Requirements for A-level exam
- Norms of matrices and vectors. Eigenvalues, eigenvectors and spectral radius of a matrix. Relation among spectral radius and norms. Attributes of matrices: symmetrical, diagonally dominant, positive definite matrices and their properties.
- Principle of iterative methods. Simple iteration method, Jacobi and Gauss-Seidel iterative method - both matrix and element-wise formulations. Convergence conditions.
- Solving system of linear equations with positive definite matrix by minimizing a functional. Method of gradient (steepest) descent. Finding the direction of the steepest descent and the optimal step-length.
- Systems of nonlinear equations. Newton‘s iterative method. Its derivation for a system of 2 nonlinear equations.
- Least squares approximation method – principle, approximation with algebraic polynomial. Quadratic deviation. Derivation of 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. Derivation of finite differences. One step Method for the Cauchy problem for a system of ODE in a normal form. Local discretisation error, global error, convergence, order of the method, explicit and implicit method. 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. Derivation of the system of equations and its properties. Consistency error and convergence of the method.
- 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, derivation of the system of equations and its properties. Consistency error and convergence of the method.
- Formulation of the mixed problem for the heat equation. Numerical solution, explicit and implicit scheme. Consistency error, convergence and stability of the schemes.
- Formulation of the mixed problem for the wave equation. Numerical solution of the mixed problem, explicit and implicit scheme. Stability of schemes.