Math 660-600--Final Exam Review
General information. The test will be held on Friday,
December 10, from 3 to 5 pm in our usual room. It will have 5 to 7
questions, some with multiple parts. You will be expected to define
terms, state and/or prove theorems, and do problems similar to those
on homework assignments. There will be no direct questions on
the material covered prior to the mid-term exam; it will come up in
answering questions on the material covered since the mid-term. Direct
questions will be over this material:
- LU factorization with partial pivoting
- Permutation matrices
- §§ 3.4.1-3.4.4
- LDMT and LDLT factorizations
- Algorithms for finding these.
- §§ 4.1.1-4.1.2
- Symmetric, positive definite systems
- Cholesky factorization
- Algorithms for finding the Cholesky factorization
- Symmetric, positive semi-definite systems
- Symmetric pivoting
- §§ 4.2.1, 4.2.3-4.2.5, 4.2.8-4.2.9
- Banded systems
- LU factorization
- Cholesky factorizations
- §§ 4.3.1-4.3.3, 4.3.5
- Householder transformations
- Householder matrices
- Algorithm for finding a Householder matrix
- Update form for multiplying by a Householder matrix
- §§ 5.1.1-5.1.6
- QR and other factorizations
- Gram-Schmidt process
- modified Gram-Schmidt process
- QR via Householder matrices
- QR via Householder with column pivoting
- Bidiagonalization
- §§ 5.2.1, 5.2.6-5.2.8, 5.4.1-5.4.3
- Least Squares Problems
- Full rank problem - §§ 5.3.1-5.3.3
- Normal equations
- QR factorization
- Rank deficient problems
- SVD and pseudoinverse - §§ 5.5.1-5.5.3
- Steepest descent and conjugate gradient methods
- §§ 10.2.1-10.2.4
- Eigenvalue problems
- Basics - definition and characteristic polynomial
- Real, symmetric matrices
- Reality of eigenvalues and orthogonality of eigenvectors
corresponding to distinct eigenvalues
- Schur decomposition
- Courant-Fischer Theorem
Updated: 12/7/99