2026/2027, winter semester:
Sparse matrices in numerical mathematics

Charles University, Prague.



Basic resources for the course:


The main resource for the course is the book that is still in the process of publication Jennifer Scott and M. Tuma: Algorithms for sparse linear systems, Birkhauser, 2023, can be downloaded via my web pages (it is an open access text by the publisher).
Course language is Czech, if everybody speaks Czech, or English, if there is someone who does not speak Czech. Slides that I use will be available throughout the course
Exercises will be based on Matlab scripts.

There exist some very good resources that the students can profit from:

  • T.A. Davis, Direct methods for sparse linear systems, SIAM, Philadelphia, PA, 2006.
  • I.S. Duff, A.M. Erisman, and J.K. Reid, Direct Methods for Sparse Matrices, Oxford University Press, London, 1986
  • A. George and J.W.H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981
  • G. Meurant, Computer Solution of Large Linear Systems, Elsevier, Amsterdam -- Lausanne -- New York -- Oxford -- Shannon -- Singapore -- Tokyo, 1999.
  • Y. Saad: Iterative methods for sparse linear systems, SIAM, 2003.

    The basic line that we follow in the course is to cover both

    Factorization for direct methods

    and

    (Algebraic) preconditioning of iterative methods

    All the slides are in the file [everything] that will be regularly updated. Note that at this moment I still consider them unfinished. The slides are significantly changed from those used in preveious years (Note in some years I was even not using any slides). I believe that this is better than making the text available by individual chapters.

    Here are the codes [codes] that I intend to use. Note that some scripts are not finished and I may ask students to finish them. I also expect that we will play with the scripts and change them online. The codes are here to show some aspects of sparse matrix transformations for eye. Therefore, in many cases, full matrix formats instead of the sparse formats are used. Also, efficiency is considered in the codes only somewhere. For me the matlab codes were always considered only as simple variations of efficient Fortran codes of Sparslab package. This is a set of selected texts intended as a source for students' presentations [presentation_sources]

    The codes are organized in chapters. In each chapter you can just stop the code after a set of instructions. Furing the exercises I will just mention/show/explain some of the codes. The rest will be up to you. Running the codes: just in the main directory set inside the script test_book.m appropriate chapter (at line 41) and run it.

    General plan of lectures and exercises in 2026/2027




  • 30.9.2026: Introduction.
            -- Motivation
            --
    7.10.2026: Factorizations.
            -- Problems, Introduction, Basic terminology for sets, graphs, matrices.
            --
    14.10.2026: Symbolic Cholesky factorization of sparse matrices
            --
    21.10.2026: Symbolic Cholesky factorization of sparse matrices
            --
    28.10.2026: Sparse matrices and data structures
            --
    4.11.2026: (Numerical) Cholesky factorization
            --
    11.11.2026: Sparse LU factorization
            --
    18.11.2026: Stability, ill-conditioning
            --
    25.11.2026: Indefinite factorizations
            --
    2.12.2026: Sparse least squares
            --
    9.12.2026: Reorderings.
            --
    16.12.2026: Algebraic preconditioning
            --
    23.12.2026: Algebraic preconditioning
            --
    6.1.2027: Summary, repetition
            --


    Slides for the lectures in January are at SNA draft.

    Example questions for the exam are at examples of questions.
                             

    There is some early material that could be considered, but its support is disconnected and I do prefer to use the book in English instead. This is the reason that the material is on demand only .
  • Řídké matice v numerické matematice (in Czech, many things should be explained in a better way) [pdf].