CONTENTS
Error Theory: Errors, floating point arithmetic, error Transmission
• Calculation Series Math Functions: Calculation Series, cutoff error, correction
• Numerical Solution of Equations: Isolation roots of nonlinear equations, Root Finding, Horner Method, Methods for solving nonlinear equations (Convergence, convergence speed) Partition Method , Regula-Falsi Method, Method of iterations, Newton-Raphson Method , Method of Chords.
• Solving Systems of Linear Equations: Direct Methods (Diagonal, Upper-Lower triangular Systems, Gauss Method, Iterative Methods (Method Gauss-Seidel, Jacobi)
• Ascending Differences: Forward, backward, central differences, transmission errors.
• Linear Interpolation: Types of Newton-Gregory interpolation, Lagrange interpolation types, the interference correction formulas.
• Numerical Integration: A method of trapezoids, Method Newton-Cotes, Simpson method, Method of Gauss
Description and Structure
The main purpose of the course is that students get the basic knowledge of Numerical Analysis using the programming language C. The main objectives of the course are: a) The introduction of the theory of errors by presenting the definitions of rounding errors and truncation, the errors of the actual conversion decimal numbers to floating point numbers (floating point) to the computer and the transmission of these errors in transactions between floating-point numbers, b) the approximate calculation of mathematical series and simulation of mathematical functions that exist in mathematical libraries of programming languages, c ) the presentation and study of approximate methods of finding the roots of nonlinear equations and polynomials and the creation of the corresponding algorithms for their implementation in the Computer, d) the presentation and study of direct and approximate methods for solving linear systems of equations, e) study and presentation polynomial interpolation methods using a table of unknown function values f) the presentation of iterative methods of finding definite integrals and the development of respective algorithms for the implementation of illustrative examples of the above methods and planned to a Computer. At the end of the semester, students should be able, by applying numerical methods, to solve math problems and write the program for every algorithm using the C language.
Evaluation
Written final exam (60%) in the theoretical part of the course that includes:
– Questions concerning the definition of storage errors and their transmission, the description of the methods of solving nonlinear equations, methods Solving Systems of Linear Equations, the polynomial interpolation finder methods, numerical integration methods and simulation algorithms of elementary mathematics of mathematical functions libraries.
– Questions related to the convergence of the methods of solving nonlinear equations and Systems of Linear Equations.
– Exercises concerning the calculation and transmission of fault storage real numbers in the PC, solving nonlinear equations and Systems of Linear Equations, finding polynomial interpolation and numerical integration.
II. Optional activities (Projects) relative to other methods known in the literature, which can not be covered within the course and be presented in the last lecture of the course, with 20% participation in the final grade of the theoretical part of the course.
III. Practice in the laboratory (40%)
– Exercises for practice on a weekly basis
– Final exam in the laboratory part of the course with exercises
Presentation of some methods not covered in the above topics, implementation by using a Programming Language such as DevC++, Oral Examination on this Project.
Objectives
Upon successful completion of the course the student will be able to:
• Understand how errors affect storage, calculations and operations between real numbers in the Computer.
• Apply Mac Laurin series for the simulation of mathematical functions that exist in mathematical libraries of programming languages and understand the truncation errors that arise.
• Implement the methods of finding roots of equations and polynomials and distinguish the advantages of each method in terms of speed and approach solutions.
• Implement the methods of solving linear systems and distinguish the advantages of each method in terms of speed and the computational cost of the instruments required for the approximation of solutions.
• Apply the interpolation methods and estimate transmission errors to difference arrays.
• Implement the methods of Numerical integration and distinguish the advantages of each method in terms of speed and approach solutions.
• Atkinson Kendall E., “An Introduction to Numerical Analysis”, New York: Willey, 1989.
• Dahlquist Germund –Bjorck Ake, “Numerical Methods”, Translated by Ned Anderson, NJ: Prentice Hall, 1982.
• Demidovitch B.P. – Maron I. A., “Computational Mathematics”, Translated by G. Yankofski: Mir Publishers, Moskcow, 1976.
• Fausett Laurene V., “Applied Numerical Analysis Using Matlab”, Upper Saddle River, NJ: Prentice Hall, 1999.
• Gill P., Murray W., Wright M., “Numerical Linear Algebra and Optimization, Volume I”, USA: Addison-Wesley, 1991.
• Kress Rainer, “Numerical Analysis”, New York, Hong Kong: Springer, 1988.
• Schwartz H.R., “Numerical Analysis : a Comprehensive Introduction”, New York: Willey, 1989.
