- distinguish and identify types of errors in numerical computing
- apply direct methods in solving systems of linear equations
- identify ill conditioned system of linear equations
- compute interpolation polynomial and cubic spline for given data and estimate the error of the approximation
- compute approximation value of a definite integral and estimate error of the approximation
- describe and apply important methods in solving problems of linear least squares
- apply derived numerical methods for solving nonlinear equations
- use fundamental methods of unconstrained optimization
- use choosen numerical software for solving problems using numerical methods
- analyze accuracy of obtained results and visualization of numerical solutions
Forms of Teaching
Lectures are held in two cycles, 3 hours per weekExercises
Exercises are held in two cycles, 1 hour per weekPartial e-learning
Students will solve online quizzes. Students can access online course materials in MoodleLaboratory
During the semester students will work on program tasks
|Type||Threshold||Percent of Grade||Threshold||Percent of Grade|
|Laboratory Exercises||0 %||15 %||0 %||15 %|
|Quizzes||0 %||20 %||0 %||0 %|
|Class participation||0 %||10 %||0 %||0 %|
|Seminar/Project||0 %||10 %||0 %||0 %|
|Mid Term Exam: Written||0 %||35 %||0 %|
|Final Exam: Written||0 %||30 %|
|Exam: Written||0 %||85 %|
Students can earn an additional 10% of points if actively participate in classes or through a seminar / project.
Week by Week Schedule
- Sources of Error in Computational Models. Floating point precision and error propagation. Finite differences.
- The Gaussian Elimination Method (GEM). LU Factorization. Applications. Pivoting Strategies.
- PLU Factorization of the matrix. Symmetric and Positive Definite Matrices. The Cholesky Factorization.
- Stability Analysis for Linear Systems. Matrix Norms. The Condition Number of the Matrix. Improving the Accuracy of GEM. Scaling. Iterative refinement.
- Lagrange Form of the Interpolation Polynomial. The Interpolation Error. Divided Differences. Newton Form of the Interpolation Polynomial.
- Approximation by splines. Linear and cubic spline. Introduction to numerical integration.
- Midpoint, Trapezoidal and Simpson rule. Composite Newton-Cotes Formulae. Richardson Extrapolation. Romberg integration. Applications.
- Midterm exam
- Linear least squares. Polynomial approximation. Overdetermined linear systems. Normal equations. Sensitivity and conditioning.
- 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
- Root finding of Nonlinear Equation . The Bisection Method. Fixed-Point Iterations Method.
- Newton’s Method. Secant method. Numerical solving of nonlinear system. Applications
- Fundamentals of unconstrained optimization. Direct Search Algorithms. Gradient Methods (the steepest descent). Line Search Techniques. Trust-Region Methods. Conjugate Gradient Method.
- Quasi-Newton Methods. Large-scale unconstrained optimization. Nonlinear Least-Squares Problems. The Gauss-Newton Method. The Levenberg-Marquardt Method.
- Final exam