TOPICS COVERED FOR MIDTERM 2 SINCE MIDTERM 1

[Stuff in brackets will not be asked about on the test].

4.2	Positive Definite Systems
4.2.1		Positive Definiteness
4.2.3		Symmetric Positive Definite Systems
                (including a basic Cholesky method).
[4.2.4]		[Gaxpy Cholesky]
[4.2.5]		[Outer Product Cholesky]
[4.2.6]		[Block Dot Product Cholesky]
[4.2.7]		[Stability of Cholesky Process]
                (You should know at least one Cholesky process).

[4.3]	[Banded Systems]
[4.3.1]		[Band LU Factorization]
[4.3.2]		[Band Triangular System Solving]
[4.3.3]		[Band Gaussian Elimination with Pivoting]

2.5	Orthogonality and the SVD.
2.5.1		Orthogonality
2.5.2		Norms and Orthogonal Transformations
2.5.3		The Singular Value Decomposition
2.5.4		The Thin SVD
2.5.5		Rank Deficiency and the SVD
[2.5.6]		[Unitary Matrices]

5	Orthogonalization and Least Squares

5.3	Full Rank LS Problem
5.3.1		Implications of Full Rank
5.3.2		The Method of Normal Equations
5.3.3		LS Solution Via QR Factorization
[5.3.4]		[Breakdown in Near Rank-Deficient Case]

5.2	QR Factorization
5.2.1		Householder QR
5.2.6		Properties of the QR Factorization
5.2.7		Classical Gram-Schmidt
5.2.8		Modified Gram-Schmidt

5.1	Householder and Givens Matrices
5.1.1		A 2-by-2 Preview
5.1.2		Householder Reflections
5.1.3		Computing the Householder Vector
[5.1.4]		[Applying Householder Matrices]
[5.1.5]		[Roundoff Properties]
[5.1.6]		[Factored Form Representation]

5.4	Other Orthogonal Factorizations
[5.4.1]		[Rank Deficiency: QR with Column Pivoting]

5.5	The Rank Deficient Least Squares Problem
5.5.1		The Minimum Norm Solution
[5.5.2]		[Complete Orthogonal Factorization and x_{LS}]
5.5.3		The SVD and the LS Problem
5.5.4		The Pseudo-Inverse
[5.5.6]		[QR with Column Pivoting and Basic Solutions] (needed for HW2)
5.5.8		Numerical Rank and the SVD