Computer numerical errors, Floating point arithmetic.
Linear Systems: Direct methods (Gauss elimination, LU factorization methods). Iterative methods (Jacobi, Gauss-Seidel, SOR methods). Eigenvalue calculation (power method).
Solving Nonlinear Equations: Bisection methods, general iterative method, Newton-Raphson, secant. Newton's method for nonlinear systems.
Interpolation and Approximation: Polynomial interpolation, Lagrange and Newton forms of the interpolation polynomial. Hermite interpolation. Interpolation with spline functions.
Least squares method.
Numerical Integration: Newton-Cotes integration formulas, simple and complex trapezoidal and Simpson formulas. Gauss integration.
Differential Equations: Initial value problems for ordinary differential equations. Single-step methods (Euler, Taylor, Runge-Kutta), Multi-step methods (Adams, Predict-Correct methods). Two-point boundary value problems, Finite difference methods.
The aim of the course is the derivation and analysis of numerical methods for solving problems in science and technology for which either no analytical solution exists or it is very difficult to calculate.
- Teacher: Βασίλειος Κοκκίνης
ECTS : 6
Language : el
Learning Outcomes : Upon successful completion of the course, students will
• Have understood the basic methods of Numerical Analysis a) for solving linear systems, nonlinear equations and differential equations b) for interpolation and approximation of data and c) for the approximate calculation of integrals.
• Be able to distinguish the differences between numerical methods and choose the most appropriate one for solving different problems
• Be able to analyse a) the asymptotic properties and behaviour of approximate models b) the numerical stability of numerical solutions and c) the algorithmic and computational properties corresponding to numerical solution methods.
• Have understood the effect of finite arithmetic errors of the computer and of method errors and be able to calculate the error bounds of approximate solutions.
• Have knowledge of basic elements of appropriate software for the implementation of various approximation methods.
• Be able to collaborate with fellow students to solve complex practical problems using the methods of Numerical Analysis.