Numerical Analysis 1

. – Numerical Analysis 1 - . – Numerical Analysis 2
. – Numerical Analysis 1
PROFESSOR MAURIZIO PAOLINI
COURSE AIMS
To teach the basics of numerical analysis, by tackling the following types of
problem from a numerical point of view: solution of nonlinear equations, linear
systems, and approximation of functions of one variable.
COURSE CONTENT
- Error theory: Absolute/relative error, conditioning of mathematical problems,
error propagation, floating point representation, and stability of algorithms.
- Linear systems: Triangular systems, Gaussian elimination, pivotal strategies, LU
decomposition, Cholesky decomposition, Jacobi, Gauss-Seidel and SOR iterative
methods, residual correction method, and stopping test.
- Nonlinear equations: Bisection, secant and Newton's methods, order of
convergence, and stopping test. Horner scheme for polynomials.
- Function approximation: Lagrange polynomial, existence and uniqueness
theorem; divided differences and Newton form of the interpolation polynomial;
Chebyshev nodes; error formula.
READING LIST
V. COMINCIOLI, Analisi Numerica. Metodi Modelli Applicazioni, McGraw Hill Libri Italia,
Milan, 1990.
A. QUARTERONI, Elementi di Calcolo Numerico, Progetto Leonardo, Bologna, 1994.
G. NALDI - L. PARESCHI - G. RUSSO, Introduzione al Calcolo Scientifico, McGraw-Hill, Milan,
2001.
TEACHING METHOD
Lectures.
NOTES
Further information can be found on the lecturer's webpage
http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice board.
at
. – Numerical Analysis 2
PROFESSOR MAURIZIO PAOLINI
COURSE AIMS
The course deals with mathematical problems concerning function approximation,
finding eigenvalues/eigenvectors, numerical integration, and the solution of the
Cauchy problem.
COURSE CONTENT
- Least squares: Least squares in the discrete and continuous contexts; property of
orthogonality; Families of orthogonal polynomials.
- Numerical integration: Interpolation formulas; Newton–Cotes formulas; overview
of the Gauss formulas;
- Eigenvalues/eigenvectors: Definition, finding methods, power and inverse power
methods, problem conditioning, Householder transformation and Givens rotation,
QR decomposition, transformation to Hessenberg Form, Sturm sequences and QR
method.
- Application of Sturm's method to the calculation of the real zeros of a
polynomial.
- Ordinary differential equations: Euler method; error analysis of Euler method;
overview of Runge–Kutta methods; linear multistep methods and Adams method;
algebraic consistency conditions and order m; root condition (weak and strong);
concept of relative stability; predictor-corrector methods.
READING LIST
V. COMINCIOLI, Analisi Numerica. Metodi Modelli Applicazioni, McGraw Hill Libri Italia,
Milan, 1990.
A. QUARTERONI - R. SACCO - F. SALERI, Matematica numerica, Springer-Verlag Italia, Milan,
1998.
K. ATKINSON, An introduction to numerical analysis, J. Wiley & Sons, New York, 1966.
TEACHING METHOD
Lectures.
NOTES
Further information can be found on the lecturer's webpage
http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice board.
at