# PřF:M7PNM1 Advanced numerical methods I - Course Information

## M7PNM1 Advanced numerical methods I

**Faculty of Science**

Autumn 2022

**Extent and Intensity**- 2/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Taught in person. **Teacher(s)**- Mgr. Jiří Zelinka, Dr. (lecturer)
**Guaranteed by**- doc. PaedDr. RNDr. Stanislav Katina, Ph.D.

Department of Mathematics and Statistics - Departments - Faculty of Science

Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science **Timetable**- Tue 8:00–9:50 M4,01024
- Timetable of Seminar Groups:

*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**- Statistics and Data Analysis (programme PřF, N-MA)

**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. Students also obtain information abou basic numerical optimization methods. 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 a suitable numerical method to find the extreme of a function **Syllabus**- Introduction (repetition of some terms, block operations with matrices - inversion and determinant, permutation matrices, Kronecker product).
- 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.).
- Optimization methods (bisection method, golden ratio, Newton's method, Nelderova-Mead method).

**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

*recommended literature***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**

- Enrolment Statistics (recent)

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