Lecture notes for "Applied Numerical Linear Algebra", Spring 2020

These notes are intended to be outlines of the material covered during each lecture, not always comprehensive notes. I will attempt to post them before lecture, but may repost them after lecture with corrections.

Jan 21: Lecture 1 - Course outline.

Jan 23: Lecture 2 - Floating point arithmetic and error analysis.

Jan 28: Lecture 3 - Norms and the SVD

Jan 30: Complete Lecture 3

Feb 4: Lecture 5 - Communication-Avoiding Algorithms

Feb 6: Complete Lecture 5, begin Lecture 6 - Gaussian Elimination

Feb 11: Complete Lecture 6 (slightly updated), begin Lecture 7 - Gaussian Elimination on matrices with structure

Feb 13: Continue Lecture 7 - Gaussian Elimination on matrices with structure

Feb 18: Complete Lecture 7, begin Lecture 9 - Least Squares

Feb 20: Continue Lecture 9 - Least Squares

Feb 25: Complete Lecture 9 - Least Squares (updated), begin Lecture 11 - Low Rank Matrices

Feb 27: Continue Lecture 11 - Low Rank Matrices (updated)

Mar 3: Complete Lecture 11 - Low Rank Matrices (updated again); begin Lecture 14 - Eigenvalue Problems

Mar 5: Continue Lecture 14 - Eigenvalue Problems

Mar 10: Continue Lecture 14 - Eigenvalue Problems (on-line)

Mar 12: Complete Lecture 14 - Eigenvalue Problems (on-line)

Mar 17: Lecture 17 - Chapter 5: Symmetric Eigenvalue Problems and the SVD (on-line) (updated, 24 Mar)

Notes for recorded lectures:

Part 1 (43 mins): Overview, Perturbation Theory for Eigenvalues (including post-lecture corrections in red)

Part 2 (37 mins): Perturbation Theory for Eigenvectors (no post-lecture corrections this time :) )

Part 3 (40 mins): Algorithms: Overview (minor post-lecture corrections/additions in red)

Part 4 (60 mins): Algorithms: QR Iteration and Divide-and-Conquer (minor post-lecture corrections/additions in red)

Part 5 (38 mins): Algorithms: Bisection, Inverse Iteration, MRRR, and Jacobi (minor post-lecture corrections/additions in red)

Apr 2: Chapters 6 and 7 - Iterative methods (updated Apr 9)

Notes for recorded lectures:

Part 1 (58 mins): Overview of iterative methods, review of Poisson Equation (including post-lecture corrections in red)

Part 2 (67 mins): Kronecker Products, Summary of methods applied to 2D and 3D Poisson Equations (including post-lecture corrections in red)

Part 3 (60 mins): Splitting Methods: Jacobi, Gauss-Seidel and Successive Overrelaxation (including post-lecture corrections in red)

Part 4 (67 mins): Splitting Methods: Convergence Analysis (including post-lecture corrections in red)

Part 5 (64 mins): Multigrid, in power point, pdf (pre-lecture), or pdf (post-lecture, including mark-up)

Part 6 (54 mins): Krylov Subspace Methods: Introduction, Arnoldi and Lanczos (including post-lecture corrections in red), including Fig 7.2 from text:

9 steps of Lanczos (pdf (pre-lecture) and pdf (post-lecture, including mark-up))

29 steps of Lanczos (pdf (pre-lecture) and pdf (post-lecture, including mark-up))

Part 7 (77 mins): Krylov Subspace Methods: GMRES and Conjugate Gradients (including post-lecture corrections in red)

Part 8 (73 mins): Chebyshev Polynomials applied to analyzing the convergence of CG and accelerating splitting methods

Typed notes

Handwritten notes from recorded lecture (including post-lecture corrections in red, begins on page 2)

Fig 6.8 from text: Decision tree for choosing an iterative algorithm (pdf (pre-lecture) and pdf (post-lecture, including mark-up))

Part 9 (52 mins): Preconditioning, to Accelerate Iterative Methods

Typed notes

Handwritten notes from recorded lecture (including post-lecture corrections in red)