Basics concepts of numerical analysis. Numerical solution of linear systems. Data and function approximation. Numerical solution of ordinary differential equations: methods for initial value and boundary value problems. Introduction to Matlab for the implementation of numerical methods.
Chapra, S., Canale, R., Numerical methods for engineers, Mc-Graw Hill.
Kahaner, D., Moler, C., Nash, S., Numerical methods and software, Prentice Hall.
W. Palm III, Introduction to Matlab for engineers, Mc-Graw Hill, 2005.
The course deals with the study and the implementation of methods for the numerical solution of mathematical problems. The course aims at presenting the basic methodologies of numerical analysis for solving mathematical problems arising in the applications, with particular attention devoted to the implementation and practical use of such methods.
The student learns classical numerical methods for solving the considered mathematical problems.
The student learns the tools understanding of the major issues related to the numerical solution of mathematical problems.
Acquired skills acquired:
The student will be able to choose the best numerical method to solve a given problem, and to understand the numerical results.
The students will also be able to develop and use simple programs in the Matlab environment in order to solve the considered mathematical problems.
Basic linear algebra: vectors, matrices, linear systems. Basic Mathematical Analysis: sequences and their convergence, limits and continuity for real functions, basic concepts of differentiation and integration for univariate and multivariate functions.
Lectures and training sessions in the computer lab.
Lectures: presentation of the theory described in the course program, with teacher-student direct interaction, to facilitate and ensure a full understanding of the subject.
Training sessions in the computer lab: training sessions to learn how to numerically solve mathematical problems in the Matlab envinronment.
The training has the goal of:
-- helping the students to develop skills to apply the theoretical knowledge;
-- encouraging criticism in the students, particularly in assessing the numerical results obtained.
Oral test: it will aimed at assessing a suitable knowledge of the topics in the course program, including the use of the Matlab environment for implementing the numerical methods.
Basic concepts: conditioning of a problem, stability and convergence of numerical methods, sources of errors in computational models, fundamental aspects of numerical algorithms.
Scientific computing: floating-point representation of numbers, machine precision and arithmetic operations, rounding errors.
Linear systems: solution of triangular systems, gaussian elimination, LU factorization and pivoting techniques.
Data and functions approximation: problem of polynomial interpolation, piecewise polynomial interpolation, basics of univariate spline functions and cubic spline interpolants. Data fitting by least squares method.
Basics of ordinary differential equations (ODEs) and systems of ODEs. Initial value and boundary value problems, theoretical and numerical solutions. Explicit one step method for initial value problems, finite difference methods for boundary value problems, convergence and implementation issues.
Basic Matlab rules and commands. Representing and saving data, basic operations and functions, creating and handling arrays. Programming with Matlab: scripts and functions, conditional and repetition instructions, logical operators. Plotting and program output. Matlab predefined (graphical) functions. Introduction to the use of tools for image analysis.