Data representation in computers and finite precision arithmetic. Formulation of solution methods via algorithms. Numerical methods for the solution of algebraic equations. Basic concepts in linear programming.
Lectures are supported and integrated by laboratory activities in Excel and Matlab environment, or in their open source counter parts.
Knowledge acquired:
The course aims to present mathematical problems and methodologies which can be used to improve teaching of mathematics in secondary schools; special focus is on a numerical approach to the solution of mathematical problems, and on ways to use computers as effective aids for teaching and learning mathematics.
Competence acquired:
Knowledge of ad-hoc numerical methods for solving algebraic equations, and of fundamentals of optimization and linear programming, of data representation in computers.
Skills acquired (at the end of the course):
Ability to use and develop simple programs in Excel and Matlab environments, or in their open source versions, to illustrate the numerical solution of mathematical problems, and prepare lectures.
Prerequisites
First level courses of Mathematical Analysis, Geometry and Linear Algebra, Numerical Analysis.
Teaching Methods
Lectures and training sessions in computer lab are planned.
Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Training Sessions in the computer lab: practicing with problem solving in Excel and Matlab envinronment.
The training sessions are conducted so to:
- help the students develop skills to apply the theoretical knowledge;
- encourage students to independently develop algorithms and organize lectures and training sessions.
Didactic seminar: each student is asked to prepare and give a lecture or a training session in computer lab on a given subject. Subjects are chosen to assess the ability of students to use their skills for teaching. In the evaluation, special attention is paid to the correctness of the exposed arguments, as well as to the ability of communicating in a simple and clear way.
Oral examination: a few questions are posed. The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of mathematical language.
Course program
Fundamentals of floating point arithmetic: representation of integer and real numbers; unit roundoff; error propagation. Formulation of solution methods via algorithms.
Algebraic equations: conditioning and localization of the roots of a polynomial; the algorithm of Horner-Ruffini; Euclide’s algorithm for finding the gratest common divisor of two polynomials; Sturm’s theorem; real roots and Newton-Horner’s method; Newton’s method as a fixed-point iteration; deflation techniques.
Basic concepts on constrained and unconstrained optimization; optimality and feasibility; formulation of a linear program (LP) and model problems; geometrical interpretation and solution of a LP; basic solutions and extreme points; the fundamental theorem of linear programming; the simplex method. Introduction to the theory of duality: primal and dual problems; weak duality theorem and strong duality; economic meaning of dual variables.
Elements of Excel and Matlab (or of their open source counter parts).