NUMERICAL ALGORITHMS

The course Numerical Algorithms fits into the curriculum of the Bachelor of Science in Computer Science as a fundamental teaching for the acquisition of skills in scientific computing and in the numerical solution of mathematical problems. It provides the theoretical and practical tools for dealing with computationally complex problems that do not admit analytic solutions or that require efficient and stable approximations to be solved by a computer.

The teaching aims to introduce students to the main numerical methods for analyzing and solving recurring mathematical problems in computer science, engineering, and applied sciences. Particular emphasis is placed on both the conceptual analysis of the algorithms (in terms of accuracy, stability, and computational complexity) and their practical implementation in a scientific computing environment (Python), in order to develop operational skills and a critical attitude in the use of numerical tools.

By the end of the course, the student will be able to consciously choose and apply appropriate numerical algorithms to different types of problems, evaluating their limits, performance and reliability. The course also aims to strengthen numerical modeling skills and lay the foundation for more advanced studies in scientific computing and numerical simulation

Attendance and participation in the training activities offered in the course and individual study will enable students to:

1. Knowledge and understanding
-- knowledge and understanding of the basic concepts of Numerical Analysis.
-- knowledge and understanding of the main numerical algorithms for solving mathematical problems

2. Ability to apply knowledge and understanding
-- ability to implement some numerical algorithms
-- ability to establish convergence conditions of numerical algorithms
-- ability to numerically solve ordinary differential equations
-- ability to approximate the solution of nonlinear equations and linear systems.

3. Assessment skills
-- interpret the results of a numerical program.

The student should know the fundamentals of Linear Algebra and Calculus in one and more real variables.

- Discretization and approximation: approximation algorithms and representation of real numbers (floating point system). Setup of a Python environment for scientific computing.
- Direct methods for the numerical solution of linear systems.
- Numerical solution of nonlinear equations: fixed point methods, Newton's, and secant methods. Extension to systems of nonlinear equations. Application to unconstrained optimization.
- Approximation and interpolation: polynomial and spline interpolation, polynomial least-squares approximation.
- Numerical integration: simple and composite Newton-Cotes methods, Gauss' method.
- Numerical solution of ordinary differential equations: Euler, Heun, Crank-Nicolson, and Runge-Kutta methods.
- Iterative methods for the numerical solution of linear systems: matrix-splitting and gradient-based methods.

The lecture notes will be distributed through the Moodle page of the course (including further bibliographical references).

Verification of learning is through an oral test, which is based on the discussion of four exercises that will be assigned during the course.

The four exercises are organized as follows:
- The solution of each exercise (i.e., implementation of the solution and a report discussing the obtained results) is to be handed in by the student via the Moodle platform by the date of the oral exam.
- To take the oral one must have handed in the solutions of all the exercises.
- Each exercise covers a topic covered in the lectures, the comparison of different methods, and their use in solving an application problem.
- The exercises will cover the following topics:
(a) Nonlinear equations
(b) Approximation and interpolation
(c) Ordinary differential equations.
(d) Solution of linear systems

oral

During the oral test, which is approximately 30 minutes long, the following will be evaluated:
- The correctness of the solutions to the exercises and the quality of the delivered reports (40% of the grade);
- The knowledge of the algorithms implemented in the solution of the exercises, the choices made with respect to the choice of parameters and variants of the algorithms themselves, and the ability to discuss the corresponding results (30% of the grade);
- The knowledge of the topics covered in the course and the ability to know how to present them formally (30% of the grade).

Lectures, theoretical exercises and computer exercises. Use of Moodle platform to propose exercises and supplementary materials.

Classes will be in English