Course detail
Numerical Methods
FSI-2NU Acad. year: 2027/2028 Summer semester
The Numerical Methods course introduces students to a core set of problems in numerical mathematics. It highlights the pitfalls of numerical computations (errors, conditioning, algorithm stability) and covers methods for solving linear and non-linear equations, interpolation, least-squares approximation, numerical differentiation and integration, and unconstrained optimization.
Language of instruction
Czech
Number of ECTS credits
4
Supervisor
Department
Entry knowledge
Students are expected to possess foundational knowledge from standard university-level mathematics courses: differential and integral calculus, basic linear algebra (vector spaces, matrices, systems of linear equations), and basic skills in programming and algorithmization.
Rules for evaluation and completion of the course
Attendance & Absence Replacement: Attendance in computer labs is compulsory and recorded. Rules for replacing missed classes are determined by the lab instructor.
Requirements for Awarding Course Credit: Mandatory and monitored attendance in computer labs. Successful completion of two written control tests during the semester (0 to 10 points per test; max. 20 points total). A minimum total score of 10 points combined across both tests is required to earn course credit.
Exam and Grading: The exam is written and consists of a practical part (numerical calculations using a calculator) and a theoretical part (testing comprehension of the covered material). Students can score between 0 and 100 points on the exam.
The final evaluation primarily reflects the result of the written exam, with consideration also given to test results during the semester and optional oral discussion.
Grading Scale: 100–90 points: A (excellent), 89–80: B (very good), 79–70: C (good), 69–60: D (satisfactory), 59–50: E (sufficient), 49–0: F (failed).
Aims
The aim of the Numerical Methods course is to introduce students to basic techniques for solving selected numerical problems and to equip them with the ability to solve these problems independently, both manually and using computers. Students will understand the essential properties of individual numerical methods, enabling them to make an effective choice of suitable methods and corresponding software products.
Graduates will gain a comprehensive overview of fundamental numerical mathematics problems and understand the pitfalls of numerical computing (rounding errors, conditioning, numerical stability). They will be able to practically solve systems of linear and non-linear equations, perform interpolation, apply the least-squares approximation, conduct numerical differentiation and integration, and solve unconstrained optimization problems.
The study programmes with the given course
Programme B-FIN-P: Physical Engineering and Nanotechnology, Bachelor's, compulsory
Programme B-MET-P: Mechatronics, Bachelor's, compulsory
Programme B-KSI-P: Mechanical Engineering Design, Bachelor's, compulsory-optional
Programme B-ZSI-P: Fundamentals of Mechanical Engineering, Bachelor's
specialization STI: Fundamentals of Mechanical Engineering, compulsory
Type of course unit
Lecture
13 hours, optionally
Syllabus
Lecture Syllabus (13 hours of direct instruction in total):
Lectures are scheduled in blocks throughout the semester according to the faculty timetable (total teaching load of 13 hours).
1. Introduction to numerical methods: Error analysis. Problem conditioning, stability of algorithms. Solving linear systems: Gaussian eliminatio, LU decomposition, pivoting.
2. Solving linear systems: Effect of roundoff errors, conditioning. Iterative methods (Jacobi, Gauss-Seidel, SOR method).
Interpolation: Lagrange, Newton and Hermite interpolation polynomial.
3. Piecewise linear and piecewise cubic Hermite interpolation. Cubic interpolating spline. Least squares method.
4. Numerical differentiation: Basic formulas. Richardson extrapolation.
Numerical integration: Basic quadrature rules (midpoint, trapezoidal and Simpson's rule). Gaussian quadrature. Composite quadrature. Adaptive quadrature.
5. Solving single nonlinear equations: bisection, Newton's method, secant method, false position method, inverse quadratic interpolation, fixed point iteration.
Solving nonlinear systems: Newton's method, fixed point iteration.
6. Unconstrained optimization of single and multi-variable functions: golden section, quadratic interpolation, Nelder-Mead, steepest descent and Newton's method.
Computer-assisted exercise
26 hours, compulsory
Syllabus
The practical content directly follows the lecture topics, focusing on algorithm implementation and manual calculations.