** No lecture ** on May 5 (SU follows Monday timetable)

** Second Test:** Friday, May 20, 09:00-11:00 (room TBA).
Covers the material of Weeks 7-13 (see below)

** Assignment 6**
here. (Due May 12, but
you should start without delay as it involves quite a bit of
computer work.)

** Class room demo ** (28 April): steepest descent
plot
and implementation

** Supplementary reading: ** Additional
notes on the CG method
(section 2.6.3 is compulsary reading)

** Supplementary reading: ** Computing the
optimal SOR parameter for the model problem
(optional but recommended reading)

Chapter 1; Lecture Slides 1 (Background Material = Recommended Reading)

Chapter 2; Lecture Slides 2 (Linear Systems)

Chapter 3; Lecture Slides 3 (Least Squares)

Chapter 4; Lecture Slides 4 (Eigenvalues)

Chapter 11 (pp.335-350); Lecture slides 11 (long) (short) (Direct and Iterative Methods for Sparse Systems)

** Week 1** (Feb 2&4):
Linear systems; existence and uniqueness of solutions;
vector and matrix norms;
intro to conditioning of linear systems.
Undergraduate slides:
English
Afrikaans

** Week 2** (Feb 9&11):
Conditioning (cont); the A = LU and PA = LU factorizations.

** Week 3** (Feb 16&18):
Instability of GE and the growth factor; Complexity of matrix computations;
Sherman-Morrison formula

** Week 4** (Feb 23&25):
Symmetric positive definite systems; Cholesky factorization;
Review of the least squares problem; Normal equations.
Undergraduate notes on least squares
English
Afrikaans

** Week 5** (Mar 1&3):
Least squares problem (cont); Orthogonal matrices;
QR factorization with Gram-Schmidt.

** Week 6** (Mar 8&10):
Gram-Schmidt; Solving the LS problem with Householder.
Geometry of a Householder transformation
diagram.
Intro to the eigenvalue problem.

** Test week & Easter break **

** Week 7** (Mar 29&31):
Eigenvalues: similarity, power iterations, Rayleigh-Quotient
iteration. Properties of symmetric/Hermitian matrices

** Week 8** (Apr 5&7):
QR iteration. Lanczos & Arnoldi algorithms.

** Week 9** (Apr 12&14):
PDE model problem.
Related undergraduate notes.
Direct methods for sparse matrices: reordering and bandwidth
reducing algorithms. Intro to iterative methods.

** Week 10 ** (Apr 19&21): Jacobi, Gauss-Seidel, SOR, Incomplete
Cholesky iterations. Convergence rates and complexity for the 2D model
problem.

** Week 11 ** (Apr 26&28): Convergence rates and complexity (cont).
Quadratic forms. Solving s.p.d. systems with
minimization methods. Method of steepest descents.

** Week 12 ** (May 3; no class on May 5): Conjugate gradient method.

Assignment 1 (handed in on Feb 18)
solutions

Assignment 2 (handed in on March 1)
solutions

Assignment 3 (handed in on March 10)
solutions

Assignment 4 (handed in on April 12)
(Additional
reading)
solutions

Assignment 5 (handed in on April 26)

Fitting a circle to data in the (x,y)-plane: class room
demo
on 25 Feb

Solving the least squares problem if QR factors are known:
class room demo
on March 1.

Reordering algorithms for direct methods for sparse
systems: class room
demo
on March 12

Iteration methods for solving sparse Ax = b: class room
demo
on April 19

JAC Weideman