Mar 11, 2002
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- review of linear algebra
	- geometry view
	- vector view

- geometry view
	- linear equations in 2D
	- linear equations in 3D

- vector view
	- linear combination of vectors
	- linear dependence and linear independence

- how singularity, solutions etc. manifest in different views
	- no solutions 
	- one solution 
	- infinite solutions

- elimination method of solving simultaneous equations Ax=b
	- why Cramer's rule is bad
	- why it is not necessary to explicitly solve
	  for inverses of matrices

- need for pivoting

Mar 13, 2002
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- Gaussian elimination method
	- formally speaking

- Pivoting and its importance
	- ways of flipping

Mar 15, 2002
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- Revisiting Gaussian elimination
	- our approach to pivoting
	   (like the book, we determine the ratios but
	  unlike the book, we revisit the divisors for the
	  ratios at each stage)

- Operations count for GE
	- for rendering in upper triangular form
		=> n^3/3
	- for solving for unknowns
		=> n^2

Mar 18, 2002
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- LU Factorization
	- L: unit lower triangular, U: upper triangular
	- what do we gain from it?

- Ax = b decomposes into
	- Ly = b
	- Ux = y

- Advantages of the decompositional approach
	- allows us to solve a whole family of problems
          (not just one)

- Finding U is trivial but where is the L in GE?
	- obtained from negatives of the multipliers
	- also need to invert a lower triangular matrix

- LU breaks down for some matrices
	- some matrices do not have an LU decomposition!
	- these are the cases where pivoting is required for GE

- Capturing pivoting in an LU decomposition
	- Instead of A=LU, do PA=LU

- Matrix decompositions in real life
	- SVD: Singular Value Decomposition
	- uses in search engines (Google, CLEVER, etc.)

Mar 20, 2002
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- How to find A^(-1) using a GE code
	- you will need this for your Assignment #7

- Recap iterative ways of solving equations
	- when are they prefered over "direct" approaches?

- Can extend simple unknowns to a vector of unknowns
	- express equations as
		x = something involving x
	- x was a single variable in Chapter 3
		(Newtons/Secants/Functional Iteration)
	- here, x is a vector

- Different ways of making Ax=b look like x=something
	- Gauss-Jacobi
	- Gauss-Seidel

- Mathematically, these two are derived from the same general scheme
	- write A in Ax=b as A=M-N
	=> x = M^(-1)Nx + M^(-1)b
	- different choices of M and N give you Gauss-Jacobi and
	  Gauss-Seidel!

- Continued for next class!

Mar 22, 2002
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- Details of iterative methods

- Write down A = D - L - U
	- In Gauss-Jacobi
		- M = D
		- N = L + U
	- In Gauss-Seidel
		- M = D - L
		- N = U

- Worked out simple examples

- Convergence criteria for iterative methods
	- when to stop
	- choice of initial vector

- Consider a new iterative scheme: SOR
	- M = 1/w (D-wL)
	- What is N?
	- What is this scheme doing?

- When w=1, SOR reduces to Gauss-Seidel

Mar 25, 2002
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- Details of iterative methods
	- convincing ourselves of what Gauss-Jacobi
	  and Gauss-Seidel are doing

- What is SOR doing?
	- a "mix" between old values and new values computed
	  using Gauss-Seidel
	- what are good values for w?

Mar 27, 2002
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- Convergence properties of iterative methods
	- if spectral radius of M^(-1)N is less than 1,
	  then the iterative method will converge
	- difficult to check for

- Alternative condition for Jacobi and Gauss-Seidel
	- if matrix A is diagonally dominant, J and GS will
	  converge (this is sufficient but not necessary)

- Alternative condition for SOR
	- if matrix A is symmetric, positive definite, SOR
	  will converge for 0<w<2 (necessary and sufficient)