


Politecnico di Torino  
Academic Year 2017/18  
01NQWMQ Numerical methods 

1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
The course is taught in Italian.
The main goal of the course is the description and analysis of the basic numerical methods, including their characteristics (applicability conditions, efficiency in terms of computational complexity and storage), and the acquisition of the necessary knowledge for the efficient solution of numerical problems, in particular by means of Matlab computing programs. 
Expected learning outcomes
Knowledge of the basic numerical methods, skill to construct and analyse, if needed, new numerical methods and to perform scientific computing using the Matlab software.

Prerequisites / Assumed knowledge
Basic notions of linear algebra, calculus and Matlab programming.

Contents
Lectures (50 hours):
1. Linear systems (12 hrs) Gauss decomposition, LU and Choleski factorizations and their applications Iterative methods: Jacobi, GaussSeidel, SOR, gradient and conjugate gradient. Preconditioning. 2. Matrix eigenvalues (6 hrs) Power and inverse power methods. Orthogonal transformations. QR factorization of a matrix. A short description of the QR method for the computation of all the eigenvalues of a matrix. Singular value decomposition: definition and properties. 3. Approximation of data and functions (12 hrs) Polynomial interpolation: Lagrange and Newton representations. Optimal choice of the interpolation nodes. Some convergence results. Piecewise polynomial interpolation: definition and convergence properties. Splines: definition and main properties; cubic splines with different types of boundary conditions. Least squares: definition, normal equation system, basics on QR and SVD methods. 4. Nonlinear equations (4 hrs) Secant and Newton methods; definition and properties. Nonlinear systems: Newton and quasiNewton methods 5. Numerical evaluation of integrals defined on intervals (6 hrs) Quadrature formulas of interpolatory type: Newton Cotes formula, orthogonal polynomials and Gaussian rules. Convergence properties. Composite rules: definition and convergence properties. 6. Initial value ordinary differential equation problems (10 hrs) Explicit and implicit onestep methods: definition consistency and convergence properties. RungeKutta methods. Explicit and implicit linear multistep methods: definition, consistency and convergence. Adams methods. Absolute stability of a method. Stiff systems and numerical methods for their solution. 
Delivery modes
The above lectures are integrated by 15 hours of exercise sessions. The role of this activity is to underline important aspects of the topics and results presented in the lecture sessions, by means of examples and the solution of some significant exercises. Numerical algorithms are also constructed. An additional computer lab activity (15 hours) is also scheduled, to improve the students ability to use the Matlab software, and, more importantly, to numerically verify the correctness of the method properties presented during the lectures and that of the algorithms constructed in the exercise sessions.

Texts, readings, handouts and other learning resources
G. Monegato, Metodi e algoritmi per il Calcolo Numerico, CLUT, Torino, 2008.
L. Scuderi, Laboratorio di calcolo numerico, CLUT, Torino, 2005. 
Assessment and grading criteria
At the end of the computer lab activities, students are asked to show that they have acquired the ability to solve, using the Matlab software, a given problem. This test lasts 1h 20m and it is worth up to 2 points.
The grade one obtains will remain valid for the three exam sessions of the current academic year. Then, in the exam sessions, students must take a written test, which lasts 2h 30m, concerning the entire course programme. This test consists of 78 questions, each one worth 3 or 4 points, some of theoretical nature and others of applied type. The maximum number of points assigned to this written test is 28. During this latter examination it is forbidden to consult notes or books and to use electronic devices. 
