Day 2, Mat360: Numerical Analysis

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Today:

Announcements
- No class on Monday: MLK, Jr. Day
- Your homework for section 1.1 will be due next Wednesday, so we can start with any questions you have on those problems.
- You have some assigned problems for section 1.2, which will be due Monday, 1/22.
Questions on anything?
Let's wrap up "preliminaries" with a discussion of "significant digits". The definition of number of significant digits is given by \[ d(\tilde{p}) = \log\left(\frac{|p|}{|\tilde{p}-p|}\right) \equiv \log(RE(\tilde{p})^{-1}) = -\log(RE(\tilde{p})) \] Let's think about why that makes sense, using the one-seventh example from the text:
Let's head into Taylor series, where, in particular, we begin with Taylor's theorem:
For some integer $n\geq0$, suppose $f(x)$ has $n+1$ derivatives on $(a,b)$, and $x_{0}\in(a,b)$ . Then for each $x\in(a,b)$, there exists a $\xi$, depending on $x$, lying between $x$ and $x_{0}$ such that \[ f(x)=f(x_{0})+\sum_{j=1}^{n}\left(\frac{f^{(j)}(x_{0})}{j!}(x-x_{0})^{j}\right)+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_{0})^{n+1}. \] The first part we call the Taylor polynomial approximation to $f$ about $x_0$, \[ T_{n}(x)=f(x_{0})+\sum_{j=1}^{n}\left(\frac{f^{(j)}(x_{0})}{j!}(x-x_{0})^{j}\right) \] and the second part is called the error term: the study of the error term may allow us to bound the error of using the Taylor polynomial as an approximation to the function $f$ in the vicinity of $x_0$: \[ |R_{n}(x)|=\left|\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_{0})^{n+1}\right| \]
Let's begin with a little recall: a few series are very easy to generate, e.g. $e^x$ expanded about $x_0=0$ -- the so-called MacLaurin series: \[ e^x=1+\sum_{j=1}^{\infty}\left(\frac{e^0}{j!}(x-0)^{j}\right) = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ldots \]
Problem #3, p. 17: Find the 36^th Maclaurin polynomial for $f(x)=e^x$.
A really beautiful trick with an imaginary number and a formula from Euler allows us to easily use that series to get the series for sine and cosine.
Euler's equation is the most beautiful in all of mathematics: \[ e^{i\pi}+1=0 \] It involves what are arguably the five most important constants in all of mathematics, and nothing more!
It comes out of Euler's formula, which is \[ e^{i\theta}=\cos(\theta)+i\sin(\theta) \] which says that the beautiful complex function $e^{ix}$ is a parameterization of the unit circle in the complex plane, with $\theta$ serving as the angle, with $\theta = 0$ along the x-axis.
Now let's use this to get the series for sine and cosine. The series for $e^x$ we found above works for complex arguments as well, so \[ e^{ix}=1+\sum_{j=1}^{\infty}\frac{(ix)^{j}}{j!} = 1+ix+\frac{(ix)^2}{2!}+\frac{(ix)^3}{3!}+\frac{(ix)^4}{4!}+\frac{(ix)^5}{5!}+\ldots \] We can separate out the real and imaginary parts, and Euler's formula says that the real parts are associated with cosine, and the imaginary parts form the series for sine: \[ e^{ix}=\cos(x)+i\sin(x)=\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots\right) + i\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots\right) \]

Problem #1a, p. 17: Find $T_3(x)$ and $R_3(x)$ for $f(x)=\sin(x)$; $x_0=0$
Problem #1g, p. 17: Find $T_3(x)$ and $R_3(x)$ for $f(x)=\cos^2(x)$; $x_0=0$
(The first part is easy, because we can multiply series; the second part is a little more complicated, because we need the fourth derivative of $f$....)
Problem #16, p. 18: Find the fourth Taylor Polynomial for $\ln(x)$ expanded about $x_0=1$.
I have an ambitious goal for today: let's create an approximation to the sine function that is accurate to within $10^{-6}$ -- everywhere! (That is, for all real numbers.)
- Our author illustrates how we can use very high powers of polynomials to get an accurate approximation to $\sin(x)$ for even very large intervals around 0.
  But, because we want an approximation everywhere, for all real numbers, we need to take a different approach.
- We start by using some analysis, and our knowledge of the sine function: What portion of domain of the sine function will suffice to construct its approximation everywhere?
- How do we compute outside of that interval?
- What approach might we use to construct the approximation on the interval of interest? (Guess that it has something to do with the Taylor Series expansion...:)
In the end, we should have something -- call it sinner(x) -- that one could plot, and that would (sinfully?) fool any calculus students...:)

Next time: Section 1.3: Speed

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