Course detail

Modern Numerical Methods

FEKT-MPC-MNMAcad. year: 2026/2027

The course deals with some numerical methods that are used to find the numerical solution of the problem that we can not or are not able to solve analytically. All methods are correctly implemented and in most cases proved. Therefore, the first we focus on the theory of errors introduced in terms of metrics and standards and their relationships. Furthermore, we discuss proceeds with Banach fixed point theorem, which is the basis of a number of numerical methods. Explanation of its action is carried out on systems of linear algebraic equations. The interpretation starts from the finite methods and iterative solution methods. Similarly, we discuss the solution of nonlinear equations, algebraic equations and their systems. We also deal with eigenvalues of the matrix and with the search for solutions to the initial and boundary value problems for ordinary differential equations and their systems and also for partial differential equations. For each numerical methods are included that guarantee convergence of the method.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Michal Novák, Ph.D.

Department

Department of Mathematics (UMAT)

Entry knowledge

We require knowledge at the level of bachelor's degree, i.e. that students must be able to work with matrices and vectors, handle the calculation of determinants, calculate the product of a matrix and inverse matrix, know the graphs of elementary functions and methods of construction, differentiate and integrate of basic functions, solve basic types of ordinary differential equations of the first order.

Rules for evaluation and completion of the course

Students may be awarded
Up to 40 points for computer exercises for a written test (10 points) and 30 points for individual homework (max. 15 points for the program and the maximum 15 points for presentation and protocol).
Up to 60 points for the written final exam. The test contains both theoretical and numerical tasks that are used to verify the orientation in the problems of numerical methods and their application. This includes tasks such as "adjust to the shape of convergence", without interpolating the end.

Computer exercises are compulsory. Properly excused absence can be replaced by individual homework, which focuses on the issues discussed during the missed exercise.
Specifications of the controlled activities and ways of implementation are provided in annual public notice.
Date of the written test is announced in agreement with the students at least one week in advance. The new term for properly excused students is usually during the credit week.

Exercises are compulsory. Duly excused absences may be made up by arrangement with the teacher. The form and method of substitution is at the sole discretion of the techer. 

Any student who obtains a non-zero number of points for the exercise is admitted to the examination.

To pass the course, a student must score a minimum of 50 points on both the exam and the practicum.

 

Aims

The aim of the course is to extend and intesify students' basic knowledge of numerical methods and their applications to solve concrete problems. Therefore, much attention is paid to the derivation of certain procedures and methods, and examples using both numerical methods and also clarify the limitations and circumscribed use of the methods.

After completing the course the student will be able to:
• Work with various matrix and vector norms and make their estimates.
• Solve systems of linear algebraic equations. Decide whether it is possible to solve the system using a given method.
• Find roots of nonlinear and algebraic equations with required accuracy.
• Solve systems of equations.
• Determine the dominant eigenvalue of a matrix.
• Find all eigenvalues. To the suitability of the specified procedure for finding eigenvalues.
• Find the numerical solution of initial value problems for ordinary differential equations and their systems with required accuracy.
• Find the numerical solution of partial differential equations. Work with boundary and internal points system.
• Explain the nature of the finite element method and know how to use it to solve problems on a computer.
• Select the appropriate method for a given type of task and estimate the rate of convergence of certain methods.
• Determine accuracy estimates for certain methods.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

BAŠTINEC, J.; NOVÁK, M. Moderní numerické metody: sbírka příkladů. Brno: FEKT, VUT v Brně, 2011. (CS)
BAŠTINEC, J., Novák, M.,: Moderní numerické metody.Brno, FEKT VUT, 2014. (CS)

Recommended reading

VITÁSEK, E., Numerické metody. SNTL Praha 1987. (CS)

Classification of course in study plans

  • Programme MPC-TIT Master's 1 year of study, summer semester, compulsory-optional, fundamental theoretical courses of the profile core
  • Programme MPC-SVE Master's 1 year of study, summer semester, compulsory-optional, fundamental theoretical courses of the profile core
  • Programme MPC-NCP Master's 1 year of study, summer semester, compulsory-optional
  • Programme MPC-MEL Master's 1 year of study, summer semester, compulsory-optional, fundamental theoretical courses of the profile core
  • Programme MPC-EKT Master's 1 year of study, summer semester, compulsory-optional, fundamental theoretical courses of the profile core
  • Programme MPC-BTB Master's 1 year of study, summer semester, compulsory-optional
  • Programme MPC-BIO Master's 1 year of study, summer semester, compulsory-optional, fundamental theoretical courses of the profile core

  • Programme MPC-AUD Master's

    specialization AUDM-TECH , 1 year of study, summer semester, compulsory-optional
    specialization AUDM-ZVUK , 1 year of study, summer semester, compulsory-optional

  • Programme MPCN-BTB Master's 2 year of study, summer semester, compulsory-optional

  • Programme MPCN-BIO Master's

    specialization MPC-BIO_TECH , 2 year of study, summer semester, compulsory-optional
    specialization MPC-SPORT_TECH , 1 year of study, summer semester, compulsory-optional

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

    1. Princip numerických metod, norma, Banachova věta o pevném bodě
    2. Soustavy lineárních algebraických rovnic
    3. Řešení rovnic
    4. Vlastní čísla a vlastní vektory
    5. Soustavy nelineárních rovnic
    6. Řešení obyčejných diferenciálních rovnic
    7. Řešení soustav ODR
    8. Řešení okrajových úloh pro ODR
    9. Metoda konečných prvků pro ODR
    10. Parciální diferenciální rovnice, klasifikace, transormace
    11. Metoda konečných diferencí pro PDR
    12. Metoda konečných prvků pro PDR

    Computer-assisted exercise

    26 hours, compulsory

    Teacher / Lecturer

    doc. RNDr. Michal Novák, Ph.D.

    Syllabus

    • The principle of numerical methods, norm, Banach theorem on a fixed point. 
    • Systems of linear algebraic equations. 
    • Solving of equations. 
    • Eigen numbers and eigen vectors. 
    • Systems of nonlinear equations. 
    • Solutions of ordinary differential equations. 
    • Solutions of system ODE. 
    • Solution boundary value problem for ODE. 
    • Finite element method for ODE. 
    • Partial differential equations, classifications, transormations. 
    • A method of finite differentials for PDE. 
    • Finite element method for PDE.
     

    Regular individual preparation for activities in the semester

    26 hours, optionally

    Teacher / Lecturer

    doc. RNDr. Michal Novák, Ph.D.

    Syllabus

    Studenti a studentky se pravidelně připravují na cvičení, aby průběžně zvládli probíranou látku, studují výukové podklady z přednášek a ze cvičení.

    Individual preparation for a final exam

    52 hours, optionally

    Teacher / Lecturer

    doc. RNDr. Michal Novák, Ph.D.

    Syllabus

    Studium výukových podkladů z přednášek a ze cvičení.