M4180 Numerical Methods I

Faculty of Science
Spring 2023
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Taught in person.
Teacher(s)
Mgr. Jiří Zelinka, Dr. (lecturer)
doc. Mgr. Jan Koláček, Ph.D. (lecturer)
RNDr. Bc. Iveta Selingerová, Ph.D. (seminar tutor)
Mgr. Jakub Záthurecký, Ph.D. (seminar tutor)
doc. PaedDr. RNDr. Stanislav Katina, Ph.D. (assistant)
Guaranteed by
doc. Mgr. Jan Koláček, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Thu 8:00–9:50 A,01026
  • Timetable of Seminar Groups:
M4180/01: Wed 16:00–16:50 M3,01023, Wed 17:00–17:50 MP1,01014, I. Selingerová
M4180/02: Tue 16:00–16:50 M3,01023, Tue 17:00–17:50 MP1,01014, J. Záthurecký
M4180/03: Tue 12:00–12:50 M3,01023, Tue 13:00–13:50 MP1,01014, J. Záthurecký
Prerequisites
!( ROCNIK ( 1 ) && PROGRAM ( B - MAT ))
Differential calculus of functions of one and more variables and integral calculus of functions of one variable. Basic knoledge of linear algebra-theory of matrices and solving systems of linear equations. Basics of programming.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
there are 7 fields of study the course is directly associated with, display
Course objectives
This course together with the course Numerical Methods II provides comprehensive introduction to the foundations of numerical mathematics as a separate discipline. The emphasis is given to the algorithmization and computer implementation. Some examples with graphical outputs help to explain even some difficult parts. At the end of course students should be able to apply numerical methods for solving practical problems and use these methods in other disciplines e.g. in statistical methods.
Learning outcomes
Student will be able to:
- to solve numerical nonlinear equations and to decide which method will be most suitable for the problem,
- use dircet and iterative methods to find solutions for linear and nonlinear systems,
- interpolate data using interpolation polynomial or spline,
- approximate the data using the least squares method,
- find a numerical approximation of the derivative and the integral,
- find numerically the minimum of the function.
Syllabus
  • Error analysis
  • Solving nonlinear equations - principle of iterative methods, order of convergence, Newton's method, method of secants, regula falsi method, solving systems of nonlinear equations, Seidel's method, Newton's method
  • Direct methods of solving the system of linear equations - Gaussian elimination method, LU decomposition, selection of pivot
  • Iterative methods for solving a system of linear equations - principle of iterative methods, convergence theorems, Jacobi iteration method, Gauss-Seidel method, SOR method
  • Polynomial interpolation - existence and uniqueness of the interpolation polynomial, Lagrange's interpolation polynomial, Newton's interpolation polynomial
  • Spline interpolation - linear splines, cubic splines
  • Polynomial approximation - Bernstein polynomials, Bézier curves
  • Least squares method
  • Numerical derivation - construction of formulas, use for numerical solution of differential equations
  • Numerical integration - construction of quadrature formulas, Newton-Cottes formula
  • Numerical optimization - simple division method, bisection, golden ratio method, Newton's method
Literature
    recommended literature
  • HOROVA, Ivana and Jiří ZELINKA. Numerické metody (Numerical Methods). 2nd ed. Brno: Masarykova univerzita v Brně, 2004, 294 pp. 3871/Př-2/04-17/31. ISBN 80-210-3317-7. info
  • MATHEWS, John H. and Kurtis D. FINK. Numerical methods using MATLAB. 4th ed. Upper Saddle River, N.J.: Pearson, 2004, ix, 680. ISBN 0130652482. info
    not specified
  • DATTA, Biswa Nath. Numerical linear algebra and applications. Pacific Grove: Brooks/Cole publishing company, 1994, xxii, 680. ISBN 0-534-17466-3. info
  • STOER, J. and R. BULIRSCH. Introduction to numerical analysis. 1st ed. New York - Heidelberg - Berlin: Springer-Verlag, 1980, 609 pp. IX. ISBN 0-387-90420-4. info
  • RALSTON, Anthony. Základy numerické matematiky. Translated by Milan Práger - Emil Vitásek. České vyd. 2. Praha: Academia, 1978, 635 s. info
Teaching methods
Lecture: 2 hours weeky, theoretical preparation. Class excercise: 2 hours weekly, Theoretical exercise (1 hour)is focused on solving of problems by methods presented in the lecture, practical exercise (1 hour) in a computer room is aimed at algoritmization and programming of presented numerical methods.
Assessment methods
Attendance of class exercises is compulsory, successful tests results or elaboration the assigned tasks is required for a credit.
Exam is written.
Grading according to the achieved results:
A: 20-22 points
B: 18-19 points
C: 16-17 points
D: 14-15 points
E: 12-13 points
F: less than 12 points
Language of instruction
Czech
Follow-Up Courses
Further Comments
Study Materials
The course can also be completed outside the examination period.
The course is taught annually.
Listed among pre-requisites of other courses
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2003, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2023, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2023/M4180