M5310p Numerické metody - přednáška

Faculty of Science
Autumn 1999
Extent and Intensity
2/0/0. Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Vítězslav Veselý, CSc. (lecturer)
Guaranteed by
doc. RNDr. Vítězslav Veselý, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Vítězslav Veselý, CSc.
Prerequisites (in Czech)
M2100 Mathematical Analysis II && M2110 Linear Algebra II
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
Syllabus
  • Introduction: flowchart of numerical problem analysis, error analysis (error sources and their propagation, numerical stability), the order of approximation and convergence.
  • Iteration methods for solving nonlinear equations f(x)=0 and x=g(x): separation of roots, convergence acceleration (Aitken's extrapolation), main principles and convergence properties of the basic iteration procedures: bracketing methods (bisection, regula-falsi), fix-point method, Newton-Raphson iterations, quasinewton methods (secant method, Steffensen's method), Seidel's and Newton's iterations for nonlinear systems of equations.
  • The solution of linear systems of equations: Direct methods: upper- and lower-triangular systems, Gaussian elimination with pivoting, LU factorization, tridiagonal systems, computational complexity; Iteration methods: Jacobi and Gauss-Seidel iterations.
  • Polynomial interpolation: basic problem statement and its solution, error terms and bounds, Runge phenomenon and Chebyshev nodes, interpolation methods for general and equally spaced nodes (Lagrange and Newton interpolation formulas), divided-difference scheme.
  • Numerical differentiation: error term, developing formulas for general and equally spaced nodes, the common 2-point and 3-point formulas as special examples, the choice of optimal node distance.
  • Numerical integration (quadrature): error term and the degree of precision, classification of quadrature formulas (closed and open formulas, Newton-Cotes and gaussian formulas) general approach to the derivation of Newton-Cotes closed and open formulas for general, symmetric and equally spaced nodes, the composite Newton-Cotes formulas, the rectangular, trapezoidal and Simpson's rule as special cases.
  • Note: The computer-aided exercises are supported by the system MATLAB.
  • See http://www.math.muni.cz/~vesely/educ/nmsylle.ps for more details.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Teacher's information
http://www.math.muni.cz/~vesely/educ_cz.html#nummet
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