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Course, academic year 2017/2018
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Fundamentals of Numerical Mathematics - NMNM201
English title: 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
Is pre-requisite for: 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 - Czech
Last update: prof. RNDr. Vít Dolejší, Ph.D., DSc. (06.10.2017)

Požadavky k zápočtu:

  • na cvičeních studenti dostanou postupně 7-8 jednoduchých úloh, které řeší doma
  • nejpozději další týden vyřešenou úlohu odevzdají (elektronicky či na papíře) cvičícímu
  • za každou úlohu mohou získat 0, 1, 2 nebo 3 body
  • k udělení zápočtu je třeba získat alespoň 14 bodů

„povaha kontroly studia předmětu“ vylučuje opakování této kontroly, POS, čl. 8, odst. 2

Literature -
Last update: G_M (27.04.2012)

Stoer J., Bullirsch R.: Introduction to Numerical Analysis, Springer, l978

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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