Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

Calculus for functions of one real variable. Riemann integral. Linear systems of equations. Basic matrix arithmetic and operations.

On completion of the course students will have learnt some basic numerical techniques, useful for tackling several types of mathematical problems commonly occurring in the engineering and physical sciences. Full mathematical proofs will not be treated in detail, the emphasis being on the logic behind each technique and the criteria and possible limits for its application. The main focus is on understanding why the methods work, what type of errors can be expected and when a method might lead to difficulties.

On completion of the course students will be able to produce approximate solutions to several types of mathematical problems, using an appropriate numerical technique. They will also be able to give a rough estimation of the associated approximation error.

In using specific numerical techniques for solving the various types of problems, students will learn to deal with the difficulties connected to putting theory into practice, which will enhance their capacity for independent learning and making correct choices.

rror analysis and convergence. Solution of non-linear equations of one variable: bisection, secant and tangent (Newton) methods. Interpolation and polynomial approximation: Lagrange polynomials, error formula, divided differences, cubic spline interpolation. Numerical differentiation formulae (two- three- and five-point); round-off error instability. Numerical integration: basic and composite quadrature rules, round-off error stability. Error formulae. Gaussian quadrature. Direct methods for the solution of linear systems: Gaussian elimination, pivoting strategies, LU matrix factorisation, techniques for special matrices. Iterative techniques for the solution of linear systems: Jacobi, Gauss-Seidel. Convergence criteria. Introduction to initial-value problems: Eulers method.

Final assessment is via a written paper and an oral examination. In the written test students must solve three exercises regarding topics discussed during the course. In the oral examination, which completes the written test, students must comment on the solution of the written test and respond to one or two questions regarding topics discussed during the course. The written test has a three-hour duration. Students are free to refer to any material they may deem useful, such as pocket calculators, personal computers, software, lecture notes, books, etc. In order to sit the oral examination, students must have obtained a minimum pass mark on all three exercises of the written test. The oral exam must be taken in the same examination session as the written test. If a student doesn't pass the oral examination, they must repeat both written and oral tests.

Students must demonstrate a good understanding of the solution procedures for the specific problems treated during the course. In particular, in the written test, students must explain the details of all operations performed for the solution of the exercises. In the oral examination, students must demonstrate an understanding of the basic theoretical principles and knowledge of the numerical solution techniques discussed during the course. In particular they must know, for each of the methods studied, its specific limitations, and must be able to make an estimation of the error range with respect to the exact analytical solution.

Students must be able to choose the solution method appropriately, together with any necessary options, and apply it correctly; more importantly, they must be able to explain the logic and meaning of the various mathematical operations in a clear and thorough way. The written test consists of three exercises, each of which, if solved correctly in every aspect and is presented in thorough detail, will be given a maximum mark of about 15. In order to pass the written test and be able to sit the oral examination, students must tackle, even if incompletely, all three exercises. The oral test will be evaluated with a maximum of 30 points, and the final mark is a weighted mean value of the written and oral exams.

The top grade (corresponding to a mark of 30/30) will be given to students who demonstrate a detailed and exhaustive understanding of all aspects of the course, and who are capable of illustrating concepts with an articulate use of technical terms and can use such acquired knowledge appropriately for solving the exercises. The minimum grade (corresponding to a mark of 18/30) will be given to students who can demonstrate a basic understanding of the principal numerical methods studied during the course, and are able to use them correctly for obtaining an approximate solution in simple exercises.

R. Burden, J. D. Faires Numerical Analysis, Brooks/Cole, 9th Ed., 2009 R. Burden, J. D. Faires Numerical Methods, Brooks/Cole, 4th Ed., 2013 A. Quarteroni, F. Saleri, Introduzione al Calcolo Scientifico, Springer-Verlag Italia, Milano 2006

- Ingegneria Informatica e dell'Automazione (Corso di Laurea Triennale (DM 270/04))

**Università Politecnica delle Marche**

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