Learning objectives
One of the main aims of the course is to provide students with basic programming constructions and the fundamentals of the most common numerical methods used to solve various applicative problems, critically presenting the main algorithms and their properties such as convergence, stability, accuracy, complexity, using examples and counter-examples to illustrate the advantages and weaknesses of the aforementioned methods. During lectures, students will be able to experiment with the presented algorithms in software environments widely used for scientific computing: MATLAB and Excel. At the end of the course, the student will be able to use computational tools to understand, analyze and solve problems of moderate difficulty in different areas of applied mathematics.
Prerequisites
Basics notions of Calculus
Course unit content
Floating-point systems – Machine arithmetics – Rounding errors - Definition of array and related algebraic operations - Introduction to Excel - Introduction to Matlab as user-friendly environment for scientific computations – Matlab as programming language - Numerical solution of nonlinear equations - Interpolation of data and functions by algebraic polynomials - Numerical integration: simple and iterated Newton-Cotes formulas - (extra) Ordinary differential equations: discrete one-step methods
Full programme
Error analysis: Representation of numbers in a computer - Rounding errors - Machine operations - Numerical cancellation - Conditioning of a problem - Stability of an algorithm.
Numerical resolution of non-linear equations: bisection algorithm and Newton algorithm - Convergence results - Stop tests.
Interpolation of data and functions: interpolation by algebraic polynomials - Lagrange interpolation formula - Interpolation error - Least square method
Numerical integration: Interpolatory quadrature rules - Newton-Cotes formulas - Error estimates - Iterated formulas - Convergence results
(extra) Numerical methods for ODEs: Explicit one-step methods - Series expansion methods - Runge-Kutta methods - Local truncation error - Stability and convergence of explicit one-step methods
Introduction to Matlab: Matlab as matrix laboratory - definition of array and algebraic array operations - Matlab as programming language: counter cycles, condition cycles, structured tests - Function files and script files - Matlab main numerical routines - Matlab for graphics
Bibliography
"Numerical analysis". L.W. Johnson, R.D. Riess. Addison-Wesley (1982).
Teaching methods
Classroom lectures and exercises. Numerical exercises with MATLAB and Excel in Computer Science lab. In the lab lessons, numerical and programming exercises will be assigned. The presentation of the solutions by the students will be taken into account for the final evaluation.
Assessment methods and criteria
Lab test with Matlab/Excel programming exercises related to simple numerical algorithms, to be done on computer, and written questions on theoretical arguments. Tests will be carried out in presence (or online if required by the the guidelines provided by the University).
Other information
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