M7PNM1 Advanced numerical methods I

Faculty of Science
Autumn 2024
Extent and Intensity
2/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
doc. Mgr. Jan Koláček, Ph.D. (lecturer)
Mgr. Jiří Zelinka, Dr. (lecturer)
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
Fri 8:00–9:50 M5,01013
  • Timetable of Seminar Groups:
M7PNM1/01: Wed 8:00–9:50 MP2,01014a, J. Zelinka
M7PNM1/02: Wed 12:00–13:50 MP1,01014, J. Zelinka
Prerequisites
Basic of calculus and linera algebra, basic numerical methods
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
Course objectives
This course follows up on the basic numerical methods, which are transmitted in courses Numerical methods I and II. Its aim is to acquaint students with the main numerical methods of linear algebra and usage of matrices in numerical solving of ordinary and partial differential equations. Emphasis is placed on methods that are used in other lectures, especially statistics. After completing the course, students should be able not only to efficiently use existing methods using existing software, but also create their own implementations of the algorithms.
Learning outcomes
Student will be able to:
- to find and apply basic matrix decompositions
- to use a suitable numerical method to find the matrix's own numbers
- to use a suitable numerical method to solve some kinds of differential equations
- to use an advanced iterative method to find the solution of the system of linear equation
Syllabus
  • Introduction (repetition of some terms, block operations with matrices, permutation matrices, ...).
  • Least Squares Method (classic approach and the approach of using Moore-Penrose pseudoinverse), non-linear least squares.
  • Matrix decomposition and their use (LU decomposition, Cholesky decomposition, singular value decomposition, QR decomposition).
  • Advanced methods for solving a system of nonlinear equations.
  • Usage of matrices in numerical solving of differential equations
  • Eigenvalues and eigenvectors.
  • Other methods (root of positive semi-definite matrix, matrix functions etc.).
  • Sparse matrices
Literature
    recommended literature
  • Speciální matice a jejich použití v numerické matematice (Orig.) : Special matrices and their applications in numerical mathematics [Fiedler, 1984]. 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
  • GOLUB, Gene H. and Charles F. VAN LOAN. Matrix computations. 3rd ed. Baltimore, Md.: Johns Hopkins University Press, 1996, xxvii, 694. ISBN 0801854148. info
  • RALSTON, Anthony. Základy numerické matematiky. Translated by Milan Práger - Emil Vitásek. České vyd. 2. Praha: Academia, 1978, 635 s. info
Teaching methods
Lectures 2h.
Exercises 2h.
Assessment methods
Oral exam
Language of instruction
Czech
Follow-Up Courses
Further Comments
Study Materials
The course is taught annually.
Listed among pre-requisites of other courses
The course is also listed under the following terms Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2024/M7PNM1