I'm returning the homework on Muller's method.
- Several of you explored divide differences (or invented
them!); some used Newton's interpolating polynomials. Others
did an expansion about one of the
. Some good ideas, at any rate.
Some spoke of Mathematica, but didn't share the code -- why not?
- When choosing the roots, some of you merely use the "new
form" of the quadratic formula (p. 65) -- you actually have to
make a choice!
Notice the line on p. 65 -- If b is positive.... We use one formula for one root, and the other formula for the second root.
We need to be selective.
- You guys really struggled with this. Mostly you spent time
doing things that Mathematica could have done much more quickly
and surely. You need to shift to a higher level, and let
Mathematica handle the lower-level stuff.
Let's have a look at my Muller's method in Mathematica.
I'd hoped that you would be making initial guesses, and computing the next best approximation. It almost never happened. For example,
- Andres tried 1,3,4=x3, and got an estimate of 1.564 for the next guess. Let's try that.
- Several of you explored divide differences (or invented
them!); some used Newton's interpolating polynomials. Others
did an expansion about one of the
- If you java folk (Eden, Aubrey, and Brian) would stop by at some point
with a laptop, we'll just do a few spot checks on your code.
And Aaron, I need a little help with your executable. I was going to run some tests, and had trouble getting it to run under my Windows....
- Your next assignment (from chapter 5), is due Thursday. We've got to keep moving!
We're getting close to wrapping up interpolation. We'll finish it on Thursday. Today's new topic (that I hope to get to) is "splines".
- A week ago we looked at four
different ways of writing the interpolating quadratic polynomial
that we consider in Muller's method:
- "The classic":
- Lagrange Polynomials
- Newton's Interpolating Polynomial
- Horner's Method
- "The classic":
- Last time we explored the Newton polynomial interpolator a little
more, and showed how to write it better using divided differences.
We saw easy it was to interpolate points using Newton's interpolating polynomial -- you simply add an additional term higher-order term, based on divided differences, to an existing interpolator.
Question for recall: how do we use Newton's method to interpolate three points?
I also emphasized that Horner's rule (just before section 5.6.2) is the proper way to evaluate polynomials. We might occasionally explore them in other forms, but to evaluate them we use Horner's telescoping notation.
- We also began exploring Taylor series more:
- The principle thing to focus on is the Theorem 5.1,
p. 172. Let's take a look at that.
You have to be careful that you're not deceived by the form of the expression to the right. Every function is not polynomial -- that's not what it's saying. All the interesting stuff is buried in that Greek letter "xi".
- An informal way of thinking about this is given just below
the theorem:
= truncated Taylor series + remainder term
- Some interesting applications of Taylor series
follow in the text. We began with
- An example:
A hint for part b: logs of products
- Example 5.1: Taylor series approximate functions
(exponentials in this case -- see Figure 5.3)
An important observation is made about the computation of these polynomials. It is the introduction of Horner's rule (or method).
This is generally how all polynomials should be evaluated.
- Example 5.2: Approximating square roots.
- Notice that we've already covered the example of
using Taylor series in the demonstration of the order
of convergence for fixed point iteration (p. 175).
- An example:
- The principle thing to focus on is the Theorem 5.1,
p. 172. Let's take a look at that.
- The next topic of interest in the early going is Big-Oh notation (section 5.1.1).
- Perhaps the best way to introduce this is through an
example, such as they give: the expansion of the geometric series
Let me point out the first error that I can recall finding in the text, p. 176:
exp(1+h) should be replaced by exp(h) in the discussion. - Our authors use an approach which I am not particularly
fond of in this section, and I may sideline a little of this
discussion. Their use of containment, on p. 176, is logical but
certainly a little unconventional.
- I will prefer that you think of Big-oh in terms of
limits (see the final comment just before "Review question" on
p. 178):
...think of
not as a set but as some specific but unknown quantity
such that
is bounded as
.
- An important example: order of convergence the Secant Method
Upshot for secant method: even though its order of convergence is lower than Newton's, the routine may run in about the same time as Newton's (depends on the problem, of course).
- Perhaps the best way to introduce this is through an
example, such as they give: the expansion of the geometric series
- Section 5.2: Existence and Uniqueness
- The interpolation problem is seen in this section as a
problem from linear algebra. Let's check out the formulation of
it that way.
Notice the increasing powers of
. Clearly there's a risk that the coefficients will be of vastly different magnitudes, which could cause severe roundoff errors. This is one of the problems with this formulation.
Many of you may have used this notion to find the coefficients of your Muller quadratic. The problem is one of solving three equations in three unknowns; and since the equations are linear, we have a linear system to solve. More generally:
- Theorem 5.2: There exists a unique polynomial of degree
that interpolates data at
distinct abscissas.
- The proof suggested by the author is from linear algebra. Let's take a slightly different look at it (p. 184).
- The interpolation problem is seen in this section as a
problem from linear algebra. Let's check out the formulation of
it that way.
Today I want to begin a discussion of splines.
We use splines to fit data points. The name "spline" comes from a tool used in design, to make pleasing curves go through prescribed constraints.
We used to use tables more than we generally do today. In the old days, if you needed the sine of an angle that wasn't in the table, then you would use neighboring values to determine the proper table value desired. We can think of this as our objective today: you need the value of sine between two points (and don't have a calculator handy).
Often cubic splines are used. These cubics allow us to fit four things, and what we fit depends on our objective. Today I want to talk about Hermite cubics, which fit two points and the slopes at those two points. Here's the picture.
Other methods are, of course, possible. Some popular splines include "natural" and "clamped" cubic splines. These achieve other objectives.
I can think of several ways of approaching the fitting of this data with the Hermite interpolant. How might you go about it?