Week Date Topic
1 8/28 Introduction; matrix multiplication
2 8/31 Caches and matrix multiply performance; norms
9/2 Vector and matrix norms; perturbing matrix-vector multiply
9/4 Condition numbers; floating point
3 9/7 Error analysis of quadratics and matrix-vector multiplies
9/9 Orthogonal matrices, SVD, low rank
9/11 Fast multiply, nonzero structure
4 9/14 Timing matrix multiply; Gaussian elimination
9/16 LU factorization and Schur complements
9/18 Triangular solves, packed storage, block LU
5 9/21 Block LU with special blocks; perturbation theory
9/23 Backward error in GE; pivoting; iterative refinement
9/25 Iterative refinement; condition estimation; scaling
6 9/28 Symmetric matrices, LDL’, and Cholesky
9/30 Stability of Cholesky; diagonal dominance; sparse LU
10/2 Matrices and graphs; tree matrices; nested dissection
7 10/5 Normal equations, QR, Gram-Schmidt, and reflections
10/7 Householder QR, storing Q and R, least squares sensitivity
10/9 Sensitivity of y = A’*x and of least squares
8 10/12 Fall break — no lecture
10/14 Sensitivity redux; sparse least squares problems
10/16 Constrained minimization, orthogonal Procrustes, rank-deficiency
9 10/19 Rank-deficient least squares; some multivariate calculus reminders
10/21 Intro to eigenvalue problems
10/23 First-order sensitivity; intro to Gershgorin
10 10/26 Gershgorin and Bauer-Fike
10/28 Power method, shift-invert, RQ iteration, and subspace iteration
10/30 Orthogonal iteration and QR
11 11/2 Orthogonal iteration, QR, and Hessenberg QR
11/4 Shift strategies for QR
11/6 Full implicit double-shift QR with deflation
12 11/9 Rayleigh quotients, minimax and interlace theorems
11/11 Interlace, inertia, and brief intro to bisection
11/13 SVD algorithms (guest lecture by Charlie Van Loan)
13 11/16 Introduction to the Lanczos iteration
11/18 More Lanczos: reorthogonalization, error estimates, convergence
11/20 Arnoldi; Krylov methods for Ax=b; stationary methods; preconditioning
14 11/23 Derivation of conjugate gradients
11/25 Thanksgiving special: a regression puzzle
11/27 Thanksgiving recess — no lecture
15 11/30 CG convergence; GMRES; a model problem
12/2 Multigrid
12/4