### Aankondigings / Announcements

Test 01 on Tuesday 14 March @ 8AM (Venue TBC).

**ASSIGNMENT 02 CORRECTIONS**

- For Problem 3(c) rather change the a 3,3 entry to
**6**. - Problem 4 is
**optional**.

### Dosent / Lecturer

- Dr Nick Hale
- Office: A410
- Email nickhale@sun.ac.za

- Prof. JAC Weideman
- Office: A315
- Email weideman@sun.ac.za

### Rooster / Timetable

- Tuesday 13:00 @ M203
- Wednesday 10:00 @ M203
- Thursday 10:00 @ M203

### Inligtingstukke / Information Sheets

Information Sheet

### Notas / Notes

MATLAB Intro and accompanying lecture slides Afrikaans / English
Chapter 1;
Lecture Slides 1
(Background Material = Recommended Reading)

Chapter 2; Lecture Slides 2 (Linear Systems)

Chapter 3;
Lecture Slides 3
(Least Squares)

(Undergraduate notes on least squares: English /Afrikaans)

Chapter 4; Lecture Slides 4 (Eigenvalues)

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

### Opdragte / Assignments

Assignment 1 (handed in on Feb 17)

Solutions 1

Assignment 2 (handed in on March 02)

### 2017 Skedule / Schedule (tentative)

** Week 1** (Jan 31, Feb 1,2):

- Fundamentals (Undergraduate slides: eng1, eng2 / afr1, afr2)
- Linear systems
- Existence and uniqueness of solutions
- Vector and matrix norms
- Intro to conditioning of linear systems. (Undergraduate slides: eng / afr)

** Week 2** (Feb 7,8,9):

- Solving linear systems
- Elimination
- LU factorization
- Gaussian elimination

** Week 3** (Feb 14,15,16):

- Instability and error growth factor
- Pivoting
- Algorithmic complexity
- Sherman-Morrison formula

** Week 4** (Feb 21,22,23):

- Special types of linear systems
- Cholesky factorization
- Chapter 3: Least squares
- Normal equations

** Week 5** (Feb 28, Mar 1,2):

- Conditioning of LS problem
- QR Factorization
- Householder transformation
- Givens rotations

** Week 6** (Mar 7,8,9):

- Gram-Schmidt
- Singular value decomposition (SVD)

### MATLAB Kodes / MATLAB Codes

- decode_ieee.m - Decodes an IEEE 754 double-precision value.
- cos_demo.m - Demonstrate cancellation error.

### 2016 Skedule / Schedule (for reference)

** 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).

** Test week & Easter break **

** Week 7** (Mar 29&31):
Eigenvalues: intro, 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.