- I haven't finished grading your last homework, but by some miracle
I've finished grading your first project.
At least for those of you who used python or Mathematica.
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....
Here are links to a couple example codes that worked fine:
- Victoria did it (still using IntegerDigits, but not as a base converter).
- Zach B. did it eliminating IntegerDigits (as discussed in class).
- Andres generalized it to do fractional parts.
- Your next programming assignment is due Tuesday. By the way, the
problem 2b is not meant to be exhaustive -- tell me as much as you
can. You might show intervals which don't result in 0 being chosen, and
which you can definitively conclude will be chosen. Graphing is
encouraged, however.
Notice that I have version of Muller's method in Mathematica that you're free to use.
- I've given you a new assignment from chapter 5, which is due next Thursday.
- Last time we looked 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
Perhaps I'm so enthralled with Muller's method because I recall a line from my first numerical analysis textbook (Conte and de Boor), in which they had this to say about Muller's:
"A method of recent vintage, expounded by Muller, has been used on computers with remarkable success. This method may be used to find any prescribed number of zeros, real or complex, of an arbitrary function. The method is iterative, converges almost quadratically in the vicinity of a root, does not require the evaluation of the derivative of the function, and obtains both real and complex roots even when these roots are not simple."
Now the third edition of C&deB was from 1980, but the first edition was written in 1965 -- before we landed on the moon -- so it might be that they meant by "recent vintage" something around that time. In fact, Muller wrote this up in 1956! Perhaps the use of "recent vintage" in textbooks is a mistake....:)
- We ended up by considering another Quadratic Method
(generalizing Newton's method), and we should take another look
at the Mathematica code to see that our results from last time
work.
- Question: How would you write the Lagrange interpolator to two points?
- 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 at
.
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.
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). 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.
We'll finish this off by having a look at the relationship between the
divided differences and the derivatives of .