Jan 23, 2002
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- New topic: finding roots of equations

- Diversion: calculator button pushing
	- (i) What happens if you keep pressing square root button?
	- (ii) Limit cycle of period 2 with 1/x button (or -x button)
	- (iii) Interesting property of the number 1.618034

- These have something to do with the topic of root finding! :)

- Simple examples from algebraic equations

- Solving painful equations 
	- e.g., worked out cos(x) cosh(x) = 1

- Iterative solution
	- start with a guess
	- keep improving it
	- if you are in the "right ballpark area", you converge!

- Bisection Method
	- analogy: finding an address on the street

- Given a function f on [a,b]
	- f(a) is -ve
	- f(b) is +ve
	- where is the zero of f?
		- what are you assuming?

- Intermediate Value Theorem

- Hi-Lo game for finding the midpoint
	- algorithm
	- error formula
	- number of iterations needed to get an error

- To use bisection
	- you need to have an iterval [a,b]
	- f() should be continous in [a,b]

- What happens with bisection for f(x) = x^-1?

- Convergence for bisection
	- at least linear 

- What happens if interval has many roots?

- Another way to play Hi-Lo
	- method of false position (regula falsi)
	- instead of midpoint, take straight line
	  intersection
	- if its a linear function, you converge in one step!

- Newton's method
	- use tangent
	- no need for ranges; just be near the root

- What happens if you are too far away from root?
	- catastrophy occurs!
	- examples of pathological cases
		- receding tangent
		- flat tangent
		- dancing points
Jan 25, 2002
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- Recap:
	- Bisection
		- linear convergence
		- asymptotic error constant = 0.5
		- can calculate #steps for given accuracy
	- Regula Falsi	
		- also linear convergence
		- asymptotic error constant < 1
		  (but difficult to determine)
		- good for linear functions

- Derivation of newton's method: 2 ways
	- using equation for tangent
	- using taylor's series linearization

- Can be used for many purposes, not just finding roots
	- e.g., finding square root (global conv.)
		- fishing rod analogy
			"throw and reel in"
	- e.g., solving with sin(x) = 0 in -Pi/2 <= x <= Pi/2
		- limit cycle for x* where tan x* = 2x*
	- e.g., solving with x^20-1 = 0, x>0
		- same fishing rod analogy
			"but really really slow here"

Jan 28, 2002
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- Revisit examples from last class in light of:
	- local convergence (e.g., the x = x - tanx iteration)
	- global convergence (e.g. square root finding)

- Revisit conditions for local convergence of Newton's method
	- starting sufficiently close to root
	- f, f', f'' are continuous in a neighborhood of root
	- f'(r) <> 0

- If you know that you have a double root
	- can modify Newton's method suitably

- Worked out problem 3.2.22 and teased out problem 3.2.10

- Secant method
	- approximate the derivative, from basic formula
	- requires values at two points
	- possibility of cancelation error
	- superlinear convergence (1.618034...)

- Fixed point iteration
	- how to spot it
	- how to figure out what it is doing
	- again, an e.g., using the square root iteration

- Worked out problem 3.3.16 from textbook

- Review of basic root finding methods
	- Bisection 
		- slow, but sure 
		- requires that f() be continuous in interval
	- Newton
		- quadratic convergence
		- requires knowledge of derivative and 
		  function at a point
		- good near the root
	- Secant
		- needs two function values
		- doesn't require explicit derivative
		- possibility of cancelation error (use carefully)