# Numerical Analysis

## Course Description:

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.

### Prerequisite Knowledge

Basic knowledge on Calculus and Linear Algebra.

### Course Units

# Title Description Hours
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

### Learning Objectives

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:

1. understand the basic methods of Numerical Analysis a) to solve linear systems, non-linear equations, and differential equations; b) to interpolate and approximate data and c) to approximate integrals;
2. know the tools and techniques of iterative methods and can effectively use the appropriate stopping criteria;
3. know the importance of using stable algorithms to ensure the reliability of the results derived by the numerical methods;
4. distinguish the differences between the methods and choose the most appropriate to solve each problem;
5. analyse a) the asymptotic properties and the behaviour of the approximate models b) the numerical stability of the numerical solutions and c) the algorithmic and computational properties corresponding to the numerical methods;
6. understand the effect of round-off errors in the computations and truncation errors of the methods and be able to calculate error bounds of approximate solutions;
7. collaborate with others to solve complex practical problems using the methods of Numerical Analysis.

### Teaching Methods

 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). Lectures on the blackboard Optional application of methods taught in the computer.

### Student Assessment

• Final written exam: 70%
• Mid-term exam: 30%

### Textbooks - Bibliography

1. Bradie B., A Friendly Introduction to Numerical Analysis, Pearson Education International, 2006.
2. Burden R. and Faires D., Numerical Analysis, 9th Edition, Brooks/Cole, Cengage Learning, 2010.
3. Sauer T, Numerical Analysis, Pearson Addison Wesley, 2006.
4. Süli E. and Mayers D., An Introduction to Numerical Analysis, Cambridge University Press, 2003.

## Lecture Time - Place:

• Thursday, 08:45 – 12:30,
Rooms:
• ΖΑμφ. Αντ. Υλ. 201

© 2017 School of Civil Engineering, ΕΜΠ