1. distinguish and identify types of errors in numerical computing
2. apply direct methods in solving systems of linear equations
3. identify ill conditioned system of linear equations
4. compute interpolation polynomial and cubic spline for given data and estimate the error of the approximation
5. compute approximation value of a definite integral and estimate error of the approximation
6. describe and apply important methods in solving problems of linear least squares
7. apply derived numerical methods for solving nonlinear equations
8. use fundamental methods of unconstrained optimization
9. use choosen numerical software for solving problems using numerical methods
10. analyze accuracy of obtained results and visualization of numerical solutions

Lectures are held in two cycles, 3 hours per week

Exercises are held in two cycles, 1 hour per week

Students will solve online quizzes. Students can access online course materials in Moodle

During the semester students will work on program tasks. Laboratory exercises are created using Jupyter notebooks, Python and associated libraries.

1. Sources of Error in Computational Models. Floating point precision and error propagation. Finite differences.
2. The Gaussian Elimination Method (GEM). LU Factorization. Applications. Pivoting Strategies.
3. PLU Factorization of the matrix. Symmetric and Positive Definite Matrices. The Cholesky Factorization.
4. Stability Analysis for Linear Systems. Matrix Norms. The Condition Number of the Matrix. Improving the Accuracy of GEM. Scaling. Iterative refinement.
5. Lagrange Form of the Interpolation Polynomial. The Interpolation Error. Divided Differences. Newton Form of the Interpolation Polynomial.
6. Approximation by splines. Linear and cubic spline. Introduction to numerical integration.
7. Midpoint, Trapezoidal and Simpson rule. Composite Newton-Cotes Formulae. Richardson Extrapolation. Romberg integration. Applications.
8. Midterm exam
9. Linear least squares. Polynomial approximation. Overdetermined linear systems. Normal equations. Sensitivity and conditioning.
10. Matrix Factorization that Solve the Linear Least Square Problem. QR and SVD Decomposition. Applications of the SVD and QR decomposition in solving linear least squares problem
11. Root finding of Nonlinear Equation . The Bisection Method. Fixed-Point Iterations Method.
12. Newton’s Method. Secant method. Numerical solving of nonlinear system. Applications
13. Fundamentals of unconstrained optimization. Direct Search Algorithms. Gradient Methods (the steepest descent). Line Search Techniques. Trust-Region Methods. Conjugate Gradient Method.
14. Quasi-Newton Methods. Large-scale unconstrained optimization. Nonlinear Least-Squares Problems. The Gauss-Newton Method. The Levenberg-Marquardt Method.
15. Final exam