Requirements for exams

• Norms of matrices and vectors. Eigenvalues, eigenvectors and spectral radius of a matrix. Relation among spectral radius and norms. Properties of matrices: symmetry, positive definiteness, diagonal dominance.
• Principle of iterative methods. Simple iteration method (Fixed point iterations). Jacobi and Gauss-Seidel methods - 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 from Taylor expansion. One step methods for the Cauchy problem for a system of ordinary differential equations in the normal form. Explicit and implicit Euler‘s method. Midpoint (Collatz's) method. Local discretisation error, global error, convergence, order of a method.
• Boundary problem for ordinary differential equation 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. Consistency error and stability of schemes.

2024