Numerical Methods (FSI-2NU)

Academic year 2026/2027
Supervisor: doc. Ing. Petr Tomášek, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Course type: departmental course
Aims of the course unit:

The aim of the course is to familiarize students with essential methods applied for solving numerical problems and provide them with an ability to solve such problems individually by hand and especially on computer. Students ought to realize that only the knowledge of substantial features of particular numerical methods enables them to choose a suitable method and an appropriate software product.
Students will be made familiar with a basic collection of numerical methods. They will make sense of errors in mathematical modelling, learn to find zeros of nonlinear equation and to solve systems of linear equations. They will master the basics of approximation including the least squares method, manage to use quadrature formulas and obtain an initial insight into the unconstrained minimization.
Learning outcomes and competences:
 
Prerequisites:

Numerical linear algebra, approximation of functions, numerical differentiation and integration, differential and integral calculus, basic programming skills.
Course contents:

Students will be made familiar with a basic collection of numerical methods. They will make sense of errors in mathematical modelling, learn to find zeros of nonlinear equation and to solve systems of linear equations. They will master the basics of approximation including the least squares method, manage to use quadrature formulas and obtain an initial insight into the unconstrained minimization.
Teaching methods and criteria:
 
Assesment methods and criteria linked to learning outcomes:

COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in seminars. Students have to pass two check tests successfully. A student can earn up to 20 points for both tests . A necessary condition for course credit acquirement is a gain of at least 10 points.
FORM OF THE EXAMINATIONS: The exam has a practical and a theoretical part. In the practical part students solve several numerical examples by hand using a scientific calculator. In the theoretical part they answer several questions to basic notions in order to check up how they understand the subject. The classification evaluation mainly includes the result of the written exam, while the results of credit tests and any discussion that follows the exam may also be taken into account.

Students will obtain 0--100 points as a result of the exam.
FINAL COURSE CLASSIFICATION: A (excellent): 100--90, B (very good): 89--80, C (good): 79--70, D (satisfactory): 69--60, E (sufficient): 59--50, F (failed): 49--0.


Attendance at seminars is checked. Lessons are planned according to the week schedules. Absence may be replaced by the agreement with the teacher.
Controlled participation in lessons:
 
Type of course unit:
    Lecture  13 × 1 hrs. optionally                  
    Computer-assisted exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture

Two-hour lessons
1. Introduction to computing: Error analysis. Computer arithmetic. Conditioning of problems, stability of algorithms.
Solving linear systems: Gaussian elimination. LU decomposition. Pivoting.
2. Solving linear systems: Effect of roundoff errors. Conditioning. Iterative methods (Jacobi, Gauss-Seidel, SOR method).
3. Interpolation: Lagrange, Newton and Hermite interpolation olynomial. 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 nonlinear equations in one dimension: bisection method, Newton's method, secant method, false position method, inverse quadratic interpolation, fixed point iteration.
Solving nonlinear systems: Newton's method, fixed point iteration.
6. Minimization of a function of one variable: golden ratio, quadratic interpolation.
Minimization methods for multivariable functions: Nelder-Mead method, steepest descent and Newton's method.
7. Minimization methods for multivariable functions: steepest descent method.
    Computer-assisted exercise

The seminars take place in a computer lab. The exercise program corresponds to the lecture topics.
Literature - fundamental:
1. HEATH, Michael T. Scientific computing: an introduction survey. 2nd ed. Boston: McGraw-Hill, 2002, 563 s. ISBN 0-07-239910-4.
2. MOLER, Cleve B. Numerical computing with MATLAB. Philadelphia: SIAM, 2004, xi, 336 s. : il. ISBN 0-89871-560-1.
3. DAHLQUIST, Germund a Ake BJÖRCK. Numerical Methods. New Jersey: Englewood Cliffs, 1974, 573 s.
4. MATHEWS, John H. a Kurtis D. FINK. Numerical methods using MATLAB. 4th ed. Upper Saddle River: Pearson Prentice Hall, 2004, ix, 680 s. : il. ISBN 0-13-191178-3.
Literature - recommended:
1. ČERMÁK, Libor a Rudolf HLAVIČKA. Numerické metody. Brno: Akademické nakladatelství CERM, 2016.
2. MOLER, Cleve B. Numerical computing with MATLAB. Philadelphia: SIAM, 2004, xi, 336 s. : il. ISBN 0-89871-560-1.
3. MATHEWS, John H. a Kurtis D. FINK. Numerical methods using MATLAB. 4th ed. Upper Saddle River: Pearson Prentice Hall, 2004, ix, 680 s. : il. ISBN 0-13-191178-3.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-ZSI-P full-time study STI Fundamentals of Mechanical Engineering -- Cr,Ex 4 Compulsory 1 1 S
B-FIN-P full-time study --- no specialisation -- Cr,Ex 4 Compulsory 1 1 S
B-MET-P full-time study --- no specialisation -- Cr,Ex 4 Compulsory 1 2 S
B-KSI-P full-time study --- no specialisation -- Cr,Ex 4 Compulsory-optional 1 3 S
C-AKR-P full-time study CLS -- Cr,Ex 4 Elective 1 1 S