Algorithms for matrix iterative methods 2025/2026

Course Journal

Last update: 12/11/25

Lab 1 - 29/09/2025

Introduction to MatLab, Krylov subspace methods, matrix computation problems

Lab1: ex 1-4

Lecture 1 - 02/10/2025

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 2 - 06/10/2025

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

Lab2: ex 1-3

Lecture 2 - 09/10/2025

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 3 - 03/10/2025

Projection methods.

Lab3: ex 1.1-1.3

Lecture 3 - 16/10/2025

Derivation of CG from the Lanczos algorithm and the D-Lanczos method. Refer to:

[SB] Sections 6.7.1.

Lab 4 - 20/10/2025

CG convergence behavior. D-Lanczos.

Lab3: ex 1.1-1.3, 2.

Lecture 4 - 23/10/2025

SYMMLQ and MINRES methods. Refer to:

SYMMLQ, [LSB] Section 2.5.4.
MINRES [LSB] Section 2.5.5.

Lab 5 - 27/10/2025

SYMMLQ, MINRES, CG, and the saddle point problem.

Lab5: ex 1.1-1.3.

Lecture 5 - 29/10/2025

CG convergence behavior in exact arithmetic. Refer to:

[LSB] Section 5.6.2-5.6.4

Lanczos algorithm and orthogonal polynomials. Refer to:

[SB] Section 6.6.

Lab 6 - 3/11/2025

CG, the moment problem, and the loss of orthogonality.

Lab6: ex 1.1-1.3.

Lecture 6 - 6/11/2025

Vorobyev's problem of moments and the moment matching property. Connections between CG (Lanczos) and Gauss quadrature. Refer to:

[LSB] Chapter 3, pp. 136-139 and 146-148 (in particular, Theorem 3.7.1).
[LSB] Corollary 5.6.2

Lab 7 - 10/11/2025

CG convergence in finite precision arithmetic.

Lab7: ex 1.1-1.3.

Questions about the projects.

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]
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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