- We do have an exam coming up next week. It will cover through
interpolation (not go into differentiation, etc.).
- You have a new assignment (still chapter 5), due next
Tuesday. This will also help you prepare for the exam. For those of you
who didn't do the last homework, you probably should do it just to get
ready for the exam.
- Your first homework on chapter 5 is returned. I'm still working on
the projects. Hope to have those by Thursday. I graded four (randomly
selected from each section):
- p. 178, #3 -- the size of the relative error must be small....
- p. 181, #6 -- you guys are funny. Use Mathematica for this! And the author wanted you to show all the approximations, so that you could see improvement.
- p. 190, #8 -- you struggled (feebly, not mightily) on this one. Ari?
- p. 221, #7 -- pretty straightforward.... But, in the end, it's nice to test your polynomial! That's the gold standard.
- There's one more spline we'll need to learn about: the Bezier spline (not today!)
I've given you a short reading assignment (this description of Bezier splines); I hope that you'll check it out before I go over the project next time.
- Before we start that, however, let's review what we talked about last
time:
- We finished off the secant demonstration, that secant is of order the golden mean. This illustrated the use of "Big-Oh" notation.
- We talked about uniqueness of the interpolator -- there is one
polynomial of
degree fitting
points.
- We talked a bit about divided differences as essentially
derivatives, which is important to the error we make in splines, say
(and other areas).
This was used to create a meaning a novel different 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
.
- We began working with splines. In particular, we investigated
Hermite cubic splines, which interpolate two points and two slopes at
those points -- and hence the derivative of the spline provides an
interpolator of the derivative of the function, as well.
This is a locally determined spline -- we simply stitch together one cubic with the others, and we have a smooth curve running through a function.
- We saw that the easiest formulation for Hermite splines is
Newton's, only we're going to repeat some abscissa:
- Finally we took a look at my implementation of this in Mathematica.
- We should say something about the error of interpolation of the
Hermite Spline.
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
.
What then is a bound on the error that we're making using my Hermite spline on the interval
for sine, given 5 equal subdivisions of the interval?
- While my Hermite spline does the job (approximating Sine for all real numbers), one might do the same job with Taylor polynomials.
You won't be surprised to learn that the Taylor Series is the key to analyzing some of the most obvious choices.
Our objective is "adaptive quadrature", section 7.5. If you understand how to do that, I'll be happy! It's very cool.
The good news is that some of this will already be familiar; the analysis part will probably be new, however.
- Section 7.1: numerical differentiation
- We begin with the one-sided forward-difference
formula, which comes out of the most important definition in
calculus (the limit definition of the derivative) by simply
dropping the
:
The authors define the truncation error as the difference of the approximation and the true value:
and the power of
The text illustrates this in Figure 7.1, p. 256, as the slope of a secant line approximation to the slope of the tangent line (the derivative at
).
We explore the consequences using the Taylor series expansion:
What is the order of accuracy of this method?
We call this a forward definition, because we think of
which comes out of
- Compare these "lop-sided" schemes with the centered-difference approximation:
which we can again attack using Taylor series polynomials. What is the order of accuracy of this method?
We can also derive it as an average of the two lop-sided schemes! We might then average their error terms to get an error term. However, what do you notice about that error term?
- This is really a good time to make use of our tools from
linear algebra, however. Let's take another crack at
deriving these methods, but from the LA perspective, and
see how matrices can make our problem so much easier, and allow
us to generalize these simple ideas quite simply.
- Now one thing you might think is that, by making
Let's think about computing with error.
It will turn out that the truncation error of a scheme like these will look like
where
is a bound on the derivative of interest on the interval of interest (second in this case), and
is a bound on the size of a truncation or round-off error.
- We begin with the one-sided forward-difference
formula, which comes out of the most important definition in
calculus (the limit definition of the derivative) by simply
dropping the