** Marks:** All marks (now including Test 2 and Assignment 6)
can be found here.
(Note that Test 2 marks were adjusted so that it counts out
of 50. Note also that these marks are tentative and
subject to change pending the outcome of an external moderation.)
Please send Email ASAP if you see mistakes.

** Assignments 5 and 6:** Have been marked and can be collected
from a blue folder on the table outside A310.

** Supplementary reading: ** Additional
notes on the CG method
(sections 2.6.3 and 2.6.4 are compulsory reading),
and the derivation of the
minmax
error bound (compulsory 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 (CG).

** Week 13 ** (May 10 & 12): Preconditioned
CG method. Error estimates for CG. Nonsymmetric variants of CG.

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)
solutions

Assignment 6 (handed in on May 13)
solutions

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

Steepest descent for Ax = b: class room
demo and
plot
on 28 April

CG and PCG demo: class room demo
on May 10

JAC Weideman