NUMERICAL METHODS

This course is one of the characterizing educational activities of the Master's Degree course and allows the student to acquire the knowledge and understanding the fundamental and applicative concepts of numerical analysis
The aim of the course is to provide basic knowledge of numerical analysis for linear algebra problems, numerical integration and differential equations
At the end of the course, the student will be able to approximate various problems of a mathematical nature and to choose the best algorithm for the problem addressed.

Knowledge and understanding
At the end of the course, students will have acquired notions and results related to methods for the numerical solution of linear systems and nonlinear equations, eigenvalue problems, for the approximation of functions, data and integrals, and for the discretization of differential equations and will have acquired techniques related to the implementation of algorithms for the efficient solution of the problems treated.

Ability to apply knowledge and understanding
Students who have passed the exam will be able to use the methodologies illustrated in the course for the numerical solution of linear systems and nonlinear equations, eigenvalue problems, for the approximation of functions, data and integrals, and for the discretization of differential equations, and will be able to predict its performance depending on the characteristics of the problem to be treated.

Autonomy of judgment
Students who have passed the exam will be able to choose the most suitable algorithms for the solution of the problem considered from all the algorithms they have studied in the course, having also acquired the tools to make the changes that may be necessary to improve its performance.

Communication skills
Students will have developed the ability to present the concepts, ideas and methodologies covered in the course.

Learning ability
The acquired knowledge will allow students who have passed the exam to tackle the study, individually or in a Master's Degree course, of more specialized aspects of numerical linear algebra and numerical modeling for differential problems, being able to understand the specific terminology and identify the most relevant issues.

Mathematical analysis I, Linear Algebra, Informatics, Mathematical analysis II

Fundamentals
- Basic concepts of Scientific Computing: model error, truncation error, rounding error, computational errors.
- Numerical methods for approximating linear systems: direct methods (Gaussian elimination, LU factorization, pivoting); iterative methods (stationary Richardson method, gradient and conjugate gradient methods, convergence conditions, stopping criteria); solution stability and accuracy.
- Numerical methods for eigenvalue approximation: power and inverse power methods with shift.
- Numerical methods for nonlinear equations: Newton’s method and fixed-point iterations; extension to the vectorial case.
- Numerical methods for interpolation: polynomial interpolation and least squares approximation.
- Numerical methods for numerical integration: quadrature formulas; Newton-Cotes (midpoint, trapezoidal, and Simpson’s rule, simple and composite) and Gaussian formulas.

Ordinary Differential Equations (ODEs)
- Theoretical analysis
- One-step (Euler, Crank-Nicolson) and higher-order methods (Runge-Kutta); consistency, zero-stability, convergence, absolute stability; extension to systems of ODEs.

Partial Differential Equations (PDEs)
- Classification of PDEs: boundary and initial conditions, physical interpretation.
- Finite Difference method for elliptic and parabolic equations.
- Finite Element method for elliptic and parabolic equations.

The course includes laboratory activities for the development of MATLAB codes.

A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, “Numerical mathematics”, Springer
A. Quarteroni, F. Saleri, P. Gervasio, “Scientific Computing with MATLAB and Octave”, Springer

The student must demonstrate that they have acquired knowledge of the most relevant topics and methods covered in the course and be able to discuss their implementation in MATLAB through a written test (consisting of multiple-choice and long-answer questions) and an optional oral discussion.

The exams scheduled in the academic calendar may be replaced by two intermediate tests conducted during the course.

written and oral

The student

18-21: has basic understanding with gaps; can apply simple numerical methods; their code implementation requires guidance.

22-23: is competent in fundamental concepts; can solve basic problems but struggles with deeper analysis and stability.

24-27: has good grasp of most numerical methods; their implemented code is mostly functional.

28-29: possesses strong theoretical and computational understanding; can analyze accuracy and stability; their code implementation is efficient.

30: masters all topics; justifies method choices rigorously; their code is optimized and well-structured.

30 cum laude: demonstrates outstanding insight and critical thinking; extends methods beyond the syllabus; their code implementation is advanced and elegant.

Standard classes (67%) - Exercises and MATLAB (33%)

English

Exam: Written test and optional oral exam