Numerical Methods I (FSI-SN1)

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:
 
Learning outcomes and competences:
 
Prerequisites:

Differential and integral calculus for functions of one and more variables. Fundamentals of linear algebra. Basic programming skills.
Course contents:
 
Teaching methods and criteria:
 
Assesment methods and criteria linked to learning outcomes:
 
Controlled participation in lessons:
 
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Computer-assisted exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture

1. Introduction to computing: error analysis, computer arithmetic, conditioning of problems, stability of algorithms.
2. Gaussian elimination method. LU decomposition. Pivoting.
3. Solution of special linear systems. Stability and conditioning. Error analysis.
4. Classical iterative methods: Jacobi, Gauss-Seidel, SOR, SSOR.
5. Generalized minimum rezidual method, conjugate gradient method.
6. Lagrange, Newton and Hermite interpolation polynomial. Piecewise linear and piecewise cubic Hermite interpolation.
7. Cubic interpolating spline. Least squares method: data fitting, solving overdetermined systems.
8. Numerical differentiation: basic formulas, Richardson extrapolation.
9. Numerical integration: Newton-Cotes formulas, Romberg's method, Gaussian formulas, adaptive integration.
10. Solving nonlinear equations in one dimension: bisection method, Newton's method, secant method, false position method, inverse quadratic interpolation, fixed point iteration.
11. Solving nonlinear systems: Newton's method, fixed point iteration.


12. QR decomposition and singular value decomposition in the least squares method.
13. Orthogonalization methods (Householder transformation, Givens rotations, Gram-Schmidt orthogonalization)
    Computer-assisted exercise

Students create elementary programs in MATLAB related to each subject-matter delivered at lectures and verify how the methods work. Furthermore students individually elaborate semester assignemets.
Literature - fundamental:
1. M.T. Heath: Scientific Computing. An Introductory Survey. Second edition. McGraw-Hill, New York, 2002.
2. C.B. Moler: Numerical Computing with Matlab, Siam, Philadelphia, 2004.
3. G. Dahlquist, A. Bjork: Numerical Methods, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1974.
4. J.H. Mathews, K.D. Fink: Numerical Methods Using MATLAB, Pearson Prentice Hall, New Jersey, 2004.
5. A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics, Springer, Berlin, 2000
Literature - recommended:
1. L. Čermák, R. Hlavička: Numerické metody, CERM, Brno, 2008.
2. L. Čermák: Vybrané statě z numerických metod. [on-line], available from: http://mathonline.fme.vutbr.cz/Numericke-metody-I/sc-1150-sr-1-a-141/default.aspx.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-MAI-P full-time study --- no specialisation -- Cr,Ex 4 Compulsory 1 3 W
C-AKR-P full-time study CZS -- Cr,Ex 4 Elective 1 1 W