Оглавление секции

  • General info and teams for the course

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    • !!! Working with Matlab in 2026 Файл
    • Course overview 2026 Файл
    • Course textbook Файл

      Yosef Saad. Iterative Methods for Sparse Linear Systems

    • Getting started with Matlab Папка
    • Teams for MSD group Опрос
    • Teams for CM group Опрос

  • Lectures

  • Classroom examples

    • Classroom examples for MSD group Папка
    • Classroom examples for CM group Папка

  • Exam

  • Module 1. Background in matrix theory and sparse linear systems

    • Lecture 1. Fundamentals of Linear Algebra. Introduction of elementary notation
    • Lecture 2. Types and structures of square matrices. Vector and matrix norms
    • Lecture 3. Subspace, range and kernel. Existence of solution. Orthogonality, Gram-Schmidt process and QR-factorization
    • Lecture 4. Matrix factorizations (QR, LU, Cholesky)
    • Lecture 5. Multiplicities of eigenvalues. Canonical forms of matrices. Diagonal form, jordan form, schur form, singular value decomposition (svd), relation between schur and svd
    • Lecture 6. Positive definite matrices. Normal and Hermitian matrices. Powers of matrices. Perturbation analysis. Errors and costs
    • Lecture 7. Graph representations of sparse matrices. Permutations and reordering
    • Lecture 8. Storage schemes for sparse matrices. Discretization of partial differential equations: overview of methods
    • Lecture 9. Discretization of partial differential equations: finite difference method

  • Practical assignment 1. Getting started with Matlab

    • Practical assignment 1. Getting started with Matlab for MSD group (dealine March 21) Задание
    • Practical assignment 1. Getting started with Matlab for CM group (dealine March 21) Задание

  • Practical assignment 2. Matrix fundamentals: types and structures

    • Practical assignment 2. Matrix fundamentals: types and structures for MSD group (dealine March 28) Задание
    • Practical assignment 2. Matrix fundamentals: types and structures for CM group (dealine March 28) Задание

  • Practical assignment 3. Vector and matrix norms. Existence of solution

    • Practical assignment 3. Vector and matrix norms. Existence of solution for MSD group (dealine Arpil 8) Задание
    • Practical assignment 3. Vector and matrix norms. Existence of solution for CM group (dealine Arpil 8) Задание

  • Practical assignment 4. Gram-Schmidt and QR-factorization

    • Practical assignment 4. Gram-Schmidt and QR-factorization for MSD group (dealine Arpil 15) Задание
    • Practical assignment 4. Gram-Schmidt and QR-factorization for CM group (deadline April 15) Задание

  • Practical assignment 5. Eigenvalues multiplicities, matrix factorizations, LU- and Cholesky factorizations

    • Practical assignment 5. Eigenvalues multiplicities, matrix factorizations, solving linear systems using LU-factorization for MSD group (deadline April 22) Задание
    • Practical assignment 5. Eigenvalues multiplicities, matrix factorizations, solving linear systems using LU- and Cholesky factorizations for linear systems for CM group (deadline April 22) Задание

  • Practical assignment 6. Normal, hermitian and positive definite matrices

    • Practical assignment 6. Normal, hermitian and positive definite matrices for MSD group (deadline May 6-10) Задание
    • Practical assignment 6. Normal, hermitian and positive definite matrices for CM group (deadline May 6) Задание

  • Practical assignment 7. Condition number, permutation and reordering, sparse formats

    • Practical assignment 7. Condition number, permutation and reordering, sparse formats for MSD group (deadline May 23) Задание
    • Practical assignment 7. Condition number, permutation and reordering, sparse formats for CM group (deadline May 23) Задание

  • Module 2. Direct and iterative methods. Krylov subspace methods for sparse linear systems and preconditioning techniques

    • Lecture 10. Discretization of PDE: overview of FEM. Direct and iterative methods for sparse linear systems: comparison. General formulation of iterative methods and convergence criterion
    • Lecture 11. Classical iterative methods. Simple iteration, Jacobi, Gauss-Seidel, Successive Over Relaxation (SOR), Symmetric Successive Over Relaxation (SSOR) and their variations
    • Lecture 12. Projection methods. 1D projection methods: Steepest Descent Method (SDM), Minimal Residual Iteration Method (MRIM), Residual Norm Steepest Descent Method (RNSD)
    • Lecture 13. Krylov subspace methods based on Arnoldi’s orthogonalization. Full Orthogonalization method (FOM)
    • Lecture 14. Krylov subspace methods based on Arnoldi’s orthogonalization. Generalized Minimal Residual method (GMRES)
    • Lecture 15. Comparison of FOM and GMRES. Methods based on Lacnzoc orthogonalization for symmetric (Hermitian) matrices:  Lanczos method for symmetric systems, Direct Lanczos, Conjugate Gradient (CG), Generalized Conjugate Residual (GCR) and biorthogonalization. Methods based on Lacnzoc orthogonalization for nonsymmetric (non-Hermitian) matrices: Lanzos method for nonsymmetric systems, Biconjugate Gradient (BiCG).
      Preconditioning techniques. Examples of preconditioners. Examples of preconditioned methods: Preconditioned conjugate gradient method (PCG), Split Preconditioned Conjugate Gradient method (Split PCG), Left preconditioned Generalized Minimal Residual method (LP GMRES), Right preconditioned Generalized Minimal Residual method (RP GMRES)

  • Practical assignment 8. Comparison of direct and iterative methods for different sparse systems

    • Practical assignment 8. Comparison of direct and iterative methods for different systems for MSD group (deadline May 30) Задание
    • Practical assignment 8. Comparison of direct and iterative methods for different systems for CM group (deadline May 30) Задание

  • Practical assignment 9 for CM group. Classical iterative methods

    • Practical assignment 9. Classical iterative methods for CM group (deadline June 13) Задание

  • Practical assignment 10 for CM group. Krylov subspace methods. FOM and GMRES

    • Practical assignment 10. Krylov subspace methods. FOM and GMRES for CM group (deadline June 13) Задание

  • Individual project for CM group

  • Individual project for MSD group