Preparing for Final Exam
Monday, May 6, 2002
Closed Book, Closed Notes
in class (McBryde 216)
2:05pm - 4:05pm

Some of you have opted to take the exam on
Saturday, May 4, 2002
Closed Book, Closed Notes
Room (to be announced)
11am - 1pm
[This will be a different question paper from the one
for Monday]

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There will be 12 questions, dealing with all the topics we have 
learnt thus far. Here's a list of things you should know
or be able to do, along with the number of questions devoted
to each topic.

number representation and errors
	- rewriting formulas so that cancelation error
	  doesn't happen
	- estimating how many terms are needed in series
	  expansions to obtain a certain accuracy

root finding
	- different methods
		- bisection
		- newton
		- secant
	- working out 1-2 steps in each method
	- convergence rates for each (if they converge)
	- conditions for convergence
	- when they fail
	- when is one preferable to another
	- fixed-point iteration
	    or functional iteration
		- newton and secant are special
		  cases of functional iteration

interpolation
	- lagrange form
	- newton form
	- shifted power form
	- power form
	- connections between
		- # nodes
		- degree of interpolating polynomial
	- divided difference table and derivatives
	- problems with equally spaced nodes
	- Chebyshev nodes
	- Errors in interpolation
		- how to estimate them

numerical differentiation
	- how to derive differentiation formulas, e.g.,
		- central-difference
		- forward-difference
	- errors in formulas
	- Problems such as 4.3.13 of your textbook
	  (this was worked out in class)

numerical integration
	- how lower and upper sums work
	- how trapezoidal rule works
	  (be able to solve problems like Problem 5.2.8)
	- the basic idea of Adaptive Simpson's rule
	  (you will not be required to work out a numerical solution)
	- different 3-point rules and upto what
	  degree of polynomials they are exact for, e.g.,
	  	- Simpson's three point rule
	  	  (equation (1) on page 222)
	  	- rules with our own choice of nodes
	  	  (e.g., Example 1 on page 230)
		- Gaussian quadrature rules (i.e., magic rules)
	- Gaussian two-point and three-point
	  quadrature rules (know these by heart)
	- should be able to solve problems like 
		- Problem 5.5.5
		- Problem 5.5.12
		- Problem 5.5.9

linear systems
	- Gaussian elimination
		- why pivoting is important
		- how to do pivoting
		- operations count (see Box on page 264)
	- LU factorization
		- when is it possible
		- connection between LU and pivoting in GE
	- how to find inverse using LU and GE
	- iterative system solution
		- general format
		- Gauss-Jacobi
		- Gauss-Seidel
		- SOR
	- general condition for convergence of iterative methods
		- using spectral radius
	- specific conditions for convergence of iterative methods
		- what they are
		- for which method are they applicable
		- whether they are necessary and/or sufficient

splines
	- basic definition of splines
	- types of smoothness possible
	- how to impose (cook-up) extra conditions so that we can 
	  solve for unknown coefficients
	- solving a problem such as from Assignment #9
	  (and the examples we worked in class)

differential equations
	- be able to write Taylor series approximations, of
	  any desired order
	 - be able to ascertain if: numerical solution of differential 
	   equation is sensitive to perturbations of initial values
	   	- when do solution curves converge?
	   	- when do solution curves diverge?
	- closed form solutions for some simple differential
	  equations (only those covered in class and notes and homework 10)
	- be able to solve system of first-order differntial equations
		by Taylor series
	- be able to solve higher-order differential equations
		by Taylor series
			=> introduce extra variables

least squares approximations
	- be able to formulate least squared error (E)
	- differentiate E w.r.t. unknowns to determine "optimal"
	  coefficients
	- solve simple curve-fitting problems
		- polynomials
		- trigonometric (e.g., see your homework 10)
		- power-laws e.g., fit y = b e^x
			=> first take log on both sides