Feb 15, 2002
------------
- Introduction to numerical integration
	- complement of the numerical
	  differentiation problem

- Simple ideas from Monte Carlo integration
	- Picasso paintings

- Basic integration rules
	- Trapezoidal rule
		- exact for linear 
	- Rectangle rule
		- exact for constant

- Composite integration rules
	- subdivide given region into smaller pieces
	- apply rules separately
	- add their results together

- General format for integration rules
	- linear combination of function evaluations
	- so, function evaluations are key

- Getting lower and upper bounds on integrals
	- Worked out Example 1 from your textbook

- Can there be magic forthcoming?
	- Exact integration rules for linear, quadratic,
	  and cubic polynomials
	  	- To evaluate Int[-1,1,f(x)], use
	  	   f(-1/sqrt(3)) + f(1/sqrt(3))
	- Exact integration rules for polynomials of
	  degree <= 5
	  	- use three function evaluations
	  	- at -sqrt(3/5), 0, and sqrt(3/5)

- will explain how magic works later (in a future class)

- Reminder: Test 1 next Friday

Feb 18, 2002
------------
- Recall
	- rectangle rules
	- trapezoidal rules
	- magic rules

- Riemann-integrable functions
	- relationship to rectangle rules

- Composite trapezoidal rule
	- rearranging summations to be more
	  efficient about function evaluations
	- error in integration using trapezoids

- Worked out example 1 from textbook

Feb 20, 2002
------------
- Started on Romberg quadrature

- Building up the Romberg array
	- pattern of entries in first column
	- recursively calculating entries in second and other columns

- Relationship between columns and error

- Revise material for Exam 1

Feb 22, 2002
------------
- Exam 1

Feb 25, 2002
------------
- Return Exam 1
	- statistics of scores
	- discuss questions and answers

- Romberg quadrature
	- recurrence formula
	- improving order of convergence for composite
	  trapezoid formulas

- Basic intro to adaptive quadrature

Feb 27, 2002
------------
- Adaptive quadrature

- Simpson's 3-point rule
	- where does this come from?
	- how it can be used to recursively construct
	  quadrature rules for subintervals
	- accurate upto cubic polynomials

- When to stop adaptive quadrature calculations
	- error estimation from successive integral
	  evaluations

Mar 1, 2002
------------
- Recap quadrature formulas
	- nodes and weights

- Creating our own quadrature formula
	- worked out example 1 on page 230

- Comparison of 3-point rules
	- Our formula: accurate upto quadratics
	- Simpson: accurate upto cubics
	- Gaussian quadrature: accurate upto quintics!
		- (magic rules)

- How Gaussian quadrature works
	- 3-point rule
	- 2-point rule

- How nodes are determined for Gaussian rules
	- roots of Legendre polynomials

- Homework 5 will be assigned today
	- due after the break