NUMERICAL ALGORITHMS

This course will enable the student to understand the meaning and potential of some fundamental numerical algorithms, with the goal of implementing them and producing a correctness analysis of the obtained results.

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.
- Initial ideas for the numerical solution of linear systems.
- Numerical solution of nonlinear equations: Picard's, Newton's, and secant methods. Extension to systems of nonlinear equations.
- Approximation and interpolation: polynomial and spline interpolation, polynomial least-squares approximation.
- Numerical integration: trapezoidal and Cavalieri-Simpson rules.
- Numerical solution of ordinary differential equations: Euler, Crank-Nicolson, and Runge-Kutta methods. Extension to systems of ODEs.
- Numerical solution of linear systems: direct and iterative methods.

A. Quarteroni, F. Saleri, e P. Gervasio. Scientific Computing with MATLAB and Octave. Springer Verlag, 2010.
Other supplementary material will be distributed via the course Moodle page.

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) Solution of linear systems
(d) Nonlinear differential equations.

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 content of the exercises and the ability to know how to discuss the 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.

English

Classes will be in English

oral