Jan 30, 2002
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- New topic: interpolation

- Examples
	- Driving from C'burg to Roanoke on I-81
	- weather maps
	- niceness and grades

- There are several choices for interpolating polynomials
	- if we insist on lowest degree <=n, we can get
	  uniqueness

- Statement of uniqueness theorem

- Two different forms for the interpolating polynomial
	- Lagrange form
	- Newton form

- Finding Newton's form by divided differences table
	- what happens if you change order of entries?
	- what does it mean when you have zeros in a column?

- Worked out exercise 4.1.15 from textbook

- Notice the difference between saying 
	- "the data comes from a cubic polynomial", and 
	- "the lowest order polynomial is a cubic"


Feb 4, 2002
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(one class in between went for MATLAB)

- Continue on interpolation and doodling

- Just more problem solving in class; solved
	- 4.1.18
	- 4.1.19
	- 4.1.22
	- 4.1.23
	- 4.1.26

Feb 6, 2002
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- Revisit Lagrange and Newton forms

- Lagrange form is not `incremental' when you get additional data

- Newton form is `incremental'

- Specializations of Newton form
	- centers are same: shifted power form
		- how do you determine the coefficients?
	- centers are same and zero: power form

- Why all these forms?
	- what is the advantage?

- Connection between divided difference table and derivatives

- Recap: interpolation finds a polynomial passing through given points
	- what about non-nodal points?

- Pathological example with 1/(1+x^2)


Feb 8, 2002
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- Continued pathological example
	- problem is equally spaced nodes

- Chebyshev nodes in [-1,1]

- Mapping Chebyshev nodes to [a,b]

- Moral of the story
	- Use Chebyshev nodes, or
		(take this on faith for now)
	- Use equally spaced nodes
		(but don't go too far from center of range)

- Errors in Interpolation

- Interpolation Errrors Theorem

- Example: error in linear interpolation

- Specializing the theorem to equally spaced nodes

- Solved problems 
	- example 1 from section 4.2
	- problem 4.2.9 (page 170)
		- but we assumed equally spaced nodes