65592 - Numeric Analysis and Algebra (2nd Cycle)

Learning outcomes

Part A:

A first course in Numerical Analysis. Covers the basic techniques of the subject and provides a foundation for the efficient numerical solution of problems in science and engineering.

Part B:

This second part of the course presents numerical methods for the solution of problems modeled by both Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE). The course discusses their analysis, applications, and computation of the solution (by first discretizing the equation, bringing it into a finite-dimensional subspace by a finite elemnt method, or a finite difference method , and finally reducing the problem to the solution of an algebraic equations)

Course contents

PART A:

1. Basics of numerical computing: floating-point arithmetic, roundoff errors, algorithms, problem conditioning, numerical stability.
2. Linear algebra: matrices, vector and matrix norm.
3. Introduction to programming using MATLAB.
4. Solving Linear Systems. Direct methods: LU factorization, pivoting, Gaussian elimination, Cholesky factorization. Basics in iterative methods.
5. Solving Systems of nonlinear equations: bisection method, Newton method, secant method.
6. Polynomial interpolation and piecewise polynomial interpolation.
7. Least Squares: models and curve fitting, norms, the normal equations, the QR factorization.
8. Quadrature: basic quadrature rules, Newton Cotes rules.

PART B:

1. Numerical Differentiation
2. Ordinary Differential Equations: one-step methods: euler and runge-kutta methods,multi-step methods: Predictor-corrector scheme,numerical solution of stiff problems
3. Boundary value Problems
4. Partial Differential Equations: classification, first order PDE , second order PDE, numerical methods: finite difference schemes, finite element schemes for parabolic-type and elliptic-type problems.

Introduction to the COMSOL computing environment . Miscellaneous problems and applications

Readings/Bibliography

Cleve Moler, Numerical Computing with MATLAB, Ed. SIAM, 2004.
Michael T. Heath, Scientific Computing: An Introductory Survey , 2nd ed., McGraw-Hill, 2002.
A.Quarteroni, F.Saleri, Calcolo scientifico: esercizi e problemi risolti con Matlab e Octave, Ed.Springer Verlag, 4a ed.,2008.
A. Quarteroni, Modellistica Numerica per problemi Differenziali, Springer, Ed. 4a, 2008.
G. Monegato, Fondamenti di Calcolo Numerico, CLUT, 1998.
Kincaid Cheney, Numerical Analysis , Brooks and Cole.,1991

Teaching methods

class hours and computational experiments in lab.

Assessment methods

Part A: Final examination in lab.

Part B: Projects where the numerical methods are used in specific applications will be assigned throughout the course.

Final discussion about the projects.

Teaching tools

Experience in Lab. is an essential part of the course. Matlab is used as problem solving environment, matrix-vector programming language, graphics.

Slides provided in the WEB site.

Links to further information

http://www.dm.unibo.it/~morigi/homepage_file/courses_file/bs1112.html

Office hours

See the website of Serena Morigi

See the website of Michele Mulazzani