Тематический план
- Общее
- Individual project
- Lecture 1
Lecture 1
Background in Linear Algebra. Basic definitions. Types and structures of square matrices.
- Lectures 2 and 3
Lectures 2 and 3
Vector and matrix norms. Range and kernel. Orthonormal vectors. Gram-Schmidt process. Eigenvalues and their multiplicities. Basic matrix factorizations and canonical forms: QR, diagonal form, Jordan form, Schur form. Basic matrix factorizations: SVD, LU, Cholessky. Existence of solution. Perturbation analysis and condition number. Errors and costs.
- Lecture 4
Lecture 4
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 5
Lecture 5
Structures and graphs representations of sparse matrices. Storage schemes for sparse matrices. Algorithms for matrix by vector multiplication.
- Lectures 6 and 7
Lectures 6 and 7
Comparison of direct and iterative methods. Overview of direct solution methods. Direct sparse methods (Gaussian elimination with partial pivoting). Iterative methods: general idea and convergence criterion. Classic iterative methods: Jacobi, Gauss-Seidel, Successive Over Relaxation (SOR), Symmetric Successive Over Relaxation (SSOR). Properties of diagonally dominant matrices, location of matrix eigenvalues. Convergence criteria for iterative methods.