78890 - Numerical Methods

Learning outcomes

At the end of the course the student has acquired the knowledge and specialized methodologies for the development of numerical algorithms, with particular reference to the solution of problems typical of the process industry.

Course contents

Prerequisites. The student knows the mathematical analysis and geometry  taught in the courses of Mathematical Analysis T-1, T-2 and Geometry and Algebra T.

All lectures will be held in Italian. It is therefore necessary to understand the Italian language to successfully attend the course and to be able to use the educational material provided.

Program:
Definition of numerical problems and main sources of error.
Conditioning of a problem. Stability of an algorithm.

Numerical solution of linear systems
Study of the Problem conditioning. Algorithm for the lower and upper triangular systems. LU factorization algorithm with and without row interchanges. Algorithm stability and pivoting methods. Direct methods for special matrices.
Iterative methods for linear systems.

Linear Least Squares Problem: QR and SVD factorizations


Numerical Methods for  non-linear equations and systems.
Problem Conditioning and error parameters.
Methods for nonlinear equations: bisections, Secant and Newton.
Methods for non-linear systems of equations.

Polynomial interpolation.
Piecewise polynomial interpolation.

Ordinary Differential Equations

Cauchy problem; existence of the solution; stability; One-step methods; error check; Multi-step Methods;
Convergence Consistency and Stability;
Stiff problems;

Readings/Bibliography

• M. T. Heath, Scientific Computing, McGrawHill 2002
• A. Quarteroni R. Sacco, F. Saleri, Numerical Mathematics, Springer, 2000.

Teaching methods

Lectures and guided laboratory exercises

Assessment methods

Lab test and written examination. The overall evaluation will be based on the average score of the tests passed.

The laboratory test requires to solve a given problem through the use of appropriate numerical methods implemented in matlab.

The written exam consists of a set of questions, of which about 50% questions of theory and 50% exercises. To be admitted to the written test it is necessary to pass the laboratory test.

For both tests are not allowed notes, codes and personal texts. Eventually allowed material will be provided during the examination.

Teaching tools

Slides, texts and library functions used in the lessons will be available  on AMS Campus.

Office hours

See the website of Fabiana Zama