Course teached as: - METODI NUMERICI PER L'INGEGNERIA Second Cycle Degree in ENVIRONMENTAL ENGINEERING (POSTGRADUATE)
Teaching Language - Part A
Course Content - Part A
Finite precision computation. Numerical solution of linear systems. Data and function approximation. Numerical integration. Numerical methods for nonlinear equations. Numerical methods for initial value problems for ordinary differential equations. Finite difference methods for boundary value problems.
M.G. Gasparo, R. Morandi, Elementi di Calcolo Numerico, McGraw Hill, 2008.
A.Quarteroni, R. Sacco, F. Saleri : Matematica Numerica. Springer-Verlag Italia,1998
Learning Objectives - Part A
The course deals with the definition and study of methods for solving mathematical problems by using computers.
Purpose of the course is to present the basic methodologies used in numerical analysis for solving mathematical problems arising in the applications with a particular attention devoted to implementation issues.
Knowledge of classical numerical methods for solving the considerd mathematical problems.
Understanding of the major issues related to the numerical solution of mathematical problems.
Skills acquired (at the end of the course):
Ability to choose the best numerical method and algorithm for solving a given problem, and to understand the numerical results.
Prerequisites - Part A
Basic linear algebra tools: vectors, matrices, linear systems. Fundamental of Mathematical Analysis.
Teaching Methods - Part A
Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Moodle learning platform: online teacher-student interaction, posting of additional notes and examples of final tests.
Further information - Part A
The course is also followed by the following classes:
- Numerical Analysis and Computer Programming (LM Ingegneria Civile)
- Advanced Numerical Analysis (LM Ingegneria Edile)
Students are evaluated as follows.
a) a written test (for Part A)
b) a Matlab practical test (for Part B)
c) an optional oral test.
a) Theoretical written test, with essay questions, to assess student's knowledge and comprehension of the theory presented in the lectures.
b) See part B.
The result obtained for test a) or, if the class requires also part B, the weighted mean of the scores obtained for the two tests, provides the exam final score in case the student does not request to hold the oral test. The cut score for passing is 18/30.
c) Oral test. It can be requested by the student after passing the previous tests. A number of questions (usually three) are posed. The test will be mainly devoted to assess the degree of knowledge and understanding of the theory presented in the course, possibly including a discussion on the written test and Matlab test (when required).
The final evaluation result will be determined on the base of the score of all the taken tests.
Course program - Part A
Numerical methods and algorithms: definitions. Errors in scientific computing: floating-point representation, machine precision and arithmetic operations; discretization error and effects of finite precision; perturbation analysis and stability.
Linear systems - Condition number of a matrix; Gaussian elimination; LU and Cholesky factorizations; pivoting strategies; error analysis.
Data and functions approximation - Polynomial and piecewise polynomial interpolation: Lagrange and Newton form of the interpolant polynomial, interpolation error, conditioning of the problem; spline functions, cubic spline interpolants. Polynomial least squares approximation.
Nonlinear equations - Iterative methods for finding the roorts of a nonlinear equation:
Bisection, Newton and Secant methods; convergence properties and implementation issues.
Differential equations - Numerical methods for initial value problems for differential equations: explicit one step methods (Euler and Runge-Kutta).
Finite difference methods for boundary value problems: central differences method and upwind method; convergence and implementation issues.