Course detail

Numerical Methods I

FSI-SN1Acad. year: 2024/2025

The course represents the first systematic explanation of selected basic methods of numerical mathematics. Passing this course, students obtain basic knowledge necessary for further study of more specialised areas of numerical mathematics.
Main topics: Direct and iterative methods for linear systems. Interpolation. Least squares method. Numerical differentiation and integration. Nonlinear equations. The students will demonstrate the acquainted knowledge by elaborating the semester assignment.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. Ing. Petr Tomášek, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Differential and integral calculus for functions of one and more variables. Fundamentals of linear algebra. Programming in MATLAB.

Rules for evaluation and completion of the course

COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in practicals. Elaboration of tasks, where the students prove their knowledge acquired. At least half of all possible 30 points in a credit test using also own programs.
FORM OF EXAMINATIONS: The exam is of test (max. 75 pts.) and oral part (max 25 pts.). As a result of the exam students will obtain 0--100 points.
FINAL COURSE CLASSIFICATION is based on the exam point 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 lectures is recommended, attendance at seminars is required. Lessons are planned according to the week schedules. Absence from lessons may be compensated for by the agreement with the teacher supervising the seminars.

Aims

The aim of the course is to familiarise students with some basic numerical methods. Substantial emphasis is also put on a computer implementation of individual methods. Students ought to understand the essence of particular methods and to realise their advantages and drawbacks. Attention is also paid to the issues of stability and conditionality of individual numerical problems. An important part of the course is independent work on assigned projects.
Students will be made familiar with the basic collection of numerical methods, namely with direct and iterative methods for systems of linear equations, with interpolation, least squares, numerical derivation and integration and with methods for nonlinear equations. Students will verify and deepen the acquired knowledge by processing several projects.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics, Springer, Berlin, 2000
C.B. Moler: Numerical Computing with Matlab, Siam, Philadelphia, 2004.
G. Dahlquist, A. Bjork: Numerical Methods, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1974.
J.H. Mathews, K.D. Fink: Numerical Methods Using MATLAB, Pearson Prentice Hall, New Jersey, 2004.
M.T. Heath: Scientific Computing. An Introductory Survey. Second edition. McGraw-Hill, New York, 2002.

Recommended reading

L. Čermák, R. Hlavička: Numerické metody, CERM, Brno, 2008.
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.

Elearning

eLearning: currently opened course

Classification of course in study plans

  • Programme B-MAI-P Bachelor's 3 year of study, winter semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CZS , 1 year of study, winter semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. Ing. Petr Tomášek, Ph.D.

Syllabus

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. QR decomposition and singular value decomposition in the least squares method.
9. Orthogonalization methods (Householder transformation, Givens rotations, Gram-Schmidt orthogonalization)
10. Numerical differentiation: basic formulas, Richardson extrapolation.
11. Numerical integration: Newton-Cotes formulas, Romberg's method, Gaussian formulas, adaptive integration.
12. Solving nonlinear equations in one dimension: bisection method, Newton's method, secant method, false position method, inverse quadratic interpolation, fixed point iteration.
13. Solving nonlinear systems: Newton's method, fixed point iteration.

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

Mgr. Jitka Zatočilová, Ph.D.

Syllabus

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.

Elearning

eLearning: currently opened course