- I haven't started grading your 2nd project (comparison of root finding methods) yet.
- Your first homework on chapter 5 is due.
- 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....
Today's new topic is "splines".
- Before we start that, however, let's review what we talked about last
time:
- we explored Muller's method (and I illustrated the
process of "deflation", whereby multiple roots can be found for a
function by each time dividing the function by
, where
is the
and latest root we've found).
- We also continued exploring Taylor series more:
- The principle thing to focus on is the Theorem 5.1,
p. 172. I think that I may have gotten the lower index
wrong on the formula when I wrote it -- it should start
with a
.
Make sure! I just had a bad feeling later, when I wrote it down myself to do some work and put
....
- We wrapped up
which illustrated how one ordinarily uses Taylor series in the
approximation of a somewhat complicated function, turning the
calculation into sums, differences, products, and quotients --
things that computers do well.
- The principle thing to focus on is the Theorem 5.1,
p. 172. I think that I may have gotten the lower index
wrong on the formula when I wrote it -- it should start
with a
- The next topic we examined was 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
.
- And were were considering an important example: order of
convergence the Secant Method, and that's where I want to pick
up today.
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
- we explored Muller's method (and I illustrated the
process of "deflation", whereby multiple roots can be found for a
function by each time dividing the function by
- So we'll start today by finishing up the secant problem. We're
almost there, but it was a little messy.
- 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.
- Divided differences and derivatives (section 5.6.5)
I want to remind you about and emphasize the relationship between the divided differences and the derivatives of
.
It's going to be important here in a bit, so take a peek at the result at the bottom of p. 219:
In particular, we want to consider a case where we compute the divided difference
, which is actually undefined if we think of it as a ratio of difference -- but this formula allows us to think of it as the appropriate derivative evaluated at
, so we define it so:
, since it must be between the minimum and maximum of
-- not much wiggle room!
- Section 5.5: Piecewise Polynomial Interpolation
Today I want to discuss 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. I've brought a spline in to show you....
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 a picture of how we might build one of these.
- One advantage of Hermite polynomials is that the derivative of the spline interpolates the first derivative!
- One disadvantage of Hermite polynomials is that we need to know the derivative of the function. Question: if we don't have a function, e.g. just data, how might we proceed?
I can think of several ways of approaching the fitting of this data with the Hermite interpolant. How might you go about it?
You might not be surprised to learn that about the easiest formulation is Newton's, only we're going to repeat some abscissa:
Since we'll need this over and over, we should go ahead and simplify those divided difference coefficients, using the assumption that
is known (use the recurrence relation at the top of p. 215 to do so).
This will have the effect of fitting the two endpoints, and the derivatives of
at the two endpoints.
I'd like to discuss my implementation of this in Mathematica.
What error are we making when we use Hermite interpolation? Proposition 5.6 gives us the answer:
which is equal to a derivatives times some stuff:
where
is some unknown number between the minimum and maximum of
.
While my Hermite spline does the job (approximating Sine for all real numbers), one might do the same job with Taylor polynomials.