- Welcome back! It's feeling spring-like. I hope that you got a
little rested up.
- Your Muller homework is due.
- Those of you who took "the easy way out" for problem one from the
exam, your additional problem is also due.
- Your next programming assignment is due Thursday.
I'll give you a new assignment on Thursday, but will leave you alone so that you can work on the programming assignment for the moment.
- More on the general overview of the difference between
- interpolation
- extrapolation
- approximation
Muller's method is discussed in the beginning of this section. It's an example of an interpolation problem: fitting three points exactly. When we think of interpolation, think of fitting something exactly -- nailing it.
Here's an example of another important problem, where approximation and estimation is more important, perhaps, and which I am working on right now:
These are ice thicknesses taken by multiple submarine probs of ice near the north pole:
I'm attempting to reproduce this result. The probes of the ice are only approximations (contain errors) themselves, so we don't worry about fitting them exactly: we're more interested in finding a simple model to estimate ice over the entire ice field.
The two models the authors have drawn are cubic polynomial fits (to part and all of the data, orange and green respectively).
We can extrapolate from the data by predicting future ice thickness using these models. As it is, we're approximating the average ice thickness for each year. We are not intentionally interpolating any values.
While classes of interpolation functions -- interpolants -- are described (e.g. polynomials, sines and cosines -- Fourier series --, exponentials, rational functions), our authors end by saying that we'll focus on polynomials, and we start with Taylor series polynomials.
- I want to begin today by looking at four different ways of
writing the interpolating polynomial that we consider in Muller's method.
- "The classic"
- Lagrange Polynomials
- Newton's Interpolating Polynomial
- Horner's Method
- Section 5.1: Taylor series
In what sense is a Taylor series polynomial an interpolant? Ordinarily we've been thinking of it as an approximation to a function.
On the other hand, we are often interpolating data (but just not points, as in the case of Muller, but rather points and slopes and higher derivatives at a single point.
So we might interpolate the point (0,0), with slope 0, and second derivative 0, by the quadratic function
This function fits all that data exactly (and passes through exactly one known point). This is the Taylor series polynomial of degree 2 subject to those constraints.
A tangent line is the Taylor series polynomial of degree 1, which fits a point and gets the slope right.
- 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. Let's check them out.
- 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).
- The derivation of Newton's method
- Another quadratic root-finder: generalize Newton's method
Muller's method is a generalization of the Secant method (which is an approximation to Newton's method).
What if we go straight from Newton's method to a quadratic method?
Given function
, suppose we know both
and
.
Given an initial guess
to the true root
. What quadratic would we use to seek the next approximation?
We might start with the "classic method", and see what conditions the a, b, and c must satisfy. We'll re-derive the Taylor series polynomial of degree 2.
- Example 5.1: Taylor series approximate functions
(exponentials in this case -- see Figure 5.3)
- An example:
A hint for part b: logs of products
- 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
- 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).
- An important example: order of convergence the Secant Method
Even though the order of convergence is less, 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.
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 is from linear algebra. Let's take a look at it (p. 184). Then we'll look at Exercise #7, p. 185.
- The interpolation problem is seen in this section as a
problem from linear algebra. Let's check out the formulation of
it that way.
- Now let's take another look at Muller's method. You've all taken a
crack at it, and I want to suggest a strategy for approaching the
problem that may have escaped you.
It's related to the concept of divided differences (section 5.6.2, p. 212); and the concept of Horner's rule (just before section 5.6.2).
We'll examine my solution to the homework in Mathematica.
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