# Old course overview (HT2016)

## Course literature

- Lecture notes in numerical linear algebra (written by the lecturer). PDF-files below.
- Parts from the book "Numerical Linear Algebra", by Lloyd N. Trefethen and David Bau. ISBN: 0-89871-361-7, referred to as [TB]. It is available in Kårbokhandeln. The chapters and recommended pages are specified in the Lecture notes PDF-files.

## Course contents:

- Block 1: Large sparse eigenvalue problems
- Literature: eigvals.pdf

- Block 2: Large sparse linear systems
- Literature: linsys.pdf

- Block 3: Dense eigenvalue algorithms (QR-method)
- Literature: qrmethod.pdf

- Block 4: Functions of matrices
- Literature: matrixfunctions.pdf

- Block 5: (only for PhD students taking SF3580) Matrix equations
- Literature: matrixequations.pdf (preliminary)

## Learning activities:

The homeworks are mandatory for completion of the course.

- Homework 1. hw1.pdf additional files: arnoldi.m. If homework 1 is completed by deadline (see below) one bonus point is awarded to the exam.
- Homework 2. hw2.pdf. If homework 2 is completed by deadline (see below) one bonus point is awarded to the exam.
- Homework 3. hw3.pdf. You will need alpha_example.m, naive_hessenberg_red.m, and schur_parlett.m. If homework 3 is completed by deadline (see below) one bonus point is awarded to the exam.
- As part of all homeworks: Course training area: wiki. A selection of the exercises suitable for exam preparation are available in selected_exercises.pdf.

## Weekly schedule:

**Week 1:**

- Lecture 1:
- Course introduction: intro_lecture.pdf (username=password=password on wiki)
- Block 1: Basic eigenvalue methods
- Additional video material:

- Lecture 2: Block 1

- Numerical variations of Gram-Schmidt. Arnoldi's method derivation
- Introduction to Arnoldi method: arnoldi_intro.pdf (username=password=password on wiki)
- Numerical variations of Gram-Schmidt orthogonalization

- Lecture 3: Block 1

- Arnoldi's method for eigenvalue problems,
- Intro to convergence of the Arnoldi method for eigenvalue problems

**Week 2:**

- Lecture 4: Block 1

- Convergence theory for Arnoldi for eigvals continued. Disk reasoning. Shift-and-invert.
- Lanczos method, Lanczos for eigenvalue problems

- Lecture 5:
- Block 2: Iterative methods for linear systems. GMRES derivation
- Deadline HW1

**Week 3:**

- Lecture 6:
- Block 2: GMRES convergence
- Block 2: Introduction to conjugate gradients (CG method)

- Lecture 7:
- Block 2: Derivation of CG method. Convergence of CG method.

- Lecture 8:
- Block 2: CG-methods for non-symmetric problems: CGN and BiCG

**Week 4:**

- Lecture 9:
- Block 2: Preconditioning
- Block 3: QR-method intro. Slides: qrmethod_lecture1.pdf

- Lecture 10:
- Block 3: QR-method. Basic QR. Two-phase approach. Improvement 1. Slides: qrmethod_lecture2.pdf
- Deadline HW2

**Week 5:**

- Lecture 11:
- Block 3: QR-method. Hessenberg QR-method. Slides: qrmethod_lecture3.pdf

- Lecture 12:
- Block 3: QR-method. Acceleration and convergence. qrmethod_lecture4.pdf

**Week 6:**

- Lecture 13:
- Block 4: Matrix functions. Definitions and basic methods. matrix_functions_lecture1.pdf

- Lecture 14:
- Block 4: Matrix functions. Methods for specialized functions. matrix_functions_lecture2.pdf
- Deadline HW3

**Week 7:**

- Lecture 15:
- Block 4: Matrix functions. Application to exponential integrators.