Algorithms for matrix iterative methods 2026/2027

Course Journal

Last update: 13/10/26

Lecture 1 - 29/09/2026

Introduction. Matrix properties (sparsity, structure, spectral information, symmetry). Refer to:

[SB] Chapter 1

From the Gram-Schmidt process to the Arnoldi and Lanczos algorithms. Refer to:

[SB] Section 6.3
[SB] Section 6.6

Lab 1 - 29/09/2026

Introduction to MatLab, Krylov subspace methods, matrix computation problems

Lab1: ex 1-4

Lecture 2 - 13/10/2026

Projection methods. Orthogonal and oblique projection methods, well-defined methods, and the finite termination property. Refer to:

[SB] Sections 5.1-5.2, Sections 6.1-6.2.
[LSB] Sections 2.1-2.2, pp. 11-22.

Lab 2 - 13/10/2026

Arnoldi and Lanczos algorithm, implementation, comparison, and loss of orthogonality.

Lab2a: ex 1-3

Projection methods.

Lab2b: ex 1.1-1.3

Exam:

The exam will cover all the material presented in lectures and practicals throughout the semester and will be conducted in oral form.

In addition, students are required to complete one homework assignment during the semester. The assignment is a team project, in which each team must reproduce, using MATLAB, the experiments described in an assigned research article.

A map of iterative methods studied in the course

Scheme MIM 2.pdf

Matlab

Here is a useful summary of Matlab's basic functions.

Here is Matlab Onramp, a 2 hours online introduction with useful exercises.

Materials and literature:

Literature:

[SB] Y. Saad: Iterative methods for sparse linear systems, SIAM, Philadelphia, 2003. (Main source) [Library]
An open free version can be downloaded here.
[LSB] J. Liesen and Z. Strakos, Krylov Subspace Methods, Principles and Analysis, Oxford University Press, 2012, 408p. [Library]
G. Meurant and P. Tichý, Error Norm Estimation in the Conjugate Gradient Algorithm, Society for Industrial and Applied Mathematics, 2024.
Barrert, R., et all: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
Higham, N.: Accuracy and stability of numerical algorithms, SIAM, Philadelphia, 2002 (2nd ed.).
Meurant, G.: Computer solution of large linear systems, Studies in Mathematics and Its Applications, North-Holland, 1999.

Material discussed in class (in order of appearance):

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