Introduction to Numerical Analysis with emphasis on methods useful for Civil Engineers. Number representation on computers. Numerical solution of Linear Systems: Direct Methods (Gauss, LU factorization methods). Stability of linear systems. Iterative methods (Jacobi method, Gauss-Seidel, SOR). Solution of Non-Linear Equations: Bisection method, Regula-Falsi method, Fixed Point iterative methods, Newton-Raphson method, Secant method. Newton method for nonlinear systems. Interpolation: Polynomial Interpolation in Lagrange and Newton form and Interpolation Error. Hermite and cubic splines interpolation. Numerical Integration: Newton-Cotes formulas, Simple and Composite Trapezoidal and Simpson integration rules, Gauss Integration. Approximation theory: Discreate least squares method, polynomial and exponential approximation. Differential Equations: Initial value Problems for ordinary differential equations. Single step methods (Euler, Taylor, Runge-Kutta). Multi-step methods (Adams, Predictor-Corrector methods). Numerical solution of systems of Differential Equations.
|1||Numerical solution of Linear Systems||Number representation in computers. Direct Methods to solve Linear Systems: Gauss elimination method, computation of the inverse and the determinant of a matrix. LU-Factorization methods. Vector norms, Matrices norms. Stability of linear systems. Iterative methods: General iterative method, Jacobi, Gauss-Seidel and Relaxation SOR methods.||5Χ4 =20|
|2||Solution of non-linear equations and systems||Detection of roots. Methods: Bisection, Regula-Falsi, Fixed Point iterative method, Newton-Raphson, and Secant. Newton-Raphson method for nonlinear systems.||2Χ4=8|
|3||Interpolation||Polynomial Interpolation in Lagrange and Newton form. Interpolation Error.||2Χ4=8|
|4||Numerical Integration||Newton-Cotes formulas, Simple and Composite Trapezoidal and Simpson integration rules. Gauss Integration.||1.5Χ4=6|
|5||Approximation theory||Discreate least squares method, polynomial and exponential approximation.||1x4=4|
|6||Numerical solution of Differential Equations||Initial value Problems for ordinary differential equations. Single step methods (Euler, Taylor, Runge-Kutta). Multi-step methods (Adams, Predictor-Corrector methods). Numerical solution of systems of differential equations||1.5Χ4=6|
The course aims at acquiring the knowledge for the solution of systems of linear equations, non-linear algebraic equations, ordinary Differential equations, interpolation and approximation of data and numerical approximation of integrals. In addition, the aim of the course is to understand the importance of numerical methods for solving scientific and technological problems for which either there is no analytical solution, or it is very difficult to calculate it. A secondary objective is to familiarize students with: (a) constructing iterative methods to approximate numerical solutions to problems; and (b) convergence of iterative methods.
Upon successful completion of the course, the students will be able to:
|Teaching methods||Lectures including theory as well as theoretical and computational exercises that focus on the application of Numerical Analysis methodologies. Solving homework exercises (individual work - optional).|
|Teaching media||Lectures on the blackboard|
|Computer and software use||Optional application of methods taught in the computer.|