M5310 Numerical methods

Faculty of Science
Autumn 2001
Extent and Intensity
2/1/0. 4 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Vítězslav Veselý, CSc. (lecturer)
Mgr. Jitka Dluhá (seminar tutor), doc. RNDr. Vítězslav Veselý, CSc. (deputy)
Guaranteed by
prof. RNDr. Ivanka Horová, 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
Course objectives
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 can also be completed outside the examination period.
The course is taught annually.
The course is taught every week.
Teacher's information
http://www.math.muni.cz/~vesely/educ_cz.html#nummet
The course is also listed under the following terms Autumn 1999, Autumn 2000.
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