Numerical Analysis

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Course Syllabus

Course Description

Introduction to Numerical Analysis. Numerical Calculations and Errors. Error Theory : Errors, Floating-point arithmetic, Error propagation, The LateX scientific paper writing environment, Fundamentals of Complexity Theory, Complexity classes, Complexity computation rules. The classes P and NP. Decision problems and optimization problems. Heuristic algorithms with examples and applications in manufacturing engineering and management science ,Calculation of series of infinite term mathematical functions (Taylor and MacLaurin), Truncation error, correction. Numerical Solution of Non Linear Equations : Roots of non linear equations, Methods of solving open and closed interval non linear equations (Convergence, speed of convergence ), Bisection method, False position, Intersection, Newton-Raphson, Fixed point iteration. Solving Systems of Linear Equations : Direct Methods (Diagonal Solving, Upper-Lower Triangular System, Gauss Elimination, Gauss – Jordan Method, LU factorization method), Iterative Methods (Gauss-Seidel Method, Jacobi Method, Sequential Superposition). Complexity of methods – comparison, Applications of methods for solving systems of linear equations in engineering science : solution of lattice structures, solution of electric circuits, dynamics – robustness registers. Sparse matrices and computational methods, Linear Interpolation : Newton interpolation formulas, Lagrange interpolation formulas, finite differences, complexity, comparison, applications. Numerical Integration : Method of Banks, Newton-Cotes Methods, Simpson Methods, Gauss Method. Application of numerical integration methods in engineering science. Numerical methods for solving ordinary differential equations. Euler’s method. Improved Euler Method. Runge-Kutta methods: 2nd, 3rd and 4th order. Numerical methods for solving ordinary differential equations. Finite difference method. Systems of ordinary differential equations. Applications of methods of solving ordinary Eqs in engineering science. Implementation of numerical methods in MATLAB, Octave environment.

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Purpose of the course

The aim of the course is for students to understand: i. The importance of numerical calculations and the theory involved in the errors involved, introducing the definitions of rounding and truncation errors, the errors of converting real decimal numbers to floating-point numbers on the computer, and the propagation of these errors in operations between floating-point numbers. ii. The approximate calculation of mathematical series (Taylor&MacLaurin) and the simulation of mathematical functions available in the mathematical libraries of programming languages, both through numerical calculations and by writing source code in C,C++ or MATLAB. iii. The approximate methods of finding the roots of nonlinear equations and polynomials and the creation of the corresponding algorithms for their implementation through C, C++ or MATLAB. iv. The direct and approximate methods of solving systems of linear equations and the complexity of each method. v. The methods of finding interpolation polynomials from a matrix of values of some unknown function, and their complexity. vi. The popular closed interval approximate methods for finding certain integrals, and their implementation in a C, C++ or MATLAB programming environment. vii. The fundamental methods for solving ordinary differential equations (Euler & Runge-Kutta).