Course detail

Modern Numerical Methods

FEKT-MPA-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

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Jaromír Baštinec, CSc.

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.

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 MPA-NCP Master's 1 year of study, summer semester, compulsory-optional

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

doc. RNDr. Jaromír Baštinec, CSc.

Syllabus

1. Principles of numerical methods, norms, Banach fixed-point theorem
2. Systems of linear algebraic equations
3. Solution of equations
4. Eigenvalues and eigenvectors
5. Systems of nonlinear equations
6. Numerical solution of ordinary differential equations
7. Numerical solution of systems of ordinary differential equations
8. Solution of boundary value problems for ordinary differential equations
9. Finite element method for ordinary differential equations
10. Partial differential equations – classification and transformations
11. Finite difference method for partial differential equations
12. Finite element method for partial differential equations 

Computer-assisted exercise

26 hours, compulsory

Teacher / Lecturer

doc. RNDr. Jaromír Baštinec, CSc.

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.

Individual preparation for a final exam

60 hours, optionally

Teacher / Lecturer

doc. RNDr. Jaromír Baštinec, CSc.

Syllabus

Independent study of materials from lectures, exercises, recommended literature, and potentially other literary sources. 

 

Regular individual preparation for activities in the semester

26 hours, optionally

Teacher / Lecturer

doc. RNDr. Jaromír Baštinec, CSc.

Syllabus

Studying lecture materials and recommended literature in order to complete the assigned task.