1. Disciplines of numerical mathematics: continuous and discrete problems, discretization order; sources of error, rounding error, computer epsilon; numerical stability.
2. Approximation and interpolation of functions: polynomial interpolation, interpolation error; least squares approximation; uniform approximation, Bernstein polynomials, spline functions; modeling curves and surfaces, Bezier curves.
3. Rootfinding for nonlinear functions: geometric approach to rootfinding; fixed-point iterations and fixed point theorem; fundamental theorem of algebra, separations and calculations of polynomial roots; Newton's method for nonlinear systems.
4. Numerical integration and derivation: numerical differentiation, Richardson extrapolation; numerical quadrature formulas, error estimation, step size control; Romberg method; Gauss formulas.
5. Numerical linear algebra: solving linear systems using LU decomposition variants, inverse matrix; eigenvalues and eigenvectors calculation, spectral decomposition; singular value decomposition, orthogonal factorization, pseudoinverse.
6. Iterative methods for solving linear systems: linear methods Jacobi, Gauss-Seidel, relaxation; nonlinear methods, steepest descent method, conjugate gradient method, preconditioning.
2. Approximation and interpolation of functions: polynomial interpolation, interpolation error; least squares approximation; uniform approximation, Bernstein polynomials, spline functions; modeling curves and surfaces, Bezier curves.
3. Rootfinding for nonlinear functions: geometric approach to rootfinding; fixed-point iterations and fixed point theorem; fundamental theorem of algebra, separations and calculations of polynomial roots; Newton's method for nonlinear systems.
4. Numerical integration and derivation: numerical differentiation, Richardson extrapolation; numerical quadrature formulas, error estimation, step size control; Romberg method; Gauss formulas.
5. Numerical linear algebra: solving linear systems using LU decomposition variants, inverse matrix; eigenvalues and eigenvectors calculation, spectral decomposition; singular value decomposition, orthogonal factorization, pseudoinverse.
6. Iterative methods for solving linear systems: linear methods Jacobi, Gauss-Seidel, relaxation; nonlinear methods, steepest descent method, conjugate gradient method, preconditioning.