Numerical Methods

• Semester 5, position 2

Acquaintance of students with the basic concepts of series theory and basic methods used in numerical calculations, as well as acquaint students with some implementations of these numerical methods in Matlab.

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

Series. Numerical series. Concept of convergence, divergence. Harmonic series. Series with positive terms. Dalambert's and Cuachy's convergence criterion. Alternate series. Leibnitz convergence criterion. Absolutely convergent series. Semiconvergent series. The Riemann-Dini theorem. Functional series. Uniform convergence. Weierstrass theorem. Properties of uniformly convergent series. Potential series. Convergence radius. Development of a function in a potential series. Taylor's and Maclaurin's series. Trigonometric series. Absolute and relative error. Representation of numbers in a computer. Floating point numbers. Significant digits. IEEE-754-2008. Single and double classes in Matlab. Machine accuracy. Arithmetic operations with approximate values. Calculation of functions with approximate values of arguments. Calculation stability. 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 method. Analysis of solution stability and matrix conditioning factor. Interpolation of functions. Lagrange interpolation. Newton's interpolation. Interpolation error and the 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 estimating. 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. Comparative convergence criterion. Dalamber's and Cuachy's convergence criterion. Alternate series. Leibnitz convergence criterion. Absolutely convergent series. Semiconvergent series. Functional series. Uniform convergence. Weierstrass theorem. Properties of uniformly convergent series. Potential series. Convergence radius. Development of a function in a potential series. Trigonometric series. Absolute and relative error. Representation of numbers in a computer. Floating point numbers. Significant digits. IEEE-754-2008 and the num2hex function. Single and double classes in Matlab. Machine precision and function eps. Loss of significant digits when calculating. Calculation of functions with approximate values of arguments. Calculation stability. Weakly conditioned systems. Norms of vectors and matrices. Systems of linear equations. Implementation of Gaussian elimination and LU factorization. The linsolve function. Inversion of matrices and operators \ and /. Selection of the main element. Conditionality of systems of linear equations. The matrix conditioning factor. Iterative methods. Implementation of the Jacobi and Gauss-Seidel methods. Convergence analysis. Interpolation. Implementation of various interpolation methods in Matlab and the interp1 functions. Interpolation error and the Lebesgue function. Numerical differentiation. Implementation of numerical differentiation in Matlab and the function diff. Methods of one-sided and two-sided differentiation. Numerical differentiation error. Nonlinear equations and systems of equations. Implementation of Newton's method. Implementation of the Newton-Kantorovich method. Convergence analysis and the order of the iterative method. Numerical integration and the function integral. Trapezoidal formula and the trapz function. Numerical integration error. Solving ordinary differential equations. Implementation of Euler and linear multistep methods and the functions ode113. Runge-Kutta methods and the function ode45.

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.S. Cvetković, M.M. Spalević, Numerical methods, Faculty of Mechanical Engineering, University of Belgrade, 2013. (Serbian). ISBN: 987-86-7083-786-7.; M. Spalević, A. Cvetković, I. Aranđelović, A. Pejčev, D. Đukić, J. Tomanović, Multiple curvilinear and sur. int. and appl., series theory, F. of M. Eng., 2015. (Serbian). ISBN: 978-86-7083-885-7.