Day 17, Mat360: Numerical Analysis

Today:

• Announcements:
  • I have finished grading the "take-home" portion of your exam.

    You should have received an electronic copy with comments and your grade.

    Recall that your exam grade is 70% in-class, and 30% take-home. So you need to weight your two grades, to get your grade for the exam: \[ grade = .7*inclass+.3*takehome \]

  • It seems that there were some struggles with the takehome, particularly with software. I believe that Danny has compiled a collection which worked as required, and I have put his files on-line which you can access and try.

    You can run "danny.m" to verify that some of the commands are working.

  • I thought that I assigned #23 for the exam, but only Stefanie and Yimin attempted it. (Good job you two!) It may have been missed because it was behind the parentheses...

    Let's take a look at that one. Yimin provideds some octave code.

  • You had a reading assignment over break: chapter 5. It will form the basis for your first project, which will be to create your own very special computer-drawn signature. We won't cover it in detail, but I'll summarize what you need for this project (next time).

    So if you haven't read it yet, please take a look!

  • Your homework over Lagrange interpolation is due next class, Wednesday.
  • You have a new homework assignment, for section 3.3, due in one week.

    I ask that you give it special attention: you are to write some octave code, and then use it to work several exercises. I need to see that you can actually code. It's a huge part of numerical analysis....

• Let's recall what we did last time:
  • We're interpolating:

    We know the value of $f$ at a selection of points, $f(x_{0})=y_{0},f(x_{1})=x_{1},\ldots,f(x_{n})=y_{n}$. The interpolating polynomial of least degree \(P_n\) passing through the $n+1$ points \[ (x_{0},y_{0}),(x_{1},y_{1}),\ldots,(x_{n},y_{n}) \] agrees with $f$ at $x_{0},x_{1},\ldots,x_{n}$ and we can say with some precision how closely this interpolating polynomial \(P_n\) agrees with $f$ at other points as well:

  • The Lagrange interpolator is \[ L_{n}(x)=\sum_{i=0}^{n}\frac{p_{i}(x)}{p_{i}(x_{i})}y_{i}, \] where \[ p_{i}(x)=\prod_{\substack{j=0\\j\neq i}}^{n}(x-x_{j}) = (x-x_{0})\cdots(x-x_{i-1})(x-x_{i+1})\cdots(x-x_{n}) \]
  • We recursively define the higher order Lagrange interpolators, based on the linear ones: \[ l(x;x_1,x_0)=\frac{x-x0}{x1-x0} \] Simple!

  • One of the most interesting facets of Lagrange interpolation is the form of the error term engendered by Lagrange interpolation:

    If $f$ has $n+1$ derivatives on $(a,b)$ and $f,f',f'',\ldots,f^{(n)}$ are all continuous on $[a,b]$, then there is a value $\xi_{x}\in(a,b)$ such that \[ f(x)-P_{n}(x)=\frac{f^{(n+1)}(\xi_{x})}{(n+1)!}(x-x_{0})(x-x_{1})\cdots(x-x_{n}). \]

  • Theorem: The polynomial, $P_{n}$ , of least degree interpolating the data $(x_{0},y_{0}),(x_{1},y_{1}),\ldots,(x_{n},y_{n})$ exists and is unique. Moreover, any interpolating polynomial of degree at most $n$ is equal to $P_{n}$.

  • As an application of interpolating polynomials, we discussed Sidi's method, summarized by the formula \[ x_{n+1}=x_{n}-\frac{g(x_{n})}{p'_{n,k}(x_{n})} \] where $p_{n,k}$ is the interpolating polynomial passing through the points \[ (x_{n},g(x_{n})),(x_{n-1},g(x_{n-1})),\ldots,(x_{n-k},g(x_{n-k})). \]
  • Lagrange's form is computationally silly, however: wasteful. We can use Neville's method to compute better.
    Why does Neville's method work? We were working out the geometric details for the first few cases, \(n=1,2,3\), and we ended on a cliff-hanger! What would become of Neville's method in the next iteration?

    You're back for the exciting conclusion!

• Today:
  • The exciting conclusion!

  • Let's check out my Mathematica code to solve several of our problems from the section.

  • Now we're moving on to Newton's form of the interpolating polynomial, and we want to begin by noting the differences:
    • While Lagrange's form \(L_{n}(x)\)is a sum of \(n+1\) \(n^{th}\) degree polynomials, Newton's method builds: it is the sum of a constant, a linear term, a quadratic term, etc., all the way up to the \(n^{th}\) degree term.
    • Newton's form can be easily "Hornerized": \[ \begin{array}{c} {N_{n}(x)= a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})(x-x_{1})+\cdots+a_{n}(x-x_{0})\cdots(x-x_{n-1})}\cr {=a_{0}+(x-x_{0})(a_{1}+(x-x_{1})(a_{2}+\cdots+(x-x_{n-2})(a_{n-1}+a_{n}(x-x_{n-1}))\ldots)} \end{array} \]

• Links:
  • Our Textbook: Tea Time Numerical Analysis (Leon Q. Brin)
  • Homework solutions:
    • 1.1 (Mathematica)
    • 1.2 (Mathematica)
    • 1.3 (Mathematica)
    • 2.1 (Mathematica)
    • 2.2 (Mathematica)
    • 2.3 (Mathematica)
    • 2.4 (Mathematica)

  • My Mathematica code for Section 2.6. Muller code, too.
  • My Octave Project for 2.6 homework.
  • Octave project for 2.5
  • Divided Differences. I use these in my Muller code, and they will prove useful when it comes to interpolating polynomials.
  • Author's octave code for these complex fractal convergence diagrams
  • Newton's method in Mathematica
  • fixed point with Aitken's acceleration in Mathematica
  • Some nice notes (and code) snippets on fixed point iteration, including these acceleration techniques.
  • fixed point in Mathematica
  • Check out how my bisection works, and how it is applied to arcsine....
  • Various sins....
    • Sins in Mathematica
    • Sins in Octave
    • ArcSins in Mathematica