- I hope that you had a good MLK, Jr. holiday.
- We're going to skip Section 1.4 -- it doesn't really seem critical, but you might check out the Octave parts, in particular. It gives some good advice for programming using Octave. So our next section will be 2.1: Bisection.
- You have a new homework assignment.
- You also owe me your first homework assignment, for section
1.1. You may have an "on-line portion", in
which case we just have to know how to share those with me.
I believe that I'm supposed to be able to see your work in Octave, if you've added yourself to my class. Stephanie, it seems that you've done that; but I'm not able to see your stuff, it seems....
- You also have some new homework problems for section 1.3, which will be due Wednesday, 1/24.
- Examined the two pieces of a Taylor's series for \(f(x)\), centered at \(x=x_0\): the polynomial approximation, and the remainder term: \[ f(x)=f(x_{0})+\sum_{j=1}^{n}\left(\frac{f^{(j)}(x_{0})}{j!}(x-x_{0})^{j}\right)+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_{0})^{n+1}. \]
- Used the series for \(e^x\) and Euler's formula to obtain the series for \(\sin(x)\) and \(\cos(x)\).
- Came up with a plan for creating a "sin" function, to trick the
calculus students into thinking that we were giving them sines,
when in fact we were computing polynomials.
We made plans to do this with an error of less than \(10^{-6}\) everywhere on the real line.
- Let's check to see if anyone has sinned well! If not, I will tell
you how I have sinned, but I'd rather hear your sins then my
own....:)
- What other functions do we get "for free", once we've sinned well?
- What functions are we close to having, but don't quite have?
I want to show how the pursuit of one of those sets us up for our next topic: root finding.
There are several facets that I want to investigate from the text today, and then we'll do a few problems.
- Suppose the sequence $\langle p_{n}\rangle$ converges to $p$: then
the order of convergence \(\alpha\) satisfies:
\[
\lim_{n\to\infty}\frac{\left|p_{n+1}-p\right|}{\left|p_{n}-p\right|^{\alpha}}=\lambda
\]
for some real number $\lambda>0$.
\(\alpha\) represents the rate at which a sequence converges, and bigger is better -- convergence is faster. And \(\alpha > 1\) -- or there is no convergence (see Crumpet 6).
Check out Table 1.4, on page 20: four sequences that converge to \(e\):
- We can estimate \(\alpha\) by checking a few of the iterates: \[ \alpha\approx\frac{\ln\left|\frac{p_{n+2}-p}{p_{n+1}-p}\right|}{\ln\left|\frac{p_{n+1}-p}{p_{n}-p}\right|} \] where does that come from?
- Let's see how the speed is borne out: look at Table 1.5, and
the formula for number of significant digits on page 22:
\[ d(p_{n+1}) \approx \alpha d(p_{n})-\log\left(\lambda|p|^{\alpha-1}\right) \] Notice that the term on the far right is independent of \(n\): it's just a constant. So ultimately, the increase in the number of significant digits at each iteration is \(\alpha\) times more.
Table 1.5 shows this very well.
But if \(\alpha = 1\), then all the increase in significant figures is in that \(\log\) term, and we'll see a constant number of digits added at each iteration. This slow rate of improvement is illustrated for the sequence \(q\):
"$\langle q_{n}\rangle$ should show each term having $-\log0.8\approx.1$ more significant digits of accuracy than the previous. More sensibly, this means the sequence will show about one more significant digit of accuracy every ten terms."
- Let's show that, if $\alpha>1$,
\[
d(p_{n_{0}+k})\approx(d_{n_{0}}-C)\alpha^{k}+C
\]
where ${\displaystyle C=\frac{\log\left(\lambda|p|^{\alpha-1}\right)}{\alpha-1}}$
as our author asserts. This says that, if $\alpha>1$, the increase in significant digits is exponential; so it might better to express it this way: \[ d(p_{n_{0}+k})-C \approx(d_{n_{0}}-C)\alpha^{k} \] or \[ b_k \approx b_0 \alpha^{k} \] (This is the theme of Crumpet 7, but I'll do it using the "expand, guess, verify" method)
- Finally let's have a look at the "Big O" notation on page 24,
under the heading "rate of convergence of a sequence":
- So, for example,
\[
\left|\frac{2n+1}{4n}-\frac{1}{2}\right|=\frac{1}{4n}\leq\frac{1}{4}\cdot\frac{1}{n},
\]
means that we can write
\[
\frac{2n+1}{4n}=\frac{1}{2}+O(\frac{1}{n})
\]
"Normally, when we find a rate of convergence, we try to find
the fastest converging sequence from our stock of simple
examples that satisfies the definition. In this case, there
is none faster."
- "Basically all the sequences studied in any depth in calculus converge with linear order."
#1a, p. 29: Find the order of convergence for ${\displaystyle \left\langle \frac{n!}{n^{n}}\right\rangle \to0}$
- "Essentially, it takes an exponentially growing exponent to converge with an order greater than 1."
#3, p. 29: Show that the sequence $p_{n}=2^{1-2^{n}}$ is quadratically convergent.
#8, p. 30: Use a Taylor polynomial to find the rate of convergence of \[ \lim_{h\to0}(2-e^{h})=1. \]
- So, for example,
\[
\left|\frac{2n+1}{4n}-\frac{1}{2}\right|=\frac{1}{4n}\leq\frac{1}{4}\cdot\frac{1}{n},
\]
means that we can write
\[
\frac{2n+1}{4n}=\frac{1}{2}+O(\frac{1}{n})
\]
"Normally, when we find a rate of convergence, we try to find
the fastest converging sequence from our stock of simple
examples that satisfies the definition. In this case, there
is none faster."
- Finally, let's look at Exercise #20: Prove that the order of convergence of a sequence is unique.
- Various sins....