Day 08 in Mat360

Last time: More Section 1.3: convergence errors

Next time: Section 2.2: Fixed Point Iteration

Today:

Announcements:
- Accounts are available for all in the Lab. Use your NKU username and password.
- Any (repaired) versions of 2, 12, 24?
- Homework 1.2 due today
- Handout for section 2.2: fixed point iteration
  - worksheet
Let's wrap up section 1.3:
- Convergence:
  Convergence basically means that you're getting close to something, and we're interested in characterizing the rate at which we're closing in. Let's look at a few examples. One thing that you'll notice is the utility of the Taylor series here.
  - 6c, p. 37
    
    If there exists a positive constant ${\kappa}$ such that
    ${|\alpha_n-\alpha|\le\kappa|\beta_n|}$
    for large values of n
    then we say that ${\{\alpha_n\}_{n=1}^{\infty}}$ converges to ${\alpha}$ with rate of convergence ${O(\beta_n)}$ .
  - 7d, p. 37
    
    Suppose that
    
    ${\lim_{h\to{0}}G(h)=0}$ and ${\lim_{h\to{0}}F(h)=L}$ .
    
    If ${\exists{K}>0}$ such that
    ${|F(h)-L|\le{K}|G(h)|,}$
    for sufficiently small h, then ${F(h)=L+O(G(h))}$ .
Section 2.1:
- section 2.1 summary
- section 2.1 worksheet
- You might try the java algorithms that are on-line
  - What's wrong with this program?
- Consider the Mathematica file (warning! These are very primitive....)
  - Same problem....
- Problems from section 2.1:
  - #3b
  - #8a

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