Errors and finite precision arithmetic; perturbation analysis and stability. Polynomial and piecewise polynomial interpolation and approximation. Interpolatory quadrature rules and composite rules; Richardson extrapolation. Nonlinear equations: fixed point iteration, bisection, secant and Newton's method. Linear systems: Gaussian elimination with partial pivoting; LU, Cholesky and QR factorizations; Jacobi and Gauss-Seidel iterations; linear least squares problems. Introduction to MATLAB.
Knowledge acquired:
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 (data and functions polynomial approximation, definite integrals, linear systems of equations, and roots of nonlinear equations) with a particular attention devoted to implementation issues.
Competence acquired:
Knowledge of classical numerical methods for solving the considerd mathematical problems.
Skills acquired (at the end of the course):
Ability to develop simple programs and to use the Matlab environment in order to solve the mathematical problems under study. Understanding of the obtained numerical results.
Prerequisites
Courses required: Mathematical Analysis I, Geometry I.
Teaching Methods
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 numerical problem solving in Matlab envinronment.
The training sessions are conducted so to:
-- help the students develop skills to apply the theoretical knowledge;
-- encourage independent judgement in the students, particularly in the understanding of the results obtained from a computer.
Moodle learning platform: online teacher-student interaction, posting of lecture notes and additional notes, supplementary exercise sheets, examples of final examinations.
Remark: suggested reading includes supplementary material that may be useful for further personal studies on the subject.
Further information
Frequency of lectures, practice and lab: Recommended
The practical Matlab test consists of two intermediate tests (or one final test): a selection of exercises is proposed (usually three for each intermediate test, four for the final one). Tests are designed to assess the ability of the students to apply their skills to problem solving. In the evaluation, special attention is paid to the correctness of the solution procedure, as well as to the ability of understanding the obtained numerical results.
Oral examination: a number of questions (usually three) 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
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.
Polynomial and piecewise polynomial interpolation: Lagrange and Newton form of the interpolant polynomial, interpolation error, conditioning of the problem, Chebyshev's abscissae; spline functions, cubic spline interpolants. Polynomial least squares approximation.
Numerical integration: Newton-Cotes formulas; composite quadrature rules; error analysis and conditioning; Richardson extrapolation; adaptive formulae.
Solution of nonlinear equations: conditioning of the problem; bisection, fixed point iteration, secant and Newton's method; convergence properties and implementation issues. Newton’s method for nonlinear systems of equations.
Direct methods for linear systems: Gaussian elimination; LU and Cholesky factorizations; Householder reflections and QR factorization; pivoting strategies; error analysis. Linear least squares problems: normal equations; solution by QR factorization.
Stationary iterative methods for large linear systems: basics; convergence analysis; metodi di splitting (Jacobi and Gauss-Seidel iterations); Richardson method.
How to use MATLAB, an interactive system for scientific computations.