Numerical methods

Introducing students to the basic concepts of queue theory and basic methods used in numerical calculations, as well as introducing students to some implementations of these numerical methods in Matlab.

Upon successful completion of the course, students are able to: - determine the convergence (divergence) of numerical and functional series, approximation methods using power series - calculate solutions to linear and nonlinear equations, interpolation problems and ordinary differential equations, in the general case and using Matlab - calculate approximations of integral and derivative values, in the general case and using Matlab - monitor the accuracy of calculations.

Sequences. Numerical sequences. The concept of convergence, divergence. Harmonic sequence. Sequences with positive terms. Dalambert and Cuachy convergence criterion. Alternative sequences. Leibnitz convergence criterion. Absolutely convergent sequences. Semiconvergent sequences. Riemann-Dini theorem. Functional sequences. Uniform convergence. Weierstrass theorem. Properties of uniformly convergent sequences. Potential sequences. Radius of convergence. Expansion of a function into a potential sequence. Taylor and Maclaurin sequence. Trigonometric sequences. Absolute and relative error. Representation of numbers in a computer. Floating-point numbers. Significant digits. IEEE-754-2008. Classes single and double in Matlab. Machine accuracy. Arithmetic operations with approximate values. Calculation of functions with approximate values ​​of arguments. Stability of computation. Weakly conditioned computations. Norms of vectors and matrices. Systems of linear equations. Gaussian elimination. LU factorization. Solving linear systems of equations in Matlab. Iterative methods. Jacobi and Gauss-Seidel methods. Analysis of solution stability and matrix condition factor. Interpolation of functions. Lagrange interpolation. Newton interpolation. Interpolation error and Lebesgue function. Numerical differentiation. Interpolation and numerical differentiation in Matlab. One-sided and two-sided methods. Numerical differentiation error. Nonlinear equations and systems of equations. Newton's method. Newton-Kantorovich method. Solving nonlinear equations in Matlab. Convergence analysis and order of methods. Numerical integration. Newton-Cotes formulas. Error estimation. Numerical integration in Matlab. Solving ordinary differential equations. Cauchy's problem. Euler's method. Explicit and implicit methods (Adams-Bashforth, Adams-Moulton). Runge-Kutta methods. Solving ordinary differential equations in Matlab.

Series. Order convergence criterion. Dalamber and Cuachy convergence criterion. Alternative series. Leibnitz convergence criterion. Absolutely convergent series. Semiconvergent series. Functional series. Uniform convergence. Weierstrass theorem. Properties of uniformly convergent series. Potential series. Radius of convergence. Expansion of a function into a potential series. Trigonometric series. Absolute and relative error. Representation of numbers in a computer. Floating-point numbers. Significant figures. IEEE-754-2008 and the num2hex function. Classes single and double in Matlab. Machine accuracy and the eps function. Loss of significant figures during calculations. Calculation of functions with approximate values ​​of arguments. Stability of calculations. Weakly conditioned systems. Norms of vectors and matrices. Systems of linear equations. Implementation of Gaussian elimination and LU factorization. The linsolve function. Matrix inversion and operators \ and /. Selection of the principal element. Conditionality of a system of linear equations. Factor conditionality of a matrix. Iterative methods. Implementation of the Jacobi and Gauss-Seidel methods. Convergence analysis. Interpolation. Implementation of various interpolation methods in Matlab and the interp1 function. Interpolation error and the Lebesgue function. Numerical differentiation. Implementation of numerical differentiation in Matlab and the diff function. Methods of one- and two-sided differentiation. Numerical differentiation error. Nonlinear equations and systems of equations. Implementation of Newton's method. Implementation of Newton-Kantorovich's method. Convergence analysis and the order of the iterative method. Numerical integration and the integral function. Trapezoidal formula and the trapz function. Numerical integration error. Solving ordinary differential equations. Implementation of Euler and linear multistep methods and ode113 functions. Runge-Kutta methods and ode45 functions.

There are no conditions.

Software: Matlab.

Total assigned hours: 75

New material: 20
Elaboration and examples (recapitulation): 10

Auditory exercises: 15
Laboratory exercises: 15
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 5
Review and grading of seminar papers: 0
Review and grading of the project: 0
Test: 5
Test: 0
Final exam: 5

Activity during lectures: 10
Test/test: 30
Laboratory practice: 30
Calculation tasks: 0
Seminar paper: 0
Project: 0
Final exam: 30
Requirement for taking the exam (required number of points): 21

A. Cvetković, M. Spalević: Numerical methods, FME, 2013. ISBN: 987-86-7083-786-7. (in Serbian); M. Spalević, A. Cvetković, I. Aranđelović, A. Pejčev, D. Đukić, J. Tomanović: Multiple, Line and Surface Integrals with Applications, Theory of Series, FME, 2015. ISBN: 978-86-7083-885-7. (in Serbian)