SubjectsSubjects(version: 850)
Course, academic year 2019/2020
  
Fundamentals of Numerical Mathematics - NMNM201
Title in English: Základy numerické matematiky
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2017 to 2020
Semester: winter
E-Credits: 8
Hours per week, examination: winter s.:4/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: prof. RNDr. Vít Dolejší, Ph.D., DSc.
prof. Ing. Miroslav Tůma, CSc.
Class: M Bc. OM
M Bc. OM > Povinné
M Bc. OM > 2. ročník
Classification: Mathematics > Numerical Analysis
Pre-requisite : {One 1st year Analysis course}
Incompatibility : NNUM105
Interchangeability : NNUM105
In complex pre-requisite: NMNM331
Annotation -
Last update: G_M (16.05.2012)
The first course of numerical analysis for students of General Mathematics.
Aim of the course -
Last update: prof. RNDr. Vít Dolejší, Ph.D., DSc. (08.06.2015)

To give a basic knowledge in numerical mathematics.

Course completion requirements -
Last update: doc. RNDr. Václav Kučera, Ph.D. (29.10.2019)

Credit requirements:

at seminars, students will be given 6 tasks, which they solve at home. They will submit the solved task (electronically or on paper) no later than one week before the beginning of their exercise to the tutor.

They can get 0 to 6 points for each task.

To obtain the credit it is necessary to obtain at least 2/3 points, ie 24.

The 'nature of the examination of the course' excludes the repetition of that examination, POS, Article 8 (2)

Literature -
Last update: doc. RNDr. Václav Kučera, Ph.D. (30.09.2019)

  • J. Duintjer Tebbens, I. Hnětynková, M. Plešinger, Z. Strakoš, P. Tichý: Analýza metod pro maticové výpočty - Základní metody, Skriptum MFF UK, 2012

  • J. Segethová: Základy numerické matematiky, Skriptum MFF UK, 2002

  • M. Feistauer, V. Kučera: Základy numerické matematiky, Skriptum MFF UK, 2014

  • L. N. Trefethen and D. Bau, III, Numerical linear algebra, SIAM, Philadelphia, PA, 1997

  • A. Quarteroni, R. Sacco and F. Saleri: Numerical mathematics, Springer-Verlag, 2000

  • D. S. Watkins: Fundamentals of Matrix Computations, Willey Interscience, New Yourk, 2010

  • Other sources at: http://msekce.karlin.mff.cuni.cz/~dolejsi/Vyuka/ZNM.html

  • http://www.karlin.mff.cuni.cz/~felcman/nm.pdf

  • http://www.karlin.mff.cuni.cz/~mirektuma/znm/znm2019.pdf

  • Videozáznamy přednášek

Teaching methods -
Last update: G_M (27.04.2012)

Lectures and tutorials in a lecture hall.

Requirements to the exam -
Last update: prof. RNDr. Vít Dolejší, Ph.D., DSc. (06.10.2017)

Examination according to the syllabus.

Syllabus -
Last update: prof. Ing. Miroslav Tůma, CSc. (09.10.2017)

1. Introduction. What is numerical mathematics.

2. Problem types and errors (forward, backward, residual). Distinguishing factorization and eigenvalue problems.

3. Schur theorem and its consequences.

4. Orthogonality. QR factorization. Time complexity of the QR factorization and its stability.

5. LU factorization and solving systems of linear equations. Growth of errors in solving systems of linear equations.

6. Singular value decomposition. Least-squares problems.

7. Iterative methods based on splittings. Power method for eigenvalue problems. Ideas behind Krylov space methods.

8. Interpolation of functions. Lagrange and Hermite polynomials. Spline functions. Least-square approximation.

9. Quadrature formulas. Gaussian and Newton-Cotes formulas.

10. Solution of Nonlinear Equations.

11. Systems of linear difference equations, homogeneous, nohomogeneous systems, fundamental system of solutions, systems with constant coefficients.

12. Numerical solution of ordinary differential equations. a) One-step methods: Examples, general one-step methods, local discretization error, accumulated discretization error, convergence, consistency, error estimates, round-off errors, aposteriori error estimate, derivation of some formulae, Runge-Kutta methods. b) Multi-step methods, general framework, convergence, stability, consistency, order of the method, error estimates, derivation of some multi-step schemes.

13. Some optimization methods. Elements of convex analysis, steepest descent methods with constant and optimal step, convergence.

Entry requirements -
Last update: G_M (27.04.2012)

basic knowledge of calculus and linear algebra

 
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