COMPUTATIONAL APPLIED MATHEMATICS
The course is an introduction to Scientific Computing, i.e., all the activities involved in solving (accurately and efficiently) technical-scientific problems by a computational approach. An essential part consists in laboratory activities using MATLAB.
Contents are related to basic topics of Complex Analysis (holomorphic functions, power series in the complex plane) and elements of Functional Analysis (Fourier series and transform).
Knowledge and understanding
Students must show to know and to understand:
• basic concepts of Linear Algebra and Calculus, with a special reference to analytical real functions and to the limit of functions of two real arguments, in both theoretical and application context;
• the MATLAB programming language and its framework.
Ability to apply knowledge and understanding
Students must show:
• to be able in using their acquired knowledge to solve advanced problems of applied mathematics computationally, even when they appear in concrete contexts and in other scientific fields;
• to be able in using the MATLAB language consciously, in order to produce scientific software that solve a variety of concrete problems.
These capabilities are also expressed in a thorough and conscious use of computational tools and advanced computer labs.
Autonomy of judgement
Students, by themselves, must be able to assess the results returned by scientific software. In addition, they should know how to identify, even using online repositories of scientific software, the most suitable algorithms and software to solve a specific problem.
Students, also working in groups, should be able to draw up a report on a numerical algorithm, to make remarks on its results, and to document its MATLAB implementation using advanced scientific computing and documentation tools. Moreover, students should be able to use the suitable terminology of applied and computational mathematics, even in English language.
Students must be able to keep up-to-date and to analyze in depth topics and specific applications of scientific computing, even accessing databases, on-line software repositories and other tools available on the web.
Basic concepts of Mathematics, Linear Algebra and Calculus. Elementary knowledge of MATLAB. These topics are covered by the following courses: Mathematics 1, Mathematics 2, Numerical Computing.
Overview of MATLAB's functions for symbolic computations (Symbolic Math Toolbox) and comparison with numerical computations. Parallel Computing in MATLAB.
Complex functions of real and complex variable, their graphical representation and geometrical interpretation.
Basic topics about holomorphic functions (Cauchy-Riemann equation) and about analytic functions (power series in the complex plane).
Trigonometric interpolation, Discrete Fourier Transform, Fourier Series and Transform. Examples of applications of Fourier analysis and synthesis.