Detailed course information

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  (preliminary)
• 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. Mobile devices can use QR-code:

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 (PDF)
• Lecture 10:
  • Block 3: QR-method. Basic QR. Two-phase approach.
  • Deadline HW2

Week 5:

• Lecture 11:
  • Block 3: QR-method. Hessenberg QR-method
• Lecture 12:
  • Block 3: QR-method. Acceleration and convergence

Week 6:

• Lecture 13:
  • Block 4: Matrix functions. Definitions and basic methods.
• Lecture 14:
  • Block 4: Matrix functions. Methods for specialized functions
  • Deadline HW3

Week 7:

• Lecture 15:
  • Block 4: Matrix functions. Application to exponential integrators. 
  • Short course summary