Day 12, Mat385

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• Announcements
  • Quiz this week is over 3.2: recurrence relations
  • Quiz returned:
    • The quiz
    • The key
    • Some observations:
      • Problem 1: after Example 5 in the section (and referenced in the highlights for the section)
      • Problem 2: after Example 6 in the section (and referenced in the highlights for the section)
      • Problem 3: Exercise #18 in the section

        You can do this one by induction, but you were told to do it directly from the definition of the Fibonacci numbers (i.e. by the recurrence relation).

      • There is a required textbook for this course. I just bought a cheap copy off the web (8 bucks).

• Last time:
  • Exercise #43 asked us for a recursive definition of a geometric progression, with constant ratio \(r\) and initial value \(a\): \[ gp(n) \equiv a + ar + ar^2 + \ldots + ar^{n-1} = \frac{a-ar^n}{1-r} \] On the right-hand side (RHS) is the closed-form solution (and we needed that in the lecture!).

    But the point of this exercise was to define the sum recursively, so we need a base case and an inductive step: \[ \left\{ \begin{array}{cc} {n=1}&{gp(1)=a}\cr {n > 1}&{gp(n)=gp(n-1)+ a r^{n-1}} \end{array} \right. \]

  • Leo had asked if Taylor Polynomials can be defined recursively, so I illustrated that with this implementation in Mathematica (pdf).

    If you follow the code, it's very similar to the definition of the geometric progression.

  • We finished section 3.2: recurrence relations

    The upshot was that we can adapt the closed-form solution for the first-order, linear, constant coefficient, non-homogeneous equation, which is appropriate for problems like the Tower of Hanoi with \[ \left\{ \begin{array}{cc} {n=1}&{C(1)=1}\cr {n > 1}&{C(n)=2C(n-1)+ 1} \end{array} \right. \] to handle "divide and conquer" problems like binary search.

  • Marked up
    • highlights from 3.2

• Today: We'll work on 3.3: Analysis of Algorithms, focusing on worst case analysis of two algorithms:
  • Binary search (with an initial merge sort), and
  • Euclid's GCD (Greatest Common Divisor)

• The Curious History of Venn Diagrams: A look at the curious history of Venn diagrams and how they blend logic with geometry (Scientific American)
  

• Some Fun, Motivating Examples from Propositional and Predicate Calculus