Numerical methods of Linear Algebra for Sparse Matrices

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  • Выбрать элемент Course overview 2025

    

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  • Выбрать элемент Course textbook

    

    Course textbook Файл

    Yosef Saad. Iterative Methods for Sparse Linear Systems

  • Выбрать элемент Working with Matlab in SFedU and getting started

    

    Working with Matlab in SFedU and getting started Папка
• Выбрать тему Individual project

  Individual project

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• Выбрать тему Module 1. Background in matrix theory and sparse linear systems. Lectures 1-7

  Module 1. Background in matrix theory and sparse linear systems. Lectures 1-7

  • Lecture 1. Background in Linear Algebra. Basic definitions. Types and structures of square matrices. Vector and matrix norms.
  • Lecture 2. Range and kernel. Existence of Solution. Orthonormal vectors. Gram-Schmidt process. Thin and full QR-factorization.
  • Lecture 3.  Eigenvalues and their multiplicities. Canonical forms by similarity transformation: diagonal form,  Jordan form, Schur form. Other matrix factorizations: SVD, LU, Cholessky. Positive definite matrices.
  • Lecture 4. Properties of normal and Hermittian matrices. Powers of matrices. Perturbation analysis and condition number. Errors and costs.
  • Lectures 5-6. Discretization of partial differential equations (PDEs). Finite differences. 1D Poisson’s equation. 2D Poisson’s equation. Overview of Finite element method. Assembly process in FEM. 
  • Lecture 7. Structures and graphs representations of sparse matrices. Storage schemes for sparse matrices. Algorithms for matrix by vector multiplication.
  • Lecture 8. Comparison of direct and iterative methods. Overview of direct solution methods. Direct sparse methods (Gaussian elimination with partial pivoting). 

  • Выбрать элемент Lecture notes for Module 1

    

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  • Выбрать элемент Practical assignment 1. Getting started with Matlab

    

    Practical assignment 1. Getting started with Matlab Задание
  • Выбрать элемент Practical assignment 2. Matrix fundamentals: types and structures

    

    Practical assignment 2. Matrix fundamentals: types and structures Задание
  • Выбрать элемент Practical assignment 3. Vector and matrix norms. Existence of solution, solving linear system in Matlab

    

    Practical assignment 3. Vector and matrix norms. Existence of solution, solving linear system in Matlab Задание
  • Выбрать элемент Practical assignment 4. Gram-Schmidt and QR-factorization

    

    Practical assignment 4. Gram-Schmidt and QR-factorization Задание
  • Выбрать элемент Practical assignment 5. Eigenvalues and their multiplicities, matrix factorizations, Hermitian and positive definite matrices

    

    Practical assignment 5. Eigenvalues and their multiplicities, matrix factorizations, Hermitian and positive definite matrices Задание
  • Выбрать элемент Practical assignment 6. Schur, LU- and Cholessky factorizations of Hermitian positive definite matrices. Solving linear systems using LU- and Cholessky factorizations.

    

    Practical assignment 6. Schur, LU- and Cholessky factorizations of Hermitian positive definite matrices. Solving linear systems using LU- and Cholessky factorizations. Задание
  • Выбрать элемент Practical assignment 7. Condition number, computational costs, well- and ill-conditioned problems

    

    Practical assignment 7. Condition number, computational costs, well- and ill-conditioned problems Задание
  • Выбрать элемент Practical assignment 8. Permutations, reordering and fill-ins

    

    Practical assignment 8. Permutations, reordering and fill-ins Задание
  • Выбрать элемент Practical assignment 9. Sparse storage and sparse formats

    

    Practical assignment 9. Sparse storage and sparse formats Задание
  • Выбрать элемент Practical assignment 10. Comparison of direct and iterative methods for different sparse systems

    

    Practical assignment 10. Comparison of direct and iterative methods for different sparse systems Задание
• Выбрать тему Module 2. Krylov subspace methods for sparse linear systems and preconditioning techniques

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

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