Requirements for B-level exam
- Norms of matrices and vectors. Eigenvalues, eigenvectors and spectral radius of a matrix. Relation among spectral radius and norms. Attributes of matrices: symmetric, diagonally dominant, positive definite matrices and their properties.
- Principle of iterative methods. Simple iteration method, Jacobi and Gauss-Seidel iterative method. Convergence conditions.
- Systems of nonlinear equations. Newton‘s iterative method. Its derivation for a system of 2 nonlinear equations.
- Principles of the Least squares approximation method. The system of normal equations and its properties.
- Initial value (Cauchy) problems for ordinary differential equations, existence and uniqueness of the solution. Substitution of derivatives by finite differences. One step Method for the Cauchy problem for a system of ODE in a normal form. Explicit and implicit Euler‘s method of the 1st order. Collatz's method.
- Boundary problem for ODE of the 2nd order in selfadjoint formulation of the equation. Existence and uniqueness of the solution. Numerical solution of the boundary value problem for ODE with Dirichlet‘s boundary condition.
- Classification of the linear partial differential equations of the 2nd order of two independent variables. Formulation of the problem for Poisson‘s equation. Numerical solution of the Poisson’s equation, approximation of Dirichlet's boundary condition.
- Formulation of the mixed problem for the heat equation. Numerical solution - explicit scheme. Stability.
- Formulation of the mixed problem for the wave equation. Numerical solution of the mixed problem - explicit scheme. Stability.